Statistics · A module in ten lessons

Basic statistics for economics students

Statistics is how economics checks itself against the data. This module teaches the ground floor, from what kind of number you are holding to variance, correlation and index numbers, using one invented district to run every example through. Every idea comes with a diagram you can move, and every lesson ends on real past-paper questions: 244 so far, from CUET, JNU, IIT JAM, Hyderabad and UGC NET.

Start at the beginning, or check where you are.All 10 lessons are ready; the full course is about 8 hours. 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, ten lessons

The running example is Ashanagar, an invented district whose sixty households have monthly incomes you can inspect, regroup and drag around. All of its figures are illustrative. The lessons say so each time, and none of it is presented as official data.

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Part 1Describing one variable
What kind of data you have, where it is centred, how far it spreads and which way it leans.
Part 2Describing two variables
How two variables move together, and the straight line that summarises the pattern.
1. 6Correlation: covariance, r and causationScatter plots, covariance as an average product, Pearson's r, rank correlation, and why correlation is not causation.50 min · 35 past questions
2. 7A first look at regression: the line of best fitLeast squares, the slope and intercept, the two regression lines and how they relate to r, R², and what a fitted line does and does not show.50 min · 35 past questions
Part 3Describing change over time
Index numbers: how prices, quantities and wages are compared across years.
1. 8Index numbers: Laspeyres, Paasche and FisherWhat inflation is, simple and weighted indices, Laspeyres against Paasche point by point, the two reversal tests, and deflating a wage.50 min · 21 past questions
Part 4Add-on: growth rates
Successive percentage changes, the compound annual growth rate and doubling time.
1. 9Add-on: growth rates, CAGR and the rule of 70Why growth multiplies, the average growth rate, continuous growth and logs, doubling time, and aggregate growth as a weighted average.45 min · 10 past questions
Part 5The hard end
The Delhi School's harder questions on all of the above.
1. 10The hard problemsThe Delhi School's harder questions on describing data, sorted by technique.40 min · 32 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 9

Which scale of measurement lets you say that one value is twice another?

Drill

Which average?

The commonest mistake in this whole topic is reaching for the arithmetic mean when another average is the right one. Six situations, five averages.

Which average? · 1 of 6

The typical household income in a district where a few families are very rich

Revise

Cheat sheet: every measure 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.

Measure	Formula	Use it for	The trap
Arithmetic mean	$x ˉ = \frac{\sum x}{n}$	Symmetric numerical data; the default	Pulled by extreme values. Never average two means without weights.
Median	$middle value$	Skewed data, ordinal data, open-ended classes	Even n: average the two middle values. Raw quartiles use (n + 1)/4, grouped use N/4.
Mode	$most frequent$	Nominal data; the commonest size	Can be absent or multiple. For continuous data group first.
Weighted mean	$\frac{\sum w x}{\sum w}$	Groups of different sizes; sector shares	Weights must reflect size, not just be present.
Geometric mean	$(x_{1} \dots x_{n})^{1/ n}$	Ratios, price relatives and growth factors	Zero if any value is zero. Always at most the arithmetic mean.
Harmonic mean	$\frac{n}{\sum 1/ x}$	Rates when the numerator is fixed (equal distance, equal spending)	Not defined with a zero. Always at most the geometric mean.
Compound growth rate (CAGR)	$(\frac{end}{start})^{1/ n} - 1$	The one steady rate that gives the same end value	Never above the arithmetic average of the yearly rates. A 20% fall needs a 25% rise.
Range, QD	$max - min \frac{Q _{3} - Q _{1}}{2}$	Quick spread; QD for outliers and open classes	The range uses two values only. Same range can hide very different variances.
Variance, SD	$σ^{2} = \frac{\sum x ^{2}}{n} - x ˉ^{2}$	The standard measure of spread	Sample: divide by n − 1. Add k: no change. Multiply by c: SD × \|c\|, variance × c².
Coefficient of variation	$\frac{σ}{x ˉ} \times 100$	Comparing spread across different units or levels	Unchanged by multiplying, changed by adding. Needs a ratio scale.
z-score	$z = \frac{x - x ˉ}{s}$	Comparing values from different distributions	Standardised data have mean 0 and SD 1.
Skewness	$S k = \frac{x ˉ - mode}{σ}$	Direction and size of the lean, free of units	Long right tail is positive, with mean > median > mode. Mode = 3 Median − 2 Mean is for moderate skew only.
Bowley's coefficient	$\frac{Q _{3} + Q _{1} - 2 M}{Q _{3} - Q _{1}}$	Open-ended classes and outliers	Always between −1 and +1. Ignores the tails.
Kurtosis	$β_{2} = \frac{m _{4}}{σ ^{4}}$	Tail weight; 3 is the normal curve	The excess is β₂ − 3. Skewness and kurtosis are separate properties.
Covariance	$\frac{\sum ( x - x ˉ ) ( y - y ˉ )}{n}$	Direction of joint movement; the building block of r	Unbounded, with units. Var(X + Y) = Var X + Var Y + 2 Cov.
Pearson's r	$r = \frac{Cov ( X , Y )}{σ _{X} σ _{Y}}$	Strength of a linear association	Between −1 and +1; unchanged by origin and positive scale; 0 is not independence; not causation.
Spearman's rank correlation	$1 - \frac{6 \sum d ^{2}}{n ( n ^{2} - 1 )}$	Ranks and ordinal data	Average the ranks of tied values.
Regression line	$b = \frac{Cov ( X , Y )}{σ _{X}^{2}}, a = y ˉ - b x ˉ$	Predicting y from x; the slope is the change in y per unit of x	b_yx b_xy = r². The two lines cross at the means. Shifting x or y leaves the slope alone.
R-squared	$R^{2} = 1 - \frac{SSE}{SST} = r^{2}$	The share of the variation in y that the line explains	Square r first: r = 0.8 explains 64%. Not comparable across different dependent variables.
Laspeyres and Paasche	$\frac{\sum p _{1} q _{0}}{\sum p _{0} q _{0}}, \frac{\sum p _{1} q _{1}}{\sum p _{0} q _{1}}$	Price indices with base-year and current-year quantities as weights	Laspeyres usually overstates, Paasche understates. Fisher is their geometric mean and passes both reversal tests.
Real value	$\frac{nominal}{price index} \times 100$	Turning rupees of different years into the same purchasing power	An index of 234.7 is a 134.7% rise. Contribution to inflation = weight × the group's own change.

Real questions

Practise across the module

JNU64

Hyderabad59

CUET-PG59

Delhi School35

IIT JAM17

UGC NET10

Questions placed in the lessons so far, by exam.

JNU sets the most: combined means, shifts and scales, the coefficient of variation, and the properties of the correlation coefficient. Hyderabad asks properties and definitions, and loves AM, GM and HM. CUET gives short computations and matching lists. The Delhi School’s harder questions are collected in Lesson 10.

Foundation

Definitions, one-step arithmetic and which-measure questions. Most CUET and Hyderabad questions sit here.

102 questionsPractise →

Standard

Grouped data, combined means, effects of shifting and scaling, and statement questions, up to JNU level.

110 questionsPractise →

Advanced

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

32 questionsLesson 10 →

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 the mean, median and mode?

The mean is the sum divided by the count, the balance point of the data. The median is the middle value once the data are sorted. The mode is the most frequent value. For symmetric data they are close together. For skewed data such as incomes the mean is pulled towards the long tail, so the median is the better description of a typical household, and the mode is the only average that works for categories such as religion or occupation.

Why does the sample variance divide by n − 1 and not n?

Deviations in a sample are measured from the sample's own mean, which is the point that makes them as small as possible, so they come out smaller on average than deviations from the true population mean. Dividing by n therefore underestimates the population variance by a factor of (n − 1)/n. Dividing by n − 1 removes the bias. Equivalently, the deviations from the sample mean add to zero, so only n − 1 of them are free.

When should I use the geometric mean instead of the arithmetic mean?

When the quantities multiply: growth rates, inflation over several years, investment returns. Convert each rate to a growth factor (1 plus the rate), take the geometric mean of the factors and subtract 1 to get the compound annual growth rate. The arithmetic average of growth rates always overstates the compound rate unless every year's rate is the same.

When is the harmonic mean the right average?

For rates such as speed or price per unit when the same amount of the numerator is used each time: equal distances at different speeds, or equal spending at different prices. A car that drives 40 km/h out and 60 km/h back over the same road averages 48 km/h for the round trip, which is the harmonic mean, not 50.

What is the coefficient of variation used for?

It divides the standard deviation by the mean, so it measures spread as a share of the average and has no units. It lets you compare the variability of two series measured in different units or at very different levels, for example the price of wheat and the price of rice. It only makes sense for data on a ratio scale, with a true zero.

What do nominal, ordinal, interval and ratio mean?

They are the four scales of measurement. Nominal data are labels with no order (religion, state). Ordinal data have an order but unmeasured gaps (grades, ranks). Interval data have meaningful differences but no true zero (temperature in Celsius, the calendar year). Ratio data have a true zero, so ratios are meaningful (income, price, a count). The scale decides which averages and measures of spread are allowed.

What is the difference between a population and a sample?

The population is every unit you want to know about, for example all households in a district. A sample is the part of it you actually observe. A number describing the population is a parameter (mean μ, standard deviation σ); a number computed from the sample is a statistic (x̄, s) and is your estimate of the parameter.

What is skewness, and how do I tell positive from negative?

Skewness describes which way a distribution leans. If the long tail runs to the right it is positively skewed and the mean is pulled above the median and mode, so mean > median > mode. If the long tail runs to the left it is negatively skewed and the order reverses. Incomes, house prices and firm sizes are typically positively skewed. Karl Pearson's coefficient, (mean − mode) ÷ standard deviation, puts a number on it.

Does a correlation of zero mean two variables are independent?

No. Pearson's correlation only measures straight-line association. If X takes the values −1, 0 and 1 with equal probability and Y = X², then Y is completely determined by X, yet the correlation between them is zero. Independent variables are always uncorrelated, but uncorrelated variables need not be independent. A correlation, however strong, also does not show that one variable causes the other.

What does R-squared tell you in a simple regression?

R² is the share of the total variation in the dependent variable that the fitted line explains: 1 minus the residual sum of squares over the total sum of squares. In a simple regression with an intercept it equals the square of the correlation coefficient, so a correlation of 0.8 gives an R² of 0.64. It describes how closely the points hug the line. It does not show that x causes y, and it is not comparable between models with different dependent variables.

Why does the Laspeyres index overstate inflation and the Paasche index understate it?

The Laspeyres index prices the base-year basket at current prices, as if households kept buying the same goods. In practice they buy less of what became dearer and more of what became relatively cheap, so pricing the old basket overstates the rise in the cost of living. The Paasche index prices the current basket, which has already shifted toward the cheaper goods, so it understates it. Fisher's ideal index, the geometric mean of the two, lies between them.

StatisticsStatistics 2: mean, median and modeThe mean as a balance point, the median and what it minimises, the mode, weighted and combined means, grouped data by step deviation, and quartiles read off an ogive.Complete guide · 55 min StatisticsStatistical inference for economics students, from joint distributions to hypothesis testsA module in thirteen lessons: joint distributions and covariance, sampling and the sampling distribution, the standard error and the central limit theorem, estimators and maximum likelihood, confidence intervals, hypothesis tests and p-values, two-sample tests, chi-square tests, analysis of variance and regression inference, on one running example with labs to move and graded past papers.Module · 635 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

Basic statistics for economics students

Measure

Formula

Use it for

The trap

Arithmetic mean

x ˉ = \frac{\sum x}{n}

Symmetric numerical data; the default

Pulled by extreme values. Never average two means without weights.

Median

middle value

Skewed data, ordinal data, open-ended classes

Even n: average the two middle values. Raw quartiles use (n + 1)/4, grouped use N/4.

Mode

most frequent

Nominal data; the commonest size

Can be absent or multiple. For continuous data group first.

Weighted mean

\frac{\sum w x}{\sum w}

Groups of different sizes; sector shares

Weights must reflect size, not just be present.

Geometric mean

(x_{1} \dots x_{n})^{1/ n}

Ratios, price relatives and growth factors

Zero if any value is zero. Always at most the arithmetic mean.

Harmonic mean

\frac{n}{\sum 1/ x}

Rates when the numerator is fixed (equal distance, equal spending)

Not defined with a zero. Always at most the geometric mean.

Compound growth rate (CAGR)

(\frac{end}{start})^{1/ n} - 1

The one steady rate that gives the same end value

Never above the arithmetic average of the yearly rates. A 20% fall needs a 25% rise.

Range, QD

max - min \frac{Q _{3} - Q _{1}}{2}

Quick spread; QD for outliers and open classes

The range uses two values only. Same range can hide very different variances.

Variance, SD

σ^{2} = \frac{\sum x ^{2}}{n} - x ˉ^{2}

The standard measure of spread

Sample: divide by n − 1. Add k: no change. Multiply by c: SD × |c|, variance × c².

Coefficient of variation

\frac{σ}{x ˉ} \times 100

Comparing spread across different units or levels

Unchanged by multiplying, changed by adding. Needs a ratio scale.

z-score

z = \frac{x - x ˉ}{s}

Comparing values from different distributions

Standardised data have mean 0 and SD 1.

Skewness

S k = \frac{x ˉ - mode}{σ}

Direction and size of the lean, free of units

Long right tail is positive, with mean > median > mode. Mode = 3 Median − 2 Mean is for moderate skew only.

Bowley's coefficient

\frac{Q _{3} + Q _{1} - 2 M}{Q _{3} - Q _{1}}

Open-ended classes and outliers

Always between −1 and +1. Ignores the tails.

Kurtosis

β_{2} = \frac{m _{4}}{σ ^{4}}

Tail weight; 3 is the normal curve

The excess is β₂ − 3. Skewness and kurtosis are separate properties.

Covariance

\frac{\sum ( x - x ˉ ) ( y - y ˉ )}{n}

Direction of joint movement; the building block of r

Unbounded, with units. Var(X + Y) = Var X + Var Y + 2 Cov.

Pearson's r

r = \frac{Cov ( X , Y )}{σ _{X} σ _{Y}}

Strength of a linear association

Between −1 and +1; unchanged by origin and positive scale; 0 is not independence; not causation.

Spearman's rank correlation

1 - \frac{6 \sum d ^{2}}{n ( n ^{2} - 1 )}

Ranks and ordinal data

Average the ranks of tied values.

Regression line

b = \frac{Cov ( X , Y )}{σ _{X}^{2}}, a = y ˉ - b x ˉ

Predicting y from x; the slope is the change in y per unit of x

b_yx b_xy = r². The two lines cross at the means. Shifting x or y leaves the slope alone.

R-squared

R^{2} = 1 - \frac{SSE}{SST} = r^{2}

The share of the variation in y that the line explains

Square r first: r = 0.8 explains 64%. Not comparable across different dependent variables.

Laspeyres and Paasche

\frac{\sum p _{1} q _{0}}{\sum p _{0} q _{0}}, \frac{\sum p _{1} q _{1}}{\sum p _{0} q _{1}}

Price indices with base-year and current-year quantities as weights

Laspeyres usually overstates, Paasche understates. Fisher is their geometric mean and passes both reversal tests.

Real value

\frac{nominal}{price index} \times 100

Turning rupees of different years into the same purchasing power

An index of 234.7 is a 134.7% rise. Contribution to inflation = weight × the group's own change.

Basic statistics for economics students

Five stages, ten lessons

A two-minute check

Which average?

Cheat sheet: every measure so far

Practise across the module

Foundation

Standard

Advanced

Frequently asked questions

Read next

Basic statistics for economics students

Five stages, ten lessons

A two-minute check

Which average?

Cheat sheet: every measure so far

Practise across the module

Foundation

Standard

Advanced

Frequently asked questions

Read next