Statistics · A module in thirteen lessons

Statistical inference for economics students

Q: What is the difference between a parameter and a statistic?

A parameter is a number that describes a whole population, such as its mean μ or standard deviation σ. It is fixed and usually unknown. A statistic is a number computed from a sample, such as the sample mean x̄ or standard deviation s. It is known once the sample is drawn, but it changes from sample to sample, so it is a random variable with its own distribution.

Q: What is a sampling distribution?

It is the distribution of a statistic over all the samples that could be drawn: the different possible values of the statistic with their probabilities. For the sample mean it is centred on the population mean μ, and its standard deviation, the standard error, falls as the sample size grows. It is a distribution of statistics, not of individual values from the population or from one sample.

Q: What is the standard error, and how does it differ from the standard deviation?

The standard deviation describes how spread out individual values are. The standard error is the standard deviation of a statistic, such as the sample mean, over repeated samples. For the mean it equals σ divided by the square root of n. It measures how precisely the sample pins down the population value, and it shrinks as the sample grows, while the standard deviation of the data does not.

Q: Why does the standard error fall with the square root of n?

Because variances of independent draws add. The variance of a sum of n independent observations is n times σ², and dividing the sum by n to get the mean divides the variance by n squared, leaving σ² over n. Taking the square root gives σ over the square root of n. In practice, quadrupling the sample size halves the standard error.

Q: What does the central limit theorem say?

For independent observations from any population with a finite mean and variance, the distribution of the standardised sample mean, (x̄ − μ) divided by σ over the square root of n, approaches the standard normal distribution as n grows. The population does not have to be normal. This is why normal-based probabilities and intervals for a mean work even when the underlying data are skewed.

Q: Does the central limit theorem need a sample of at least 30?

No. The theorem describes a limit and sets no minimum sample size. Thirty is a rough rule of thumb for moderately skewed data. A symmetric population gives a close-to-normal sample mean with very few observations, while a heavily skewed or rare-event population can need hundreds before the normal curve is a good fit.

Q: How do I calculate the sample size for a survey?

To estimate a mean within a margin E at a given confidence, use n = (zσ/E)², where z is 1.96 for 95% confidence, and round up. To estimate a proportion, use n = z²p(1 − p)/E², taking p = 0.5 when you have no prior estimate, since that gives the largest, safest answer. For a margin of ±3 percentage points at 95% confidence this is 1,068 respondents, whatever the size of the population.

Q: What is the difference between stratified and cluster sampling?

In stratified sampling the population is split into groups that differ from one another, and you sample from every group. This removes the differences between groups from the error, so it gains precision. In cluster sampling you choose some whole groups at random and survey units within them. It saves travel and cost but usually loses precision, because units in a cluster tend to resemble each other.

Q: What is the finite population correction?

When a sample is drawn without replacement from a finite population, the variance of the sample mean is σ²/n multiplied by (N − n)/(N − 1). The factor is below 1, so the sample mean is more precise than with replacement, and it falls to zero when the sample is the whole population. It matters when the sample is more than about 5% of the population and is otherwise ignored.

Q: If the covariance of two random variables is zero, are they independent?

No. Independence means every cell of the joint table, or the whole joint density, factorises into the product of the margins. It forces the covariance to be zero, but the converse fails. If X takes the values −1, 0 and 1 with equal probability and Y = X², then Y is fixed by X, yet the covariance is exactly zero. Covariance only detects straight-line association.

Inference is the step from the households you measured to the ones you did not. This module teaches it in the order an economist meets it: joint distributions, sampling, the standard error and the central limit theorem, then estimation, confidence intervals and tests. The same invented district, Ashanagar, runs through every lab, and every lesson ends on real past-paper questions: 122 so far, from CUET, JNU, IIT JAM, Hyderabad, GATE, UGC NET and the Delhi School.

Start at the beginning, or check where you are.All 13 lessons are ready; the full course is about 11 hours. It assumes the basic statistics module: means, variances, covariance and the idea of a distribution. Each lesson opens with a real exam question, builds the idea with something you can move, and ends with that question solved and a graded set of past papers.

Start Lesson 1 →Two-minute check

The path

Five stages, thirteen lessons

The running example is Ashanagar, the invented district from the basic statistics module, whose sixty households have monthly incomes you can sample from again and again. Because the whole population is known, every lab can show the truth next to the estimate. All figures are illustrative.

0 of 13 doneYour progress is saved on this device.

Part 1From pairs to samples
Two variables at once, how a sample is drawn, and how far a sample mean strays from the truth.
Part 2Estimating
What makes an estimator good, how maximum likelihood finds one, and how to put an interval around it.
Part 3Testing
Hypotheses, the two errors, power and p-values, then the tests you are asked to run.
1. 7Hypothesis testing: errors, power and p-valuesNull and alternative, the two errors, size and power, one and two tails, and what a p-value is not.50 min · 22 past questions
2. 8The tests you will be askedz and t tests for means and proportions, two samples, paired data, F for variances, chi-square, and a one-way ANOVA.55 min · 8 past questions
Part 4Comparing and relating
Two groups, counts in a table, several means at once, and the line through a scatter.
Part 5The hard end
The Delhi School's harder questions on all of the above.
1. 13The hard problemsThe Delhi School's harder questions on inference, sorted by technique.45 min · 24 past questions

Where to start

A two-minute check

One question from each lesson that is ready. Your first wrong answer tells you where to begin; get them all and go straight to the cheat sheet.

Question 1 of 12

In a joint distribution, Cov(X, Y) = 0. Which statement is certainly true?

Drill

Which sampling design?

Examiners describe a survey in a sentence and ask you to name the design. Five scenarios, five designs. Look for what is chosen at random, and in what order.

Which sampling design? · 1 of 5

A district survey must travel to villages. The cost of reaching a village is large, so the team chooses 20 villages at random and interviews households inside them.

Revise

Cheat sheet: every result so far

The formula, when to use it, and the trap examiners set on it. Each row links to the lesson that teaches it. Rows are added as lessons are published.

Result	Formula	Use it for	The trap
Joint table rule	$\sum_{x} \sum_{y} p (x, y) = 1$	Finding an unknown constant k, a or b	Add every cell. For a product formula over a rectangle, the total is a product of totals.
Marginal	$P (X = x) = \sum_{y} p (x, y)$	One variable on its own	Margins hide the dependence. Two tables can share margins.
Conditional	$P (Y = y ∣ x) = \frac{p ( x , y )}{P ( X = x )}$	The distribution once X is known	Divide by the marginal of the variable you condition on. For densities, integrate over the right limits.
Covariance	$E [X Y] - E [X] E [Y]$	Direction of joint movement; variances of sums	Var(aX + bY) has 2ab Cov. Zero covariance is not independence.
Independence	$p (x, y) = P (X = x) P (Y = y)$	Checking every cell, or factorising a density	One zero cell with non-zero margins breaks it. A non-rectangular support breaks it.
Number of samples	$(n N), N^{n}$	Counting possible samples	C(N, n) without replacement and unordered. N to the n with replacement, ordered.
Sample mean	$E (X ˉ) = μ, Var (X ˉ) = \frac{σ ^{2}}{n}$	The centre and spread of the sampling distribution	Without replacement multiply the variance by (N − n)/(N − 1).
Standard error of a mean	$\frac{σ}{n}$	How far x̄ strays from μ	Quadruple n to halve it. Not the SD of the data.
Standard error of a proportion	$\frac{p ( 1 - p )}{n}$	Surveys, polls, coin tosses	Largest at p = 0.5. Estimate with p̂ when p is unknown.
Central limit theorem	$\frac{X ˉ - μ}{σ / n} \to N (0, 1)$	Probabilities for a mean from any population	Needs a finite variance, not normality. Sets no minimum n.
Sample size	$n = (\frac{z σ}{E})^{2}, \frac{z ^{2} p ( 1 - p )}{E ^{2}}$	Planning a survey	Round up. Use p = 0.5 when nothing is known. The population size does not enter.
Unbiased, MSE	$E (θ^) = θ, MSE = Var + Bias^{2}$	Judging an estimator at a fixed n	Unbiased is not the same as good. S² is unbiased for σ², S is biased for σ.
Consistent	$P (∣ θ^_{n} - θ ∣ < ε) \to 1$	Judging an estimator as n grows	Bias and variance both going to zero is enough. Neither implies the other property.
Best combination	$w_{1} = \frac{σ _{2}^{2}}{σ _{1}^{2} + σ _{2}^{2}}$	Two independent unbiased estimators	The more precise estimator gets the larger weight.
Maximum likelihood	$θ^= ar g max_{θ} \prod f (x_{i}; θ)$	Building an estimator: log, differentiate, set to zero	Normal variance MLE divides by n (biased). Uniform: the sample maximum, no derivative.
Exponential and Bernoulli MLE	$λ^= \frac{n}{\sum x}, p^= \frac{k}{n}$	The two commonest closed forms	λ̂ is the reciprocal of the mean. One Bernoulli draw gives 0 or 1.
Confidence interval for a mean	$x ˉ \pm t_{α /2, n - 1} \frac{s}{n}$	A range of plausible values for μ	95% describes the method, not this interval. Use t when σ is estimated.
Interval for a variance	$[\frac{( n - 1 ) S ^{2}}{χ _{α /2}^{2}}, \frac{( n - 1 ) S ^{2}}{χ _{1 - α /2}^{2}}]$	Normal samples	Lopsided. The larger point gives the lower limit.
Type I, Type II and power	$α = P (reject ∣ H_{0}), 1 - β = P (reject ∣ H_{1})$	Describing a test's two errors	Lower α alone lowers power. Size is computed under H₀, β under H₁.
p-value	$p = 1 - Φ (z) (upper), 2 (1 - Φ (∣ z ∣))$	How surprising the data are if H₀ is true	Not the probability that H₀ is true. Reject when p ≤ α.
Sample size for power	$n = (\frac{z _{α} + z _{power}}{d})^{2}$	Planning a test	Halve the effect, quadruple the sample.
t, z and proportion tests	$\frac{x ˉ - μ _{0}}{s / n}, \frac{p ^ - p _{0}}{p _{0} ( 1 - p _{0} ) / n}$	One mean, one proportion	t has n − 1 df. The test of a proportion uses p₀ in the standard error.
Standard error of a difference	$\frac{σ _{1}^{2}}{n _{1}} + \frac{σ _{2}^{2}}{n _{2}}$	Two independent means	The variances add. Do not add the standard errors, and do not subtract the variances.
Two-sample t	$\frac{x ˉ - y ˉ}{s _{p} 1/ n _{1} + 1/ n _{2}}$	Independent samples, equal variances	df = n₁ + n₂ − 2. With unequal spreads use Welch.
Paired t	$\frac{d ˉ}{s _{d} / n}$	The same units measured twice	n is the number of pairs, df = n − 1. Pairing helps when the two columns are positively correlated.
Two proportions	$\frac{p ^ _{1} - p ^ _{2}}{p ˉ ( 1 - p ˉ ) ( 1/ n _{1} + 1/ n _{2} )}$	Two polls, two rates	Pool p̄ for the test, do not pool for the interval.
F for two variances	$\frac{s _{1}^{2}}{s _{2}^{2}} \sim F_{n_{1} - 1, n_{2} - 1}$	Equal spreads in two normal samples	A ratio, so F, not chi-square. Sensitive to non-normality.
Chi-square test	$χ^{2} = \sum \frac{( O - E ) ^{2}}{E}, E = \frac{R C}{N}$	Goodness of fit and independence in a table	df = k − 1 (minus 1 per estimated parameter), or (r − 1)(c − 1). Expected counts of at least 5. The statistic grows with N, so look at Cramér's V too.
ANOVA mean squares	$F = \frac{MSB}{MSW}, E (MSW) = σ^{2}$	Several means in one test; then Bonferroni α/m for the pairs	A significant F does not say which mean differs. With two groups F = t².
Slope in simple regression	$SE (b) = \frac{s}{S _{xx}}, t = \frac{b}{SE ( b )}$	Is the slope different from zero? Interval for β	df = n − 2, not n − 1. In simple regression F = t². The interval for one new case is wider than for the average.
Chi-square and ANOVA	$χ^{2} = \sum \frac{( O - E ) ^{2}}{E}, F = \frac{SSB / ( k - 1 )}{SSW / ( N - k )}$	Association between attributes; several means	Chi-square df = (r − 1)(c − 1). ANOVA's F has (k − 1, N − k) df.

Real questions

Practise across the module

Delhi School31

JNU20

Hyderabad19

IIT JAM18

UGC NET16

CUET-PG11

GATE7

Questions placed in the lessons so far, by exam.

Inference questions are spread thinly across the exams, so this module is shorter on past papers than the basic statistics one, and the lessons say so rather than pad. CUET and Hyderabad ask definitions and classifications. JNU sets joint tables and conditional expectations. IIT JAM and GATE ask for densities, integrals and sample-size reasoning. The Delhi School’s harder questions are collected in Lesson 13.

Foundation

Definitions and one-step calculations. Most CUET and Hyderabad questions sit here.

43 questionsPractise →

Standard

Joint tables, standard errors and statement questions, up to JNU and IIT JAM level.

55 questionsPractise →

Advanced

The Delhi School of Economics' harder questions, collected in the last lesson.

24 questionsLesson 13 →

These are questions from the TestStop bank. A free account answers 10 a day, each with its worked solution; All-Access opens every set with no daily limit.

Quick answers

Frequently asked questions

What is the difference between a parameter and a statistic?

A parameter is a number that describes a whole population, such as its mean μ or standard deviation σ. It is fixed and usually unknown. A statistic is a number computed from a sample, such as the sample mean x̄ or standard deviation s. It is known once the sample is drawn, but it changes from sample to sample, so it is a random variable with its own distribution.

What is a sampling distribution?

It is the distribution of a statistic over all the samples that could be drawn: the different possible values of the statistic with their probabilities. For the sample mean it is centred on the population mean μ, and its standard deviation, the standard error, falls as the sample size grows. It is a distribution of statistics, not of individual values from the population or from one sample.

What is the standard error, and how does it differ from the standard deviation?

The standard deviation describes how spread out individual values are. The standard error is the standard deviation of a statistic, such as the sample mean, over repeated samples. For the mean it equals σ divided by the square root of n. It measures how precisely the sample pins down the population value, and it shrinks as the sample grows, while the standard deviation of the data does not.

Why does the standard error fall with the square root of n?

Because variances of independent draws add. The variance of a sum of n independent observations is n times σ², and dividing the sum by n to get the mean divides the variance by n squared, leaving σ² over n. Taking the square root gives σ over the square root of n. In practice, quadrupling the sample size halves the standard error.

What does the central limit theorem say?

For independent observations from any population with a finite mean and variance, the distribution of the standardised sample mean, (x̄ − μ) divided by σ over the square root of n, approaches the standard normal distribution as n grows. The population does not have to be normal. This is why normal-based probabilities and intervals for a mean work even when the underlying data are skewed.

Does the central limit theorem need a sample of at least 30?

No. The theorem describes a limit and sets no minimum sample size. Thirty is a rough rule of thumb for moderately skewed data. A symmetric population gives a close-to-normal sample mean with very few observations, while a heavily skewed or rare-event population can need hundreds before the normal curve is a good fit.

How do I calculate the sample size for a survey?

To estimate a mean within a margin E at a given confidence, use n = (zσ/E)², where z is 1.96 for 95% confidence, and round up. To estimate a proportion, use n = z²p(1 − p)/E², taking p = 0.5 when you have no prior estimate, since that gives the largest, safest answer. For a margin of ±3 percentage points at 95% confidence this is 1,068 respondents, whatever the size of the population.

What is the difference between stratified and cluster sampling?

In stratified sampling the population is split into groups that differ from one another, and you sample from every group. This removes the differences between groups from the error, so it gains precision. In cluster sampling you choose some whole groups at random and survey units within them. It saves travel and cost but usually loses precision, because units in a cluster tend to resemble each other.

What is the finite population correction?

When a sample is drawn without replacement from a finite population, the variance of the sample mean is σ²/n multiplied by (N − n)/(N − 1). The factor is below 1, so the sample mean is more precise than with replacement, and it falls to zero when the sample is the whole population. It matters when the sample is more than about 5% of the population and is otherwise ignored.

If the covariance of two random variables is zero, are they independent?

No. Independence means every cell of the joint table, or the whole joint density, factorises into the product of the margins. It forces the covariance to be zero, but the converse fails. If X takes the values −1, 0 and 1 with equal probability and Y = X², then Y is fixed by X, yet the covariance is exactly zero. Covariance only detects straight-line association.

What does a 95% confidence interval actually mean?

It describes the method, not the one interval you computed. If you repeated the sampling many times and built an interval each time in the same way, about 95% of those intervals would contain the true population value. The true value is a fixed number, so it is not correct to say there is a 95% probability that it lies in a particular interval: that interval either contains it or does not.

When do I use the t distribution instead of the normal for a confidence interval?

Use t when the population standard deviation is unknown and replaced by the sample standard deviation, which matters most in small samples from a roughly normal population. The degrees of freedom are n − 1. The t critical value is larger than the normal one, for example 2.571 against 1.96 at 95% with 5 degrees of freedom, which makes the interval wider to allow for the extra uncertainty in s. As n grows the two converge.

What makes a confidence interval wider or narrower?

Its width is twice the critical value times the standard error. A higher confidence level raises the critical value and widens the interval. A larger sample size cuts the standard error by the square root of n, so four times the sample halves the width. A more variable population widens it. The sample mean itself does not change the width.

What is the difference between an unbiased and a consistent estimator?

An unbiased estimator is right on average over repeated samples at a fixed sample size. A consistent estimator converges on the true value as the sample size grows. Neither implies the other: the first observation of a sample is unbiased but not consistent, since its variance never falls, while the sum of the observations divided by n plus 1 is biased in every sample but consistent. A sufficient condition for consistency is that both the bias and the variance go to zero.

Why is the sample standard deviation a biased estimator even though the sample variance is unbiased?

Unbiasedness is not preserved by a non-linear transformation. The square root is a concave function, so by Jensen's inequality the expected value of S is strictly less than the square root of the expected value of S squared, which is σ. S therefore underestimates σ on average, by about 20% when n is 2. It is still a consistent estimator, because it converges to σ as n grows.

What is maximum likelihood estimation?

It chooses the parameter value that makes the observed data most probable. Write the likelihood as the product of the density of each observation, take logs to turn it into a sum, differentiate with respect to the parameter and set the result to zero. For a Bernoulli sample this gives the sample proportion, for an exponential sample the reciprocal of the sample mean, and for a normal sample a variance that divides by n, which is biased but consistent.

What is the difference between a Type I and a Type II error?

A Type I error is rejecting a null hypothesis that is true, a false alarm, and its probability is the significance level α. A Type II error is failing to reject a null hypothesis that is false, a miss, with probability β. Power is 1 − β, the probability of rejecting a false null. For a fixed sample size, lowering α raises β, so cutting false alarms on its own makes the test less powerful.

What does a p-value mean?

The p-value is the probability, calculated assuming the null hypothesis is true, of getting a test statistic at least as extreme as the one observed. A small p-value means the data would be unusual under the null. You reject the null when the p-value is no larger than the significance level. It is not the probability that the null is true, and it does not measure the size of an effect.

How do I choose between a z test and a t test?

Use z when the population standard deviation is known, or for proportions and large samples where the normal approximation holds. Use t when you estimate the standard deviation from the sample, with n − 1 degrees of freedom for one mean and n₁ + n₂ − 2 for two pooled means. The t critical value is larger for small samples and converges to the normal one as the sample grows.

When is a paired t test used instead of a two-sample t test?

Use a paired test when the two measurements are linked unit by unit, such as the same households before and after a programme. Take the difference for each unit and run a one-sample t test that the mean difference is zero. A two-sample test assumes the groups are independent and, applied to paired data, throws away the pairing and most of the power.

What is the chi-square test of independence?

It tests whether two categorical variables are related, using a table of counts. For each cell it computes the expected count under independence, row total times column total over the grand total, then adds (observed − expected) squared over expected across cells. The statistic is compared with a chi-square distribution with (rows − 1) × (columns − 1) degrees of freedom. A large value means the table departs from independence.

What does one-way ANOVA test?

It tests whether the means of three or more groups are all equal. It splits the total variation into variation between group means and variation within groups, and the F statistic is the ratio of the two mean squares. A large F means the group means differ by more than ordinary scatter allows. It says at least one mean differs, not which one, and it avoids the inflated false-alarm rate of running many pairwise t tests.

StatisticsInference 1: joint distributions, marginals, covariance and independenceThe joint table and density, the normalising constant, marginal and conditional distributions, expectation of a function of two variables, covariance, why zero covariance is not independence, and the minimum-variance portfolio.Complete guide · 50 min StatisticsBasic statistics for economics students, from zero to exam levelA module in ten lessons: types of data, the three averages, the geometric and harmonic means, variance and standard deviation, then shape, correlation, regression and index numbers, with a short add-on on growth rates, on one running example, with graded past papers and the Delhi School’s hard problems.Module · 465 min StatisticsProbability, from zero to exam levelA module of lessons that starts at "what is a sample space" and ends at the DSE's hardest questions: counting, the rules, conditional probability, Bayes' theorem, random variables, the discrete and continuous distributions, and a final lesson of hard problems, each with its own interactives and graded past-paper sets.Module · 315 min

Statistical inference for economics students

Result

Formula

Use it for

The trap

Joint table rule

\sum_{x} \sum_{y} p (x, y) = 1

Finding an unknown constant k, a or b

Add every cell. For a product formula over a rectangle, the total is a product of totals.

Marginal

P (X = x) = \sum_{y} p (x, y)

One variable on its own

Margins hide the dependence. Two tables can share margins.

Conditional

P (Y = y ∣ x) = \frac{p ( x , y )}{P ( X = x )}

The distribution once X is known

Divide by the marginal of the variable you condition on. For densities, integrate over the right limits.

Covariance

E [X Y] - E [X] E [Y]

Direction of joint movement; variances of sums

Var(aX + bY) has 2ab Cov. Zero covariance is not independence.

Independence

p (x, y) = P (X = x) P (Y = y)

Checking every cell, or factorising a density

One zero cell with non-zero margins breaks it. A non-rectangular support breaks it.

Number of samples

(n N), N^{n}

Counting possible samples

C(N, n) without replacement and unordered. N to the n with replacement, ordered.

Sample mean

E (X ˉ) = μ, Var (X ˉ) = \frac{σ ^{2}}{n}

The centre and spread of the sampling distribution

Without replacement multiply the variance by (N − n)/(N − 1).

Standard error of a mean

\frac{σ}{n}

How far x̄ strays from μ

Quadruple n to halve it. Not the SD of the data.

Standard error of a proportion

\frac{p ( 1 - p )}{n}

Surveys, polls, coin tosses

Largest at p = 0.5. Estimate with p̂ when p is unknown.

Central limit theorem

\frac{X ˉ - μ}{σ / n} \to N (0, 1)

Probabilities for a mean from any population

Needs a finite variance, not normality. Sets no minimum n.

Sample size

n = (\frac{z σ}{E})^{2}, \frac{z ^{2} p ( 1 - p )}{E ^{2}}

Planning a survey

Round up. Use p = 0.5 when nothing is known. The population size does not enter.

Unbiased, MSE

E (θ^) = θ, MSE = Var + Bias^{2}

Judging an estimator at a fixed n

Unbiased is not the same as good. S² is unbiased for σ², S is biased for σ.

Consistent

P (∣ θ^_{n} - θ ∣ < ε) \to 1

Judging an estimator as n grows

Bias and variance both going to zero is enough. Neither implies the other property.

Best combination

w_{1} = \frac{σ _{2}^{2}}{σ _{1}^{2} + σ _{2}^{2}}

Two independent unbiased estimators

The more precise estimator gets the larger weight.

Maximum likelihood

θ^= ar g max_{θ} \prod f (x_{i}; θ)

Building an estimator: log, differentiate, set to zero

Normal variance MLE divides by n (biased). Uniform: the sample maximum, no derivative.

Exponential and Bernoulli MLE

λ^= \frac{n}{\sum x}, p^= \frac{k}{n}

The two commonest closed forms

λ̂ is the reciprocal of the mean. One Bernoulli draw gives 0 or 1.

Confidence interval for a mean

x ˉ \pm t_{α /2, n - 1} \frac{s}{n}

A range of plausible values for μ

95% describes the method, not this interval. Use t when σ is estimated.

Interval for a variance

[\frac{( n - 1 ) S ^{2}}{χ _{α /2}^{2}}, \frac{( n - 1 ) S ^{2}}{χ _{1 - α /2}^{2}}]

Normal samples

Lopsided. The larger point gives the lower limit.

Type I, Type II and power

α = P (reject ∣ H_{0}), 1 - β = P (reject ∣ H_{1})

Describing a test's two errors

Lower α alone lowers power. Size is computed under H₀, β under H₁.

p-value

p = 1 - Φ (z) (upper), 2 (1 - Φ (∣ z ∣))

How surprising the data are if H₀ is true

Not the probability that H₀ is true. Reject when p ≤ α.

Sample size for power

n = (\frac{z _{α} + z _{power}}{d})^{2}

Planning a test

Halve the effect, quadruple the sample.

t, z and proportion tests

\frac{x ˉ - μ _{0}}{s / n}, \frac{p ^ - p _{0}}{p _{0} ( 1 - p _{0} ) / n}

One mean, one proportion

t has n − 1 df. The test of a proportion uses p₀ in the standard error.

Standard error of a difference

\frac{σ _{1}^{2}}{n _{1}} + \frac{σ _{2}^{2}}{n _{2}}

Two independent means

The variances add. Do not add the standard errors, and do not subtract the variances.

Two-sample t

\frac{x ˉ - y ˉ}{s _{p} 1/ n _{1} + 1/ n _{2}}

Independent samples, equal variances

df = n₁ + n₂ − 2. With unequal spreads use Welch.

Paired t

\frac{d ˉ}{s _{d} / n}

The same units measured twice

n is the number of pairs, df = n − 1. Pairing helps when the two columns are positively correlated.

Two proportions

\frac{p ^ _{1} - p ^ _{2}}{p ˉ ( 1 - p ˉ ) ( 1/ n _{1} + 1/ n _{2} )}

Two polls, two rates

Pool p̄ for the test, do not pool for the interval.

F for two variances

\frac{s _{1}^{2}}{s _{2}^{2}} \sim F_{n_{1} - 1, n_{2} - 1}

Equal spreads in two normal samples

A ratio, so F, not chi-square. Sensitive to non-normality.

Chi-square test

χ^{2} = \sum \frac{( O - E ) ^{2}}{E}, E = \frac{R C}{N}

Goodness of fit and independence in a table

df = k − 1 (minus 1 per estimated parameter), or (r − 1)(c − 1). Expected counts of at least 5. The statistic grows with N, so look at Cramér's V too.

ANOVA mean squares

F = \frac{MSB}{MSW}, E (MSW) = σ^{2}

Several means in one test; then Bonferroni α/m for the pairs

A significant F does not say which mean differs. With two groups F = t².

Slope in simple regression

SE (b) = \frac{s}{S _{xx}}, t = \frac{b}{SE ( b )}

Is the slope different from zero? Interval for β

df = n − 2, not n − 1. In simple regression F = t². The interval for one new case is wider than for the average.

Chi-square and ANOVA

χ^{2} = \sum \frac{( O - E ) ^{2}}{E}, F = \frac{SSB / ( k - 1 )}{SSW / ( N - k )}

Association between attributes; several means

Chi-square df = (r − 1)(c − 1). ANOVA's F has (k − 1, N − k) df.

Statistical inference for economics students

Five stages, thirteen lessons

A two-minute check

Which sampling design?

Cheat sheet: every result so far

Practise across the module

Foundation

Standard

Advanced

Frequently asked questions

Read next

Statistical inference for economics students

Five stages, thirteen lessons

A two-minute check

Which sampling design?

Cheat sheet: every result so far

Practise across the module

Foundation

Standard

Advanced

Frequently asked questions

Read next