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ImpactMojo 101 Series · Free Forever

Bivariate
Analysis
101

How Two Variables Move Together — Cross-Tabs, Correlation, Group Comparisons & Simple Regression for Development Practitioners in South Asia

Research-BackedSouth Asia Focus~90 SlidesFree Access

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Agenda

What We Cover

01

What Bivariate Analysis Is

Slides 3–10

02

Choosing a Method by Variable Type

Slides 11–17

03

Cross-Tabulation & Contingency Tables

Slides 18–25

04

Visualising Relationships

Slides 26–34

05

Correlation

Slides 35–44

06

Comparing Two Groups

Slides 45–53

07

Comparing Several Groups (ANOVA)

Slides 54–61

08

Association Between Categories (Chi-Square)

Slides 62–71

09

Simple Linear Regression

Slides 72–81

10

Pitfalls

Slides 82–91

11

Reporting, Practice & Tools

Slides 92–99

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01

Section One

What Bivariate Analysis Is

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Definition

From one variable to two

Univariate analysis describes one variable at a time — the average household size, the spread of incomes. Bivariate analysis is the very next step: it asks how two variables relate to each other.

Bivariate analysis

The study of the relationship between two variables — whether they move together, how strongly, and in what direction. 'Bi' = two; 'variate' = variable.

Most interesting development questions are bivariate at heart: does this go with that? Does schooling go with earnings? Does the programme go with better outcomes?

Univariate asks	Bivariate asks
What share of children are underweight?	Does it differ by wealth quintile?
What is median household income?	Does it differ between migrant and non-migrant households?
How many women completed secondary school?	Is completion linked to age at marriage?

Bivariate analysis is where programme questions actually live. Nobody funds a programme because 18 per cent of children are underweight; they fund it because the 18 per cent is concentrated somewhere, and finding where is a two-variable question.

It is also where overclaiming begins. A univariate statistic is hard to misread; a bivariate one invites a causal sentence, and Section 9 exists because that sentence is usually written before the analysis supports it.

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The Step After Describing

Where it sits in the analysis ladder

01

UNIVARIATE: describe one variable (centre, spread, shape)

→

02

BIVARIATE: relate two variables (direction, strength)

→

03

MULTIVARIATE: many variables at once (control, adjust)

This course lives squarely in the middle rung. Master it before reaching for regression with twenty controls — the two-variable picture is where most reasoning errors are caught.

Level	Answers	Typical tool
Univariate	What is typical, how spread out	Mean, median, histogram
Bivariate	Do these two move together	Cross-tab, scatter, r, t-test
Multivariate	Does it hold once other things are held constant	Multiple regression

The third row is where a bivariate finding usually goes to die. Almost every two-variable relationship in development data shrinks once wealth or education is held constant, which is not a failure of the bivariate method — it is the method doing its job as a first look.

Do not skip the first row. Analysts who jump straight to relationships miss the impossible values, the spike at zero and the missingness that would have changed everything downstream.

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Two Questions

Direction and strength

Direction

Do they move the same way (positive) or opposite ways (negative)? More literacy, fewer births — that is a negative direction.

Strength

How tightly do they track each other? A loose cloud is weak; a near-straight line is strong.

Almost every bivariate tool you will meet is just a precise way to answer these two questions — plus a third: could this be chance?

Question	What answers it	What does not
Which way does it go?	The sign of r; the direction of the difference	A p-value
How strong is the pattern?	The size of r; how far apart the group means are	Significance alone
How big is the effect?	The slope, in real units	r, which is unitless
Could it be chance?	The p-value or the confidence interval	The size of r

These four questions are distinct and are constantly conflated. Strength, size and significance answer different things, and a report that supplies only one of them has left the reader unable to judge the other two.

Report direction, magnitude and uncertainty together. “Three percentage points lower (95% CI 1 to 5)” answers all four rows in one clause; “significantly lower” answers one.

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Vocabulary

Explanatory and response variables

Explanatory variable (X)

The variable you think does the explaining or predicting — also called the independent or predictor variable. Conventionally on the horizontal axis.

Response variable (Y)

The outcome you want to understand or predict — also called the dependent variable. Conventionally on the vertical axis.

Naming X and Y does not prove X causes Y. It only states which one you are treating as the outcome. The causal claim must be earned separately.

Also called	Role
Explanatory, independent, predictor, X	The thing you think does the explaining
Response, dependent, outcome, Y	The thing you are trying to account for

The labels are a decision you make, not a property of the data. Nothing in a dataset says which variable is the cause; calling one “independent” is an assumption you are importing, and the arithmetic will run identically if you swap them.

Which is why the naming deserves a moment’s thought. Regressing income on toilet ownership and toilet ownership on income both produce a line; only one of them matches a story anyone believes, and the software has no opinion.

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Why It Matters

The questions practitioners actually ask

Do villages with self-help groups have higher women's savings?
Is anaemia more common among Adivasi women than others?
Does distance to a health centre predict institutional delivery?
Did test scores differ between the treated and control schools?
Is caste associated with whether a household has a toilet?

Each is a relationship between two variables — and each maps to a specific bivariate method, which Section Two helps you choose.

Programme question	Variable pair	Method
Do poorer households use the service less?	Wealth quintile × used/not	Cross-tab, chi-square
Do trained farmers get higher yields?	Trained/not × yield	Difference in means, t-test
Does distance predict attendance?	Km × days attended	Scatter, correlation, regression
Do outcomes differ across five blocks?	Block × score	ANOVA, then pairwise

Every one of these is a real question a programme asks in its first year, and each has exactly one appropriate first method — determined by the variable types, not by preference. That mapping is Section 2, and it removes most of the anxiety about “which test”.

Note what none of them establishes. Trained farmers differ from untrained farmers in more than training, so the second row measures a difference between two self-selected groups — useful, and not an impact estimate.

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Warning Up Front

A relationship is a clue, not a verdict

Bivariate analysis can reveal a relationship, quantify its strength, and tell you whether it is likely real or just noise. What it cannot do, on its own, is prove that one variable causes the other.

The plural of anecdote is not data; and the presence of correlation is not the presence of cause.

— a working principle of careful analysis

Bivariate analysis can show	It cannot show
That two things move together	That one causes the other
Where to look next	What to conclude
That a difference is bigger than noise	That the difference is due to your programme
A pattern worth explaining	The explanation

A bivariate relationship is a clue. In observational development data it almost always has at least one plausible confounder, and often several — wealth, education and location are correlated with nearly everything else we measure.

That does not make it worthless. Clues direct effort: they tell you which blocks to visit, which subgroup to investigate and which hypothesis is worth the cost of testing properly. The error is treating the clue as the verdict.

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Roadmap

How this course is built

Tools

Cross-tabs for category × category
Correlation for number × number
t-tests and ANOVA for comparing groups
Chi-square and simple regression

Judgement

Choosing the right method
Reading effect size, not just p-values
Spotting confounders and fallacies
Reporting a relationship honestly

Examples come from India and the wider region — the data you actually meet at work.

Section	Method it gives you
2–3 · Choosing and cross-tabs	Pick a method from variable types; read a contingency table
4 · Visualising	Scatter, box plot, grouped bars — and when each lies
5 · Correlation	Pearson, Spearman, and what r is not
6–7 · Comparing groups	t-test, ANOVA, effect size
8–9 · Chi-square, regression	Association between categories; a line and its slope
10–11 · Pitfalls and reporting	What to check, and how to write it honestly

Sections 10 and 11 are the ones that change practice. The methods are standard and available in any software; what separates a defensible analysis from a misleading one is the checking and the wording, which is where most training stops.

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02

Section Two

Choosing a Method by Variable Type

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The Master Move

First, classify both variables

The single most useful habit in bivariate analysis: before choosing any method, ask what kind of variable each of your two is — categorical or numeric. The pair of answers points straight to the right tool.

Categorical

Labels or groups — sex, caste, religion, district, yes/no. Includes ordered categories (wealth quintile, Likert scale).

Numeric

Counts and measurements — age, income, test score, fertility rate, distance in km. You can meaningfully average them.

Variable	Type	Note
District, caste, sex	Categorical (nominal)	No order
Wealth quintile, Likert scale	Categorical (ordinal)	Ordered, unequal gaps
Age, income, yield	Numeric (continuous)	Ratios meaningful
Number of children	Numeric (count)	Discrete; often skewed
Used the service (yes/no)	Categorical (binary)	The commonest outcome in our work

Classify both variables first and the method chooses itself. Nearly all confusion about “which test” is confusion about variable types, and it is resolved by looking at the codebook rather than by reading more about tests.

Ordinal is the awkward middle. Wealth quintile is ordered but its gaps are not equal, so treating it as numeric assumes something the variable does not carry. Treating it as categorical is always defensible; treating it as numeric sometimes is.

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The Decision Table

Method by the two variable types

X type	Y type	Describe with	Test with
Categorical	Categorical	Cross-tab, % & bars	Chi-square test
Categorical (2 groups)	Numeric	Group means, box plots	t-test
Categorical (3+ groups)	Numeric	Group means, box plots	One-way ANOVA
Numeric	Numeric	Scatter plot	Correlation / regression

Pin this table to your wall. Nine times out of ten, classifying your two variables tells you exactly which method to reach for.

X	Y	Start with
Categorical	Categorical	Cross-tab with row %; chi-square
Categorical (2 groups)	Numeric	Box plots; difference in means; t-test
Categorical (3+ groups)	Numeric	Box plots; ANOVA, then pairwise
Numeric	Numeric	Scatter; correlation; simple regression
Numeric	Binary	Beyond this deck — logistic regression

Every row begins with a picture, not a test. The plot is not a courtesy to the reader; it is how you discover the curve, the outlier or the two clusters that would make the test meaningless.

The last row is worth knowing exists. A yes/no outcome with a numeric predictor is extremely common in our work — did she deliver in a facility, given distance — and linear regression handles it badly. That is a signal to seek help, not to proceed.

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Cat × Cat

Two categories: is membership linked?

When both variables are categories — say caste group and has a household toilet (yes/no) — you ask whether knowing one tells you anything about the other.

Describe it with a cross-tabulation and percentages; test it with a chi-square test of independence. Covered in Sections Three and Eight.

Question	Cross-tab of
Do SC/ST households use the scheme less?	Social group × used/not
Is dropout linked to sex?	Sex × enrolled/dropped
Does facility delivery vary by wealth?	Wealth quintile × place of delivery
Is complaint type linked to block?	Block × complaint category

Cross-tabulation is the most used and least respected tool in this list. Most equity findings a programme will ever produce are a two-way table with row percentages, and no more sophisticated method is required to see them.

Always print the counts alongside the percentages. A row showing 67 per cent uptake is a different claim when the row holds 300 households and when it holds three.

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Cat × Num

A group label and a number: compare means

When one variable is a group label and the other is numeric — treatment vs control and test score — you compare the numeric outcome across the groups.

Two groups

Difference in means — the t-test idea. Section Six.

Three or more groups

One-way ANOVA compares all the group means at once. Section Seven.

Report	Why
The mean or median for each group	The quantity anyone actually wants
N per group	Small groups produce unstable means
Spread within each group	Two groups can differ in variability, not level
The difference, in units	“3.2 kg lower” beats “significantly lower”

Use the median when the numeric variable is skewed, which for income, expenditure and landholding it always is. The comparison of medians is less standard and more honest, and a box plot shows both.

The third row is under-reported and often the finding. A programme that raises average yield while widening the spread has helped some farmers and left others where they were — visible in the spread, invisible in the mean.

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Num × Num

Two numbers: do they track each other?

When both variables are numeric — female literacy and fertility rate across districts — plot one against the other on a scatter and measure how tightly they move together.

Describe the strength with correlation (Section Five); model the line with simple linear regression (Section Nine).

Step	What you are looking for
Plot the scatter	Shape: line, curve, clusters, nothing
Look for outliers	Points far from the cloud, which will dominate any fit
Compute r	Only if the shape is roughly linear
Fit a line, if the question needs a rate	Slope in real units, with its uncertainty

The order matters and is routinely reversed. Computing r first and plotting afterwards means the number frames how you read the picture; plotting first means the picture tells you whether the number is meaningful at all.

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A Caution

The method serves the question, not the reverse

Do not pick a fancy test and then hunt for variables to feed it. Start from the real question, classify the two variables it involves, and let the decision table hand you the method.

Far better an approximate answer to the right question than an exact answer to the wrong one.

— John Tukey

Wrong order	Right order
“Let us run a regression”	“What are we trying to find out?”
Method chosen for familiarity	Method chosen by variable types
Recode a variable so the test runs	Choose the test that fits the variable
Report whatever the software printed	Report what answers the question

The third row is the most damaging habit. Collapsing a five-category variable into two so a familiar test applies throws away information and usually flatters the result, because the collapse point is chosen after seeing the data.

Write the question in a sentence before opening the software. It takes a minute, it determines the method, and it is the sentence the report will need anyway.

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03

Section Three

Cross-Tabulation & Contingency Tables

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The Workhorse

Counting two categories together

Cross-tabulation (contingency table)

A table that counts how many cases fall into each combination of two categorical variables — rows for one variable, columns for the other.

It is the most-used tool in applied development analysis: a single table that shows, at a glance, how two group memberships line up.

A contingency table shows	Report alongside
Counts in each combination of categories	Row or column percentages
Row and column totals (marginals)	The total N
Where cells are unexpectedly full or empty	Any cell with fewer than five cases

The marginals are the part people skip and the part that carries the context. Without the totals, a reader cannot tell whether an imbalanced table reflects a relationship or simply that one category is much larger than the other.

Small cells are the standard trap. Once any cell drops below about five expected cases, chi-square becomes unreliable — and the table will still print a percentage that looks like all the others.

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A 2×2 Table

Toilet ownership by location (raw counts)

	Has toilet	No toilet	Total
Rural	320	280	600
Urban	340	60	400
Total	660	340	1,000

Illustrative figures. The four inner cells are the joint counts; the right and bottom margins are the marginal totals for each variable on its own.

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Why Percentages

Raw counts mislead when groups differ in size

Rural and urban have different totals (600 vs 400), so comparing raw counts is unfair. To compare fairly, convert to percentages — but in which direction?

This is the single most common cross-tab error: percentaging the wrong way and reading a relationship backwards. Get the direction right and the table tells the truth.

Raw count comparison	Why it misleads
“More non-users are rural”	Most of the population is rural
“Most dropouts are boys”	There may be more boys enrolled
“This block has the most cases”	It also has the most people

Percentages are how a table becomes comparable when the groups differ in size — which they always do. A count table answers “how many”; only a percentage table answers “is membership linked”.

Keep both in the published table. Percentages for the comparison, counts so the reader can see the base — the standard format is “62.4% (n=187)” and it answers both questions in one cell.

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Row Percentages

Percent within each row

	Has toilet	No toilet	Row total
Rural	53%	47%	100%
Urban	85%	15%	100%

Each row sums to 100%. This answers: of rural households, what share have a toilet? 53% rural vs 85% urban — a clear location gap. (Illustrative.)

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Column Percentages

Percent within each column

	Has toilet	No toilet
Rural	48%	82%
Urban	52%	18%
Column total	100%	100%

Each column sums to 100%. This answers a different question: of households without a toilet, what share are rural? 82%. Same table, different story.

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The Rule

Percentage in the direction of the cause

Convention: percentage within categories of the explanatory variable, then compare across them. If location (X) might shape toilet ownership (Y), percentage within rural and within urban — that is row percentaging here.

01

Put X in the rows

→

02

Percentage so each row = 100%

→

03

Compare the same Y-column across rows

If your question is	Percentage within
“Do rural households use it less than urban?”	Location — so each location sums to 100
“Among users, how many are rural?”	Use status — so users sum to 100
“Does uptake differ by wealth?”	Wealth quintile

Percentage in the direction of the presumed cause, so that each category of the explanatory variable sums to 100 and the comparison across those categories is the finding. Getting this backwards produces a table that answers a question nobody asked.

The two directions answer genuinely different questions, and both can be true and interesting. State which one you have used in the table heading; readers assume whichever suits their expectation otherwise.

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Reading a 2×2

What a difference in percentages means

85% − 53%

= 32 percentage-point gap in toilet ownership, urban vs rural

Direction

Urban households far more likely to have a toilet — a positive urban–toilet link

A gap in the table suggests an association. Whether it is bigger than chance is what the chi-square test in Section Eight decides.

Way to express a 2×2 difference	Example
Percentage-point difference	62% vs 48% — a gap of 14 points
Ratio of proportions	62/48 = 1.3 times as likely
Relative change	29% higher

All three describe the same table and land very differently on a reader. Reporting only the largest-sounding one is the commonest way an honest table becomes a misleading sentence.

Give the percentage-point gap and the two underlying percentages. From those a reader can compute any of the others, and cannot be led by the choice of framing.

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04

Section Four

Visualising Relationships

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Plot First

Always look before you compute

Before any coefficient or test, draw the relationship. A picture reveals direction, strength, curvature, clusters and outliers that a single number can hide entirely.

The greatest value of a picture is when it forces us to notice what we never expected to see.

— John Tukey

The plot reveals	The statistic hides it
A curve rather than a line	r near zero suggests “no relationship”
One extreme point driving everything	r looks strong
Two distinct clusters	A line is fitted through empty space
A ceiling or floor in the data	The fit is distorted at the ends

Every row above produces a plausible-looking number. The software does not warn you, and none of these conditions is rare in field data — ceilings and clusters are the normal shape of programme datasets.

Plot first, every time, even when you are confident. It costs one line of code and it is the only step in this deck that catches all four failures at once.

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Match Plot to Types

Which chart for which pair

Variable pair	Best plot	Shows
Numeric × numeric	Scatter plot	Direction, strength, shape
Categorical × numeric	Box plot by group	Spread & median per group
Categorical × numeric	Grouped / clustered bars	Mean per group
Categorical × categorical	Stacked / grouped bars	Shares within groups

Pair	Plot	Avoid
Numeric × numeric	Scatter	Two lines on one time axis
Categorical × numeric	Box plot; strip plot for small N	A bar of means with no spread
Categorical × categorical	Grouped or stacked bars	Pies, especially several
Ordered category × numeric	Box plots in order	Alphabetical ordering

The second row’s “avoid” is the most common chart in this sector. A bar chart of group means shows the centres and conceals everything else — the spread, the N, the outliers — and it is what most reporting templates default to.

With fewer than about twenty points per group, show the points. A box plot summarising six observations implies more than it knows; the raw dots are both more honest and easier to read.

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Scatter Plots

The workhorse for two numbers

A scatter plot puts the explanatory variable on the X-axis, the response on the Y-axis, and one dot per case. The cloud's tilt shows direction; its tightness shows strength.

Read it like this: upward cloud = positive; downward = negative; round blob = no linear relationship; tight line = strong; fat cloud = weak.

When reading a scatter, ask	Because
Is the pattern linear?	Everything that follows assumes it
How tight is the cloud?	That is strength, and it is visible
Are there points far from the rest?	They will dominate any fitted line
Does the spread change along X?	Fanning out breaks a standard assumption
What is the unit — person, village, district?	Decides what the pattern can be about

The last row is the one that becomes the ecological fallacy in Section 10. A scatter of state averages is a statement about states; nothing in the picture prevents a reader from taking it as a statement about people, and most do.

With large N, overplotting hides density. Thousands of points become a solid mass; transparency, jitter or a hexbin plot restores the information that the raw scatter has already lost.

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See It

Three scatters: positive, negative, none

Same axes, three relationships

Illustrative

Green rises, red falls, indigo wanders. Your eye reads direction and strength instantly — before any number.

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Real-Feeling Data

Female literacy vs child mortality, by state

Female literacy (%) vs under-5 mortality (per 1,000), major states

Illustrative, patterned on Census 2011 & NFHS-5

A strong negative pattern: states with higher female literacy tend to have lower child mortality. Illustrative, but it mirrors the real Census–NFHS picture closely.

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Box Plots

Comparing a number across groups

For a categorical X and numeric Y, a box plot per group beats a single average. Each box shows the median (the line), the middle 50% (the box), the range (the whiskers) and outliers (the dots).

Box plots let you see not just whether the centres differ between groups, but whether the spreads do — two distributions with the same mean can look completely different.

The box plot shows	What that tells you
The median line	The typical value — robust to skew
The box (Q1 to Q3)	Where the middle half sits
The whiskers	The bulk of the range, by a stated rule
Points beyond them	Candidates for investigation, not deletion
Box width, if drawn to scale	Relative group size

A box plot shows five numbers where a bar chart shows one, in the same space, and it is the single best default for comparing a numeric outcome across groups. Overlapping boxes are also a visual signal that a difference in means may not survive testing.

State the whisker rule in the caption. Different software uses different conventions — 1.5×IQR is common but not universal — and an unlabelled box plot is ambiguous about which points count as outliers.

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Grouped Bars

A mean compared across groups

Mean monthly earnings by education level (illustrative, ₹000)

Illustrative, patterned on PLFS-style data

A clean grouped bar shows the mean of a numeric outcome rising across ordered categories — the visual form of a category × number relationship.

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Honest Charts

The same rules still apply

Start bar axes at zero — a truncated axis fakes a gap
Label both axes with units
Note the denominator and sample size
Put the source and date on the chart
Do not let one outlier dominate a scatter unremarked

A bivariate chart is an argument about a relationship. Truncated axes and missing denominators turn an honest pattern into a misleading one.

Rule	Bivariate-specific version
Axis from zero, or say why not	Truncating a Y-axis exaggerates a group gap
Label units	“Score” on what scale, out of what?
Show N	Per group, not just overall
Order categories meaningfully	By value or by natural order, never alphabetically
Do not imply causation in the title	“Yields by training status”, not “Training raises yields”

The last row is where most causal overclaiming actually enters a report. Nobody writes “training causes yields” in the analysis section; the chart title says it instead, and the chart is what gets screenshotted into the presentation.

Titles are the cheapest place to be disciplined. A neutral title costs nothing and removes the claim you cannot support, while leaving the finding entirely visible.

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05

Section Five

Correlation

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The Number

Putting a value on 'move together'

Correlation coefficient (r)

A single number summarising how strongly and in which direction two numeric variables move together in a straight-line sense. It ranges from −1 to +1.

Where a scatter shows the relationship, r measures it — one number for direction and strength combined.

r tells you	r does not tell you
Direction of a linear association	Which variable causes which
How tightly points follow a line	How steep that line is
A property of this sample	Anything about an individual case
Nothing about units	The size of the effect in real terms

The second row is the distinction most worth holding. Correlation measures scatter around a line, not the line’s gradient — so a very strong correlation can accompany a practically negligible effect, and a modest one can accompany a large effect with noise around it.

When the question is “how much”, you want a slope, not r. “Each additional kilometre is associated with 0.4 fewer days attended” is usable by a programme; r = −0.62 is not.

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The Scale

r runs from −1 to +1

−1

Perfect negative — points fall exactly on a downward line

0

No linear relationship — a shapeless cloud

+1

Perfect positive — points fall exactly on an upward line

The sign gives direction; the distance from zero gives strength. r = −0.8 and r = +0.8 are equally strong, opposite in direction.

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Pearson's r

The default: Pearson correlation

The everyday correlation is Pearson's r. Crucially, it measures only the linear (straight-line) component of a relationship between two numeric variables.

If the true relationship is curved — rising then falling — Pearson's r can be near zero even when the two variables are tightly related. Pearson sees lines, not curves.

Pearson’s r assumes	Broken when
Both variables are numeric	One is an ordinal scale
The relationship is linear	The pattern curves or plateaus
No extreme outliers	One district is far from the rest
Roughly symmetric distributions	Income, landholding, firm size

Three of the four are broken routinely in development data, which is why Spearman’s rank correlation is often the better default here. It is not a weaker method; it is a method that assumes less.

Compute both and compare. If Pearson and Spearman disagree sharply, something in the four rows above is happening — usually an outlier or a curve — and the disagreement is the diagnostic.

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Rough Guide

Interpreting the size of r

\|r\|	Rough strength	What the scatter looks like
0.0 – 0.1	Negligible	Shapeless cloud
0.1 – 0.3	Weak	Faint tilt
0.3 – 0.5	Moderate	Clear tilt, wide scatter
0.5 – 0.7	Strong	Tight tilt
0.7 – 1.0	Very strong	Near a straight line

These bands are conventions, not laws. In messy social data, an r of 0.3 can be a genuinely important signal.

\|r\|	Loosely	Caveat
0.0–0.2	Negligible to weak	These bands are conventions, not facts. What counts as strong depends entirely on the field: an r of 0.3 is large in social data and trivial in a physical measurement
0.2–0.4	Weak to moderate
0.4–0.7	Moderate to strong
0.7–1.0	Strong to very strong

Be suspicious of a very high r in survey data. Above about 0.9 between two supposedly distinct social variables, the usual explanation is that they measure the same thing — or that one was derived from the other during cleaning.

Report r with N and the scatter. The band label is the least informative part; a reader who sees the plot and the sample size can judge the strength without any adjective at all.

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Worked Example

Literacy and fertility, quantified

Female literacy (%) vs total fertility rate, major states

Illustrative, patterned on Census 2011 & NFHS-5

This cloud has a clear downward tilt — an r near −0.9 (illustrative). Strong and negative. But strength is not the same as cause: literacy may proxy for income, health and much else.

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Spearman

When to use rank correlation instead

Spearman's rank correlation (ρ)

Pearson's correlation computed on the ranks of the data rather than the raw values. It measures whether two variables move together in the same order, not strictly in a straight line.

Use it for ordinal data — wealth quintiles, Likert scales
Use it for skewed data — income, landholding
Use it when the relationship is monotonic but curved
It is robust to outliers that would swing Pearson's r

Use Spearman when	Because
A variable is ordinal	Ranks are all the variable actually carries
The data is heavily skewed	Ranking removes the influence of the long tail
There are extreme outliers	An outlier becomes just the top rank
The relationship is monotonic but curved	Spearman detects consistent direction without linearity

Spearman is Pearson computed on the ranks, which is why it inherits the interpretation and drops the linearity requirement. It answers “do they move in the same direction consistently” rather than “do they follow a straight line”.

The cost is information. Ranking discards magnitude, so a Spearman correlation cannot tell you that the gap between the first and second districts is enormous while the rest are close together.

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Pearson vs Spearman

Two correlations, side by side

	Pearson r	Spearman ρ
Measures	Linear association	Monotonic (rank) association
Best data	Numeric, roughly symmetric	Ordinal or skewed numeric
Outlier-sensitive?	Yes — one point can swing it	No — uses ranks
Range	−1 to +1	−1 to +1

When Pearson and Spearman disagree sharply, suspect non-linearity or an outlier — and go back to the scatter.

If they agree	If they differ substantially
The relationship is roughly linear and outlier-free	Look for an outlier, a curve, or heavy skew
Report either; say which	Report Spearman, and explain why
Nothing further needed	The disagreement is itself worth a sentence

Running both is a two-line diagnostic that catches most of the ways a correlation misleads, and it costs nothing. Where they diverge, the plot will show you why within seconds.

Never choose between them after seeing which is larger. That is the forking-paths problem in miniature: decide by the variable types and the shape of the data, and state the rule you used.

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Strength ≠ Significance

A big r and a small p are different things

Strength (effect size)

How big is the relationship? That is what r itself tells you — the practical magnitude.

Significance (p-value)

How sure are we it is not zero by chance? That depends heavily on sample size.

With thousands of cases, a tiny, uninteresting r = 0.05 can be 'statistically significant'. Always report the size, not just the star.

Situation	r	p	Reading
N = 20,000	0.03	Tiny	Real, and far too small to matter
N = 25	0.55	Not significant	Possibly important; sample too small to tell
N = 400	0.42	Small	The useful case

Strength and significance answer different questions, and sample size drives the second far more than the first. With a large enough N almost any correlation becomes significant, which makes “significant correlation” a nearly empty phrase in big datasets.

The second row is the one that gets discarded wrongly. A non-significant result in a small sample is not evidence of no relationship; it is an absence of evidence, and reporting it as “no association” is a real error.

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r-squared Preview

Square it for 'variance explained'

Square the correlation and you get R² — the share of the variation in one variable that is statistically accounted for by the other. r = 0.7 means R² = 0.49: about half the variation is shared.

We return to R² properly in the regression section — it is the natural bridge from correlation to a fitted line.

r	r²	Shared variation
0.3	0.09	9%
0.5	0.25	25%
0.7	0.49	49%
0.9	0.81	81%

Squaring is deflating, which is the point of doing it. A correlation of 0.5 sounds like half the story and accounts for a quarter of the variation — a useful corrective when a moderate r is being described as a strong finding.

“Variance explained” is a statistical phrase, not a causal one. It means the line accounts for that share of the scatter; it does not mean the X variable produces that share of the outcome, and the wording invites exactly that reading.

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06

Section Six

Comparing Two Groups

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The Setup

One group label, one number

A categorical X with two values (treatment/control, girls/boys, SHG/non-SHG) and a numeric Y (score, income, weight). The question: do the two group means differ — and is the difference real?

01

Split the data into two groups by X

→

02

Compute each group's mean of Y

→

03

Ask: is the gap bigger than chance?

Before comparing two groups, check	Why
How the groups were formed	Self-selected groups differ in more than the label
N in each	An unbalanced comparison is dominated by the small group’s noise
The distribution in each	Skew or bimodality makes a mean the wrong summary
Whether observations are independent	Households within a village are not

The first row decides what the comparison can mean. Randomly assigned groups support a causal reading; groups that chose themselves — trained farmers, scheme participants, migrants — differ in motivation, resources and information as well as in treatment.

The last row is quietly violated in almost all field data. Cluster sampling makes households within a village correlated, which makes a standard t-test overstate precision — the p-value is smaller than it should be.

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Difference in Means

The quantity of interest

Mean test score: control vs treatment schools (illustrative)

Illustrative

The raw difference is 7 points (61 − 54). But two questions remain: could a 7-point gap arise by chance, and is 7 points big enough to matter?

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The t-test Idea

Signal divided by noise

t-test

A test of whether the difference between two group means is larger than we would expect from sampling variation alone — essentially, the size of the gap relative to how noisy the data are.

Intuitively, t ≈ difference in means ÷ uncertainty in that difference. A big, clean gap gives a big t; a small gap drowned in scatter gives a small t.

Component	Effect on the t statistic
Larger difference between means	Larger t — more signal
Larger spread within groups	Smaller t — more noise
Larger sample	Larger t — the noise estimate tightens

That is the whole idea: signal divided by noise, scaled by how much data you have. Everything else about the t-test is machinery for turning that ratio into a probability, and understanding the ratio is enough to read a result critically.

It also explains the two disappointments. A real difference in a noisy, small sample fails to reach significance; a trivial difference in a huge sample reaches it easily. Both are the formula working correctly.

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What Drives It

Three things make a gap convincing

A larger difference between the two means
Less spread (variation) within each group
A larger sample in each group

The same 7-point gap is unconvincing with 20 noisy pupils per arm, but compelling with 2,000 tightly clustered ones. Sample size and spread decide the verdict.

Makes a gap convincing	In practice
A big gap	The part you cannot control
Low variability within groups	Better measurement; more homogeneous comparison
A large sample	The part you can plan for, at design stage

Only the third row is under your control after the fact, and it is decided before any data is collected. This is why power calculations belong at design and are useless at analysis, when the sample is whatever it is.

The second row is undervalued. Reducing measurement error — better questions, trained enumerators, a cleaner instrument — raises your ability to detect a difference just as surely as adding respondents, and usually costs less.

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The p-value

What a p-value does and doesn't say

p-value

The probability of seeing a difference at least this large if there were truly no difference between the groups. Small p (say < 0.05) means the gap is unlikely to be pure chance.

A p-value is not the probability the effect is real, nor a measure of its size. It only addresses 'could this be chance?' — nothing about importance.

A p-value is	A p-value is not
The probability of data this extreme if there were no real difference	The probability that there is no difference
A statement about the data, given a hypothesis	A statement about the hypothesis, given the data
Sensitive to sample size	A measure of importance
Compared to a threshold someone chose	A natural boundary at 0.05

The first row is the definition and is almost universally misstated, including in published papers. The conditional runs one way: it assumes no difference and asks how surprising the data is, not the reverse.

0.05 is a convention, not a discovery. Treating 0.049 and 0.051 as categorically different results is arithmetic superstition, and reporting the actual value with a confidence interval avoids the whole problem.

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Effect Size

How big, in plain terms

Significance asks whether there is a difference; effect size asks how big. Report it in real units (7 marks, ₹400/month) or as a standardised measure (Cohen's d) so others can judge whether it matters.

p-value

Is the gap likely real, not chance?

Effect size

Is the gap big enough to act on?

Way to state the size	Example
Raw difference in units	“0.4 kg heavier on average”
With a confidence interval	“0.4 kg (95% CI 0.1 to 0.7)”
Standardised (Cohen’s d)	“d = 0.3” — for comparing across studies
In programme terms	“About one additional child per 20 households”

The last row is the one decision-makers use and the one analysts most often omit. Translating an effect into the unit a programme operates in — households, children, rupees per beneficiary — is part of the analysis, not a communications afterthought.

Prefer raw units with an interval to a standardised measure. Cohen’s d is useful when comparing across studies with different scales; within one programme it converts something concrete into something abstract.

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The Trap

Significant but trivial — and vice versa

Significant but trivial

Huge sample → a 0.3-mark difference is 'significant' but means nothing for any child.

Real but 'not significant'

Tiny pilot → a promising 9-point gap fails the test purely for lack of sample. Absence of evidence is not evidence of absence.

Always read the p-value and the effect size together. Neither alone tells the whole story.

Case	What to write
Significant, tiny	“A difference of 0.2 points, statistically detectable and too small to act on”
Large, not significant	“A gap of 8 points that this sample cannot confirm; worth a larger study”
Large and significant	The straightforward case — report both numbers
Small and not significant	“No evidence of a difference” — not “no difference”

The fourth row’s distinction is not pedantry. “No evidence of a difference” leaves open that the study was too small to see one; “no difference” asserts something the data cannot support, and it is how negative findings get overstated.

Reporting the confidence interval resolves all four rows at once. An interval from −0.1 to 0.3 says “small, whatever else”; one from −5 to 12 says “we do not know”, and neither can be misread as the other.

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Assumptions

When the simple t-test is appropriate

The two groups are independent (different people)
Y is roughly symmetric within each group (or n is large)
For paired data — same people before/after — use a paired t-test instead
For severely skewed data, consider a rank-based alternative

Assumption	Check	If broken
Independent observations	How was the sample drawn?	Account for clustering
Roughly normal, or large N	Histogram per group	Mann–Whitney, or transform
Similar spread in both groups	Box plots side by side	Welch’s t-test — often the better default
No extreme outliers	Look at the plot	Investigate before deciding

Welch’s version is a sensible default for group comparisons in field data, because it does not assume equal variances and loses very little when they are equal. Many packages now use it automatically; check which one yours ran.

The first assumption is the one that cannot be patched at analysis. If the design was clustered, the analysis must account for it — and no choice of test statistic repairs a p-value computed as though villages were households.

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07

Section Seven

Comparing Several Groups (ANOVA)

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The Setup

When there are three or more groups

A categorical X with three or more values — four caste groups, five wealth quintiles, six districts — and a numeric Y. We want to know whether the group means differ overall.

Tempting shortcut: run a t-test on every pair. Don't. With many pairs, the chance of a false 'significant' result piles up fast — the multiple-comparisons problem.

Why not just run many t-tests?
Five groups gives ten pairwise comparisons
Each carries its own chance of a false positive
At 5% each, the chance of at least one false positive is around 40%
So “one pair differed” means very little without adjustment

This is the multiple-comparisons problem, and it is why ANOVA exists. One overall test asks whether the groups differ at all, at a controlled error rate, before you go looking for which pair.

The same arithmetic applies to subgroup analysis generally. Testing an outcome across twelve districts and reporting the one that reached significance is the forking-paths problem with a respectable name.

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The Idea

One-way ANOVA, conceptually

One-way ANOVA

Analysis of Variance — a single test of whether the means of three or more groups differ by more than chance, by comparing variation BETWEEN groups to variation WITHIN groups.

Despite the name, ANOVA is about means. It uses variances as the yardstick for deciding whether those means are really apart.

ANOVA answers	It does not answer
Is there a difference somewhere among the groups?	Which groups differ
At a controlled overall error rate	By how much
Under stated assumptions	Whether the difference matters

A significant ANOVA is a permission slip, not a finding. It licenses you to look for which pairs differ, using a method that adjusts for having looked — and on its own it tells a programme nothing it can act on.

Report the group means and their intervals whatever the F test says. Those are the numbers a reader needs; the test statistic is a gatekeeping device, not the result.

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Between vs Within

The heart of ANOVA

Between-group variation

How far apart the group means are from each other — the signal.

Within-group variation

How much individuals scatter inside each group — the noise.

The F-ratio = between ÷ within. If groups sit far apart relative to their internal scatter, F is large — evidence the means genuinely differ.

Source of variation	What it captures
Between groups	How far the group means sit from the overall mean
Within groups	How much individuals vary around their own group’s mean
F = between ÷ within	Large when the groups are far apart relative to internal spread

It is the t-test’s signal-to-noise idea generalised to several groups. Nothing conceptually new is happening; the ratio simply compares variation attributable to group membership with variation that has nothing to do with it.

Which is why within-group spread matters so much. Five quintiles with widely different means but enormous internal scatter will not produce a large F — and looking at the box plots shows you that before the test does.

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See It

Mean nutrition score by wealth quintile

Mean child height-for-age z-score by wealth quintile (illustrative)

Illustrative, patterned on NFHS-5

Five group means rising steadily across quintiles. ANOVA asks: taken together, is this spread of means bigger than the scatter within each quintile? (Illustrative; the real gradient is well documented.)

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After ANOVA

It says 'somewhere', not 'where'

A significant ANOVA tells you the groups are not all equal — but not which ones differ. To locate the differences, follow up with post-hoc pairwise comparisons that correct for multiple testing.

Think of ANOVA as the smoke alarm: it tells you there is a fire somewhere, then post-hoc tests find the room.

After a significant ANOVA	Why
Run a post-hoc test (Tukey, Bonferroni)	Adjusts for the many comparisons you are about to make
Report every group mean with its interval	The pattern matters, not just the significant pair
Say how many comparisons were made	A reader cannot judge otherwise
Look at the ordering	A monotonic gradient across quintiles is a stronger finding than one odd pair

The last row is the one that carries real weight in our work. If nutrition scores rise steadily from the poorest quintile to the richest, that gradient is far more informative than any single pairwise test, and it is visible without one.

Do not run unadjusted t-tests after ANOVA. It reintroduces exactly the error rate the ANOVA was protecting you from, and it is the commonest way this sequence is misused.

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Assumptions

When one-way ANOVA fits

Groups are independent
Y is roughly symmetric within each group (or n is large)
Group spreads are not wildly different
For badly skewed data, a rank-based alternative (Kruskal–Wallis) may suit

ANOVA is the natural extension of the two-group t-test to many groups — same logic, scaled up.

Assumption	If broken
Independent observations	Account for clustering; ANOVA cannot fix it
Roughly normal within groups	Kruskal–Wallis, the rank-based alternative
Similar spread across groups	Welch’s ANOVA
Groups defined before looking	Otherwise the test is meaningless

The fourth row is not usually listed as an assumption and matters more than the other three. Groups formed after inspecting the outcome — “the blocks that did badly” — guarantee a difference, because the difference is how the groups were chosen.

Kruskal–Wallis is a safe fallback for the skewed outcomes common here, and costs little. As with Spearman, choose it from the shape of the data rather than from which result you prefer.

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Recap

Comparing groups, at a glance

Groups	Y type	Method
2 (independent)	Numeric	Two-sample t-test
2 (same units, paired)	Numeric	Paired t-test
3 or more	Numeric	One-way ANOVA
3+, skewed data	Numeric	Kruskal–Wallis

Groups	Parametric	Rank-based
Two	t-test (Welch by default)	Mann–Whitney
Three or more	One-way ANOVA	Kruskal–Wallis
Then	Post-hoc pairwise, adjusted	Adjusted pairwise ranks

Pick the row from the number of groups and the column from the shape of the data. That is the entire decision, and it removes almost all of the anxiety about choosing a test for group comparisons.

Whichever cell you land in, report the group medians or means with intervals. The test decides whether to believe the pattern; the summary statistics are the pattern.

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08

Section Eight

Association Between Categories (Chi-Square)

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The Setup

Two categories: are they independent?

Back to two categorical variables and a cross-tab. The chi-square test of independence asks: is the pattern in this table bigger than we would expect if the two variables were completely unrelated?

Independence

Two variables are independent if knowing one tells you nothing about the other — the distribution of Y is the same within every category of X.

Chi-square tests	Typical use here
Whether two categorical variables are independent	Social group × scheme uptake
Using counts, not percentages	Feed it the raw table
Against what independence would predict	The expected counts on the next slide

Chi-square answers a yes/no question and is silent on strength. It tells you the pattern is unlikely under independence; the percentage-point gap in the cross-tab tells you whether it matters, and only one of those belongs in a recommendation.

Feed it counts, never percentages. Running chi-square on a table of percentages produces a number that depends on whether you scaled to 100 or to 1,000 — a mistake that is easy to make and invisible in the output.

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The Core Idea

Observed vs expected counts

Chi-square compares what you observed in each cell with what you would expect if the two variables were independent. Big gaps between observed and expected = evidence of association.

01

Compute expected counts (if independent)

→

02

Compare with observed counts, cell by cell

→

03

Sum the scaled squared gaps → χ²

→

04

Large χ² → reject independence

Step	What happens
Compute what each cell would hold if the variables were independent	Row total × column total ÷ grand total
Compare with what was observed	Cell by cell
Square the gaps, scale by expected, add up	Large when observed departs from expected
Refer to the distribution	Gives the p-value

The scaling in step three is what makes the statistic sensible. A gap of ten cases matters enormously in a cell expected to hold twelve and hardly at all in one expected to hold two thousand.

Look at the individual cell contributions, not just the total. Most software will report them, and they tell you which combination is actually driving the result — which is the finding a programme can use.

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Expected Counts

What 'no relationship' would predict

Each cell's expected count is (row total × column total) ÷ grand total. It is the count you'd see if Y were distributed identically across every category of X.

Example: with 660 of 1,000 households owning a toilet (66%), independence predicts 66% of the 600 rural households — 396 — would own one. We observed only 320. That gap is the signal.

Expected count for a cell	Meaning
Row total × column total ÷ N	What that cell would hold if the two variables were unrelated
Uses only the margins	It preserves how many are in each category overall
Rarely a whole number	That is fine; it is an expectation, not a count

Expected counts are the whole idea of the test made explicit. Independence is not an abstraction here: it is a specific table you can compute, and the statistic simply measures how far the real table sits from it.

Check them before trusting the p-value. The validity rule on the next slides is about expected counts, not observed ones — a cell can hold seven real cases and still be too sparse if only two were expected.

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Observed vs Expected

Side by side (illustrative)

Cell	Observed	Expected	Gap
Rural, toilet	320	396	−76
Rural, no toilet	280	204	+76
Urban, toilet	340	264	+76
Urban, no toilet	60	136	−76

Rural areas have far fewer toilets than independence predicts, urban far more. Consistent gaps like these push χ² up. (Illustrative figures.)

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Reading the Result

What chi-square tells you

Small p

Reject independence — the two variables ARE associated

Large p

No evidence of association — consistent with independence

Chi-square tells you whether there is an association, not how strong it is. For strength, report a measure like Cramér's V alongside the test.

Chi-square gives you	You still need
Evidence against independence	The cross-tab, to see the direction
A p-value	A measure of strength — Cramér’s V, or the percentage gap
Nothing about which cell	Cell contributions, or the percentages
Nothing causal	Everything in Section 10

Reporting a chi-square result alone is close to reporting nothing. “Uptake was associated with social group (p < 0.01)” tells a reader that a relationship exists and not which group is disadvantaged or by how much — the only two facts a programme needs.

Lead with the table. “Uptake was 61% among general-category households and 44% among SC/ST households (p < 0.01)” carries the test and the finding in one sentence.

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Validity

When chi-square is valid

Each case counted once (independent observations)
Cells hold counts, not percentages or means
Expected count ≥ 5 in (almost) every cell
Categories are mutually exclusive and exhaustive

The expected-count rule is the one most often broken: with thin cells the chi-square approximation fails. Use Fisher's exact test for small tables instead.

Requirement	If it fails
Counts, not percentages or means	The statistic is meaningless
Independent observations	Clustering inflates the statistic
Expected count of about 5 or more in most cells	Combine categories, or use Fisher’s exact test
Each case in exactly one cell	Multiple-response questions break this

The last row catches people with survey data. A “tick all that apply” question puts one respondent in several cells, which violates the test’s basic structure — and the software will happily produce a p-value anyway.

Combining sparse categories is legitimate if the combination is defensible on substantive grounds and decided before seeing the result. Merging categories until the p-value crosses a threshold is not.

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Common Mistakes

Three ways chi-square goes wrong

Feeding it percentages instead of raw counts
Ignoring tiny expected counts in sparse cells
Reading a significant result as proof of causation
Forgetting it says nothing about the strength of association

Mistake	Consequence
Running it on percentages	The result scales with your choice of base
Ignoring sparse cells	An unreliable p-value that looks normal
Reporting significance without direction	A finding nobody can act on
Reading it as causal	The error the whole of Section 10 addresses

All four are avoidable by looking at the cross-tab first. The table shows the counts, the sparse cells and the direction; the test adds one number to a picture you should already have understood.

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In Practice

Chi-square in a development workflow

It is the natural partner to the cross-tab: you describe the two categories with row percentages, then test whether the visible gap is bigger than chance with chi-square.

Worked flow

Caste × toilet ownership

Build the cross-tab → percentage within caste → eyeball the gap → check expected counts ≥ 5 → run chi-square → report the test and a strength measure → resist the leap to 'caste causes…'.

Step	Output
Build the cross-tab with counts	The raw table
Add row percentages in the causal direction	The comparison
Check expected counts	Whether the test is valid
Run chi-square	Evidence against chance
Report percentages, gap, N and p	Something a programme can use
List plausible confounders	Honest framing

Steps one and two do most of the work, and are frequently skipped in favour of going straight to the test. The table is the finding; the test is a check on whether the table is worth discussing.

Step six costs a sentence and buys credibility. Naming the obvious confounder yourself — usually wealth or location — is what distinguishes an analyst who understands the limits from one who has not thought about them.

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Recap

The category × category toolkit

Step	Tool	Answers
Describe	Cross-tab + %	What does the pattern look like?
Test	Chi-square	Is it bigger than chance?
Quantify strength	Cramér's V	How strong is it?
Small table?	Fisher's exact	Same question, thin cells

Need	Use
See the pattern	Cross-tab with row percentages
Test independence	Chi-square, on counts
Small or sparse table	Fisher’s exact test
Strength of association	Cramér’s V, or the percentage-point gap
Two binary variables, risk framing	Risk ratio or odds ratio

The last row is worth knowing for health work. Risk ratios and odds ratios are the standard language of epidemiology, and they are not interchangeable — odds ratios exaggerate when the outcome is common, which is a frequent source of misreporting.

For most programme reporting, the percentage-point gap is enough. It is understood without training, it is in the units of the decision, and it does not need a footnote explaining what it means.

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09

Section Nine

Simple Linear Regression

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From Cloud to Line

Drawing the best line through a scatter

Correlation gives a number for a numeric–numeric relationship. Simple linear regression goes one step further: it fits the single straight line of best fit through the scatter, summarising the relationship as an equation.

Simple linear regression

Modelling a numeric response Y as a straight-line function of one numeric predictor X: Y = intercept + slope × X, plus error.

Regression adds to correlation	Why it matters
A slope in real units	“0.4 fewer days per extra kilometre”
A prediction for any X	Fitted values, with their uncertainty
An explicit direction	You must nominate X and Y
Residuals to inspect	Where the model fails, and on which cases

The line is chosen to minimise squared vertical distances, which is why extreme points pull it so hard: an error twice as large counts four times as much. That single fact explains most of the outlier behaviour in Section 10.

The fourth row is the underused output. Plotting residuals against fitted values reveals curvature, changing spread and influential cases in one picture — and takes one line of code that almost nobody runs.

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See It

A regression line over a scatter

Years of schooling vs monthly earnings, with line of best fit

Illustrative

The red line is the one that sits closest to all the points at once — it minimises the total squared vertical distance from points to line (least squares). Illustrative data.

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The Equation

Y = a + bX

a (intercept)

Predicted Y when X = 0 — where the line crosses the vertical axis

b (slope)

Change in Y for each 1-unit rise in X — the rate of the relationship

The slope carries the action. Its sign matches the correlation's sign; its size is in real units of Y per unit of X.

Term	Reads as
a (intercept)	Predicted Y when X is zero
b (slope)	Change in Y for a one-unit rise in X
Residual	What the line got wrong for this case
Fitted value	What the line predicts for this case

The slope is the number that belongs in the report, because it is in the units the programme thinks in. Everything else in the output is machinery for deciding how much to trust it.

Write the units into the sentence. “b = −0.42” means nothing on its own; “0.42 fewer days attended per additional kilometre” is the same number and a usable finding.

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Intercept Care

When the intercept is meaningless

The intercept is the predicted Y at X = 0 — but X = 0 may be nonsensical or far outside your data. The earnings at zero years of schooling may be a mathematical artefact, not a real group.

Interpret the intercept only when X = 0 is both meaningful and within the range of your data. Otherwise treat it as a mathematical anchor for the line.

X	Is X = 0 meaningful?
Distance to facility	Yes — living at the facility
Years of schooling	Yes — no schooling
Mother’s age	No — there are no newborn mothers
Year (e.g. 2015–2024)	No — the intercept refers to year zero

Where X = 0 is impossible, the intercept is arithmetic rather than a finding. It is required to position the line and should not be interpreted, and reporting it as though it means something is a small but common embarrassment.

Centring fixes it. Subtract the mean of X before fitting, and the intercept becomes the predicted Y at the average X — a quantity that exists and is worth quoting.

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Reading the Slope

What b means in plain words

In our example the line rises about ₹1,250 per extra year of schooling (illustrative). So b ≈ 1.25 in ₹000: ‘each additional year of schooling is associated with about ₹1,250 more monthly earnings.’

Note the phrase ‘associated with’, not ‘causes’. Regression fits a line; it does not, by itself, license a causal claim.

Say	Do not say
“Associated with 0.4 fewer days”	“Causes 0.4 fewer days”
“On average, across this sample”	“For any given household”
“Within the observed range of X”	“At any distance”
“Before accounting for wealth and location”	Nothing about what was not controlled

A slope from observational data is an association per unit, not an effect of intervening. If you halved everyone’s distance tomorrow, the change in attendance would not be the slope — because distance is entangled with everything else that differs between near and far households.

The second row matters for targeting. A slope describes an average across the sample; individual households vary around it enormously, and a prediction for one household carries far wider uncertainty than the line suggests.

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R-squared

How much variation the line explains

R² (coefficient of determination)

The proportion of the variation in Y that is explained by X through the fitted line. It runs from 0 (line explains nothing) to 1 (line explains everything). In simple regression, R² = r².

R² = 0.8

Line explains 80% of the variation in Y; 20% is left unexplained

R² = 0.1

Line explains just 10% — X is a weak guide to Y

R²	Reads as
0.05	The line accounts for 5% of the variation — typical for social data
0.30	A substantial share, by the standards of this field
0.95	Suspicious in survey data — check for a definitional link

Low R² is normal and not a failure. Human behaviour varies for reasons no two-variable model captures, and a slope can be precisely estimated, important and worth acting on while the model explains a small fraction of the scatter.

The third row is a warning, not a triumph. An R² that high between two social variables usually means one was computed from the other — income and income-per-capita, or a score and one of its own components.

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R-squared Caution

High R² is not always good

A high R² does not prove the model is correct or causal
In social data, an R² of 0.2 can still be a useful finding
A low R² with a clear slope can still matter for policy
R² says nothing about whether a straight line fits

Report R² for context, but never let it crowd out the slope, its uncertainty, and a look at the scatter.

High R² can come from	Which is not a good sign
Both variables trending over time	Almost any two series will fit well
Aggregated data — districts, states	Averaging removes individual noise mechanically
One variable derived from the other	A definitional relationship, not a finding
A few extreme points	Leverage, not fit

The second row explains why state-level scatters look so convincing. Aggregation averages away individual variation, so district or state data routinely produces R² values that would be impossible at household level — and the tighter fit says nothing about individuals.

Judge a model by the residual plot, not by R². A high R² with a clear curve in the residuals is a worse model than a low R² with residuals scattered evenly.

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Prediction vs Explanation

Two different goals for one line

Explanation

Understand the relationship: how does Y change with X, and how strong is it? The slope is the prize.

Prediction

Estimate Y for a new X you haven't measured. Accuracy, not interpretation, is the prize.

Be clear which you are doing — the same line, read for different purposes, demands different cautions.

	Prediction	Explanation
Goal	Guess Y accurately	Understand how X relates to Y
Cares about	Fit and out-of-sample accuracy	The slope and its uncertainty
Confounders	Do not matter if fit holds	Matter completely
Useful for	Targeting, forecasting	Deciding what to change

The third row is the distinction that matters most. A model that predicts well can be built on pure proxies and still be useful for finding households to visit; the same model says nothing about what would happen if you changed one of those variables.

Trouble arrives when a predictive model is read as an explanatory one. A targeting model that flags households by roof type is not a finding that roofs cause deprivation, and acting on it as though it were produces roof-replacement programmes.

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Extrapolation

Don't predict beyond your data

Extrapolation — using the line far outside the range of X you actually observed — is one of regression's classic traps. A line fitted on 0–16 years of schooling says nothing reliable about 30 years.

The relationship may bend, plateau or break entirely outside your data. Predict within the range you measured, and flag any step beyond it.

Why extrapolation fails	Example
The relationship may curve outside the range	Dose response that plateaus
No data means no evidence of the shape	The line continues because it is a line
Predictions become physically impossible	Negative attendance; percentages above 100
Uncertainty widens sharply at the edges	And keeps widening beyond them

State the range of X your model was fitted on, and refuse predictions outside it. This is a one-sentence discipline and it prevents the most confident category of error — a number computed by the software with nothing behind it.

Time-series projection is the commonest case. A trend fitted to five years and projected to twenty is arithmetic, not forecasting, and the intervals around it — if anyone computed them — would usually be wide enough to include almost anything.

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10

Section Ten

Pitfalls

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The Big One

Correlation is not causation

A relationship between two variables can arise for several reasons, only one of which is ‘X causes Y’. This is the single most important caution in all of bivariate analysis.

Reverse causation: Y might cause X
Confounding: a third factor C drives both
Selection: how the sample was chosen creates the link
Chance: with enough variables, some correlate by luck

Before writing a causal sentence, rule out
Reverse causation — could Y have produced X?
Confounding — what third factor drives both?
Selection — how did units end up in each group?
Chance — how many relationships were examined?
Measurement — do both variables share a common error?

The exercise is to say each aloud and explain why it is unlikely here. If you cannot, the finding is an association and should be written as one — which is not a weaker claim, only an accurate one.

The fifth row is rarely considered. If the same enumerator recorded both variables, or both come from one respondent’s recall, a shared bias can create a correlation with no relationship behind it at all.

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Confounding

The lurking third variable

A confounder is a variable linked to both X and Y that creates — or distorts — the relationship between them. It is the commonest reason a bivariate link is misleading.

01

Household wealth (confounder C)

→

02

drives girls' schooling (X)

→

03

AND drives child survival (Y)

→

04

so X and Y correlate — partly via C

Apparent bivariate link	The usual confounder
Toilet ownership and child height	Household wealth
Training attendance and yields	Who chooses to attend
Private schooling and test scores	Parental education and income
Distance and facility delivery	Remoteness, which brings poverty and fewer roads

Wealth is the confounder in most development data, because it is correlated with nearly everything we measure. A bivariate relationship that has not been checked against wealth has not really been checked.

The cheapest partial fix is stratification. Repeat the analysis within each wealth quintile: if the relationship survives inside every stratum, it is not purely a wealth artefact — and if it disappears, you have learned that in an afternoon.

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Why It Matters Here

Bivariate links almost always hide confounders

Recall our literacy–fertility scatter. Female literacy really does track lower fertility — but richer states also have better health systems, later marriage and more contraception. Wealth and development confound the simple two-variable picture.

This is exactly why bivariate analysis is a starting point. To isolate one variable's effect you need to control for confounders — the job of multivariate regression.

What bivariate analysis is good for	What it should not be used for
Describing who is worse off	Claiming why they are worse off
Finding where to look next	Deciding what to change
Screening many candidate relationships	Concluding from the one that survived
Communicating a pattern simply	Attributing it to a programme

The left column is genuinely valuable and often undersold. Equity analysis — who is being reached, who is not — is descriptive by nature, needs no causal claim, and is the most useful thing most programmes could do with their own data.

Moving to the right column requires design, not statistics. A comparison group, randomisation, or a credible natural experiment — none of which can be added at analysis stage.

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Ecological Fallacy

Group patterns ≠ individual truths

Ecological fallacy

Wrongly inferring something about individuals from a relationship observed only at the group (district, state) level.

States with higher average literacy have lower average fertility — that does not mean the literate women within a state are the ones with fewer children. A relationship across districts need not hold across people.

Unit of the scatter	What a pattern can support
States	A statement about states
Districts	A statement about districts
Households	A statement about households
Individuals	A statement about individuals

The rule is that simple and is broken constantly. A famous state-level scatter of female literacy against fertility is a fact about states; every reader takes it as a fact about women, and the two can even point in opposite directions.

Put the unit in the chart title. “By state” or “by household” costs two words and is the only thing standing between an honest chart and an ecological inference.

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Outliers & Leverage

One point can swing everything

A single extreme point can dominate a correlation or drag a regression line toward itself — high leverage. The whole ‘relationship’ may rest on one unusual case.

Always check: does the pattern survive if I remove the most extreme point? If r or the slope collapses, the finding was that one point, not a real trend.

Term	Meaning
Outlier	Far from the cloud in Y
Leverage	Far from the cloud in X — can swing the slope hard
Influence	Both: removing it changes the conclusion

Because the line minimises squared distances, one far-out point can dominate the fit. In a scatter of Indian states this is routine — a very large or very small state sits alone at one end and quietly determines the slope.

The honest test is to refit without it and report both. If the conclusion changes, that is the finding: your result depends on one observation, and the reader is entitled to know which.

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Non-linearity

Straight-line tools miss curves

Pearson's r and linear regression assume a straight line. A U-shaped or saturating relationship — common in real data — can show a near-zero correlation while being strongly, but non-linearly, related.

The fix is the cheapest in statistics: plot the scatter. A curve is obvious to the eye and invisible to r.

Common non-linear shapes here	Example
Diminishing returns	Each extra year of schooling adds less
Threshold	Nothing happens until a minimum dose is reached
Ceiling	Coverage cannot exceed 100%, so gains flatten
U-shape	Benefit rises then falls — r near zero

Straight-line tools will fit a straight line to any of these and report a slope, an r and a p-value. Only the scatter reveals that the summary is describing a shape the data does not have.

Diminishing returns is the most common and the most consequential. A linear model averages the steep early gains with the flat later ones, understating what a programme achieves at the bottom and overstating it at the top.

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Anscombe

Four datasets, identical statistics

Anscombe-style: same r, four very different shapes

After F. J. Anscombe (1973)

Anscombe's quartet (1973): four datasets with near-identical means, variances and correlation, yet utterly different shapes. The lesson, still true: always plot your data.

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Simpson's Paradox

A trend can reverse when you split

Simpson's paradox: a relationship in the pooled data can flip direction within every subgroup. A scheme can look worse overall yet be better in every district — if districts differ in size and baseline.

Always disaggregate before concluding. The aggregate two-variable relationship can point the opposite way to the truth.

Aggregate says	Every subgroup says	Because
Hospital A does worse	A does better in each severity band	A takes the severe cases
Scheme B looks weaker overall	B is stronger in every district	B operates where uptake is hardest

Neither view is wrong; they answer different questions. Which one you want depends on the decision — a patient choosing a hospital and a regulator assessing one need different tables, and the choice has to be argued rather than assumed.

Always split an aggregate result by the obvious grouping variable once. A reversal is a finding worth understanding; discovering it after publication is not.

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Pitfall Checklist

Before you believe a relationship

Did I plot it — is the shape really linear?
Could a confounder explain it?
Does it survive removing the outlier?
Is this a group pattern I'm reading onto individuals?
Does it reverse when I disaggregate?
Is there a plausible mechanism, or just a coincidence?

Check	Catches
Did I plot it?	Curves, clusters, outliers
What is the unit of analysis?	Ecological fallacy
What is the obvious confounder?	Wealth, almost always
Does it survive stratification?	Spurious associations
Does one point drive it?	Leverage
How many relationships did I test?	Forking paths
How big is it, in units?	Significant-but-trivial

Seven checks, none requiring new statistics. Running them on every bivariate finding before it leaves your desk is what separates analysis that survives scrutiny from analysis that is quietly withdrawn later.

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11

Section Eleven

Reporting, Practice & Tools

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Report Honestly

How to report a relationship

State the direction and strength in plain words
Give the effect size in real units, not just a p-value
Report the uncertainty — confidence interval or range
Name the sample size and how it was drawn
Flag the obvious confounders you could not rule out

Report	Example
The pattern in plain words	“Uptake was lower among SC/ST households”
The numbers, with units	“44% vs 61% — a gap of 17 points”
N, per group	“(n = 312 and 908)”
Uncertainty	“95% CI for the gap: 11 to 23 points”
The obvious confounder	“Not adjusted for wealth or block”

All five fit in two sentences, and a report that contains them cannot easily be misread. The version that says only “uptake was significantly lower (p < 0.01)” has withheld everything a reader needs to judge it.

Naming the confounder yourself is a strength, not a weakness. It is what a competent reviewer will ask first, and answering it in advance is the difference between careful and defensive.

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Causal Language

Mind your verbs

Safe phrasing

‘is associated with’
‘tends to go with’
‘is correlated with’

Reserve for evidence

‘causes’
‘leads to’
‘increases / reduces’

Causal verbs need a causal design — an experiment or a careful quasi-experiment. A bivariate correlation does not earn them.

Safe with observational data	Reserve for causal designs
Associated with, linked to	Causes, produces, leads to
Higher among, lower in	Raises, reduces, drives
Accompanied by, varies with	Impacts, results in
Predicts (statistically)	Improves, prevents

Verbs are where causal claims enter a report unnoticed. The analysis section says “associated”, the executive summary says “improves”, and the press release says “proves” — each step made by someone who did not run the analysis.

“Predicts” is the treacherous one. It has a precise statistical meaning and an everyday causal one, and a reader outside the field will hear the second. Where it matters, spell out “statistically predicts, without implying cause”.

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Common Sins

What to avoid in write-ups

Reporting a p-value with no effect size
Claiming causation from a correlation
Hiding the scatter behind a single coefficient
Testing many pairs and reporting only the one that worked
Using a state-level relationship to make individual claims

Sin	Correction
Reporting p without an effect size	Give the gap in units, with an interval
“No difference” from a non-significant result	“No evidence of a difference”
Reporting only the relationship that worked	Say how many were examined
Causal verbs on observational data	Use the left column of the previous slide
Percentages with no N	Always “62% (n = 187)”
A chart title that states a cause	Name the variables, not the mechanism

None of these is dishonesty; all of them are habits. They survive because they make findings sound stronger and nobody in the review chain is assigned to catch them — which is why a personal checklist is the practical remedy.

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A Worked Habit

A reliable bivariate workflow

01

CLASSIFY: what type is each variable?

→

02

PLOT: scatter, box plot or bars

→

03

DESCRIBE: direction, strength, shape

→

04

TEST: the matching method

→

05

INTERPRET: effect size + uncertainty

→

06

CAUTION: confounders, fallacies

Six steps, every time. The discipline is what turns a number into a defensible finding.

Step	Output
Write the question in one sentence	Determines everything after
Classify both variables	Chooses the method
Plot it	Shape, outliers, clusters
Describe it — means, medians, percentages, N	The finding
Test, if the question needs a test	Evidence against chance
Check the pitfall list	What could still be wrong
Write it with hedged verbs and the caveat	Something defensible

Steps three and four are the analysis; step five is a check on it. Reversing that order — running the test first and looking afterwards — is how most of the errors in Section 10 survive to publication.

Step one prevents the most waste. An analysis without a written question expands to fill the available time and produces a folder of tables nobody can act on.

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Tools

Software for bivariate analysis

Tool	Good for	Note
Excel / Google Sheets	Cross-tabs, scatter, CORREL, t-test	Start here; pivot tables for cross-tabs
R	Every test, publication graphics	Free; cor.test, t.test, aov, chisq.test, lm
Python (pandas, scipy, statsmodels)	Cleaning + tests + regression	Free, general-purpose
Jamovi / JASP	Point-and-click stats	Free, friendly for learners
Stata / SPSS	Survey data, weights	Common in research shops

Tools matter less than habits: classify, plot, then test. A clear scatter in a spreadsheet beats a misread coefficient in any package.

Tool	Good for
Spreadsheets	Cross-tabs, simple charts, sharing with non-analysts
R	Everything here, reproducibly; free
Python	Same, if the team already uses it
Stata / SPSS	Where your collaborators already work

Every method in this deck exists in all four. The reason to move beyond a spreadsheet is reproducibility — being able to rerun the whole analysis from raw data after a correction — not sophistication.

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Read	For
Wheelan, Naked Statistics	These methods, informally and well
Reinhart, Statistics Done Wrong	Sections 6, 7 and 10, in detail
Huff, How to Lie with Statistics	Old, short, and still the best on charts
Pearl & Mackenzie, The Book of Why	Why correlation is not causation, properly
Your survey’s methodology note	What your data can actually support

If you remember five things	Section
Classify both variables; the method follows	2
Plot it before you compute anything	4
Report the effect size, not just the p-value	6
Name the confounder yourself	10
Match the verb to the design	11