Why bother with descriptive stats?

Imagine you’re a doctor. Before you diagnose someone, you take their temperature, check their pulse, look at their weight. You don’t just jump straight to surgery. You describe what’s going on first.

Descriptive statistics is exactly that for economists. Before you run any regressions or complicated models, you summarise your data. You look at the shape of it. You spot the weird values. You understand what you’re working with.

And here’s the thing — sometimes the descriptive stats alone tell you everything you need to know. When researchers found that the average US household income was $105,000 but the typical household earns closer to $74,000, that gap alone revealed something huge about inequality. No regression required.

Measures of central tendency — where is the middle?

The Mean (average)

The mean is what most people call “the average.” It’s dead simple to calculate. But here’s its weakness: it’s easily dragged around by extreme values.

Classic example: imagine a room with 9 ordinary people earning £30,000 a year and one billionaire earning £1 billion. The average income in that room is over £100 million. Does that represent anyone in the room? Nope. The billionaire is pulling the whole thing upward.

The Median (the true middle)

The median ignores the extremes. It just finds the person in the middle of the line. In that same room? The median is £30,000 — which actually represents what most people earn.

This is why governments and economists often use the median household income rather than the mean. The mean flatters inequality.

The Geometric Mean (for growth rates)

You use this for growth rates over time. If your investment grew by 50% one year and dropped by 33% the next, the arithmetic average is +8.5%. But your actual return is 0% — because 1.5 × 0.67 = 1.0. The geometric mean correctly captures compound growth. Most financial returns and GDP growth figures should use this.

Measures of spread — how much variety is there?

Variance and Standard Deviation

The mean tells you the centre. But how spread out are the values around that centre? That’s what variance and standard deviation measure.

Think of two countries with average incomes of £30,000. In Country A, almost everyone earns between £28,000 and £32,000. In Country B, half the population earns £5,000 and the other half earns £55,000. The mean is the same, but Country B has massively more inequality. The standard deviation reveals this.

Why divide by (n−1) instead of n? Because with a sample (not the full population), dividing by n gives you a slightly too-small estimate. Using (n−1) corrects for this. It’s called Bessel’s correction. Just remember it — it won’t come up often, but examiners love to ask about it.

Worked example — step by step

Inflation rates over 5 years: 2%, 4%, 3%, 6%, 5%.

1. Mean: (2+4+3+6+5)/5 = 4%
2. Subtract mean from each: (−2, 0, −1, +2, +1)
3. Square each: (4, 0, 1, 4, 1)
4. Add them up: 4+0+1+4+1 = 10
5. Divide by (n−1) = 4: s² = 2.5
6. Square root: s = 1.58%

So on average, inflation was 4% per year, give or take about 1.58 percentage points.

Coefficient of Variation — comparing apples and oranges

If wages in the UK have a mean of £35,000 and SD of £10,000, and wages in Pakistan have a mean of £8,000 and SD of £5,000, which country has more wage inequality? You can’t just compare the SDs — the scales are different. The CV lets you compare: UK = 28.6%, Pakistan = 62.5%. Pakistan has proportionally more wage inequality.

Shape of the distribution — skewness and kurtosis

Skewness — is the distribution lopsided?

A perfectly symmetric distribution (like a bell curve) has skewness = 0. But economic data is rarely perfectly symmetric.

• Positive skew (right-skewed): Most values are low, but there’s a long tail of very high values dragging the mean up. Income, wealth, house prices — these are all right-skewed. A few millionaires pull the average way up.
• Negative skew (left-skewed): Most values are high, but there’s a tail of low values pulling the mean down. Think: scores on a very easy exam where most students score highly but a few score very badly.

Kurtosis — how fat are the tails?

Kurtosis measures whether extreme values (outliers) happen more or less often than you’d expect if the data were normally distributed.

• Normal distribution: Excess kurtosis = 0. Your baseline.
• Fat tails (excess kurtosis > 0): Extreme events happen more than expected. Stock markets are like this — “once in a century” crashes happen every decade or so. The 2008 financial crisis was labelled a “25-sigma event” under normal assumptions. That should happen once in the lifetime of the universe. It didn’t — because financial returns have fat tails, not normal ones.
• Thin tails (excess kurtosis < 0): Extreme values are very rare.

📋 Case Study: UK Household Income — Why the Average Misleads

✏️ Practice Questions

🎯 Summary

• Always explore your data with descriptive statistics before running any model.
• The mean is the average, but it’s sensitive to outliers. The median is more robust — use it for skewed data like income.
• Standard deviation tells you how spread out the data is from the mean.
• The coefficient of variation lets you compare spread across datasets with different scales.
• Positive skewness means a long right tail — income, wealth, and asset prices are classic examples.
• High kurtosis means fat tails — extreme values happen more often than normal — common in financial returns and a key reason the 2008 crisis was underestimated.

📚 References & Further Reading

1. Atkinson, A.B. (2015). Inequality: What Can Be Done? Harvard University Press. — Excellent accessible read on income distribution.
2. Piketty, T. & Saez, E. (2003). Income inequality in the United States. Quarterly Journal of Economics, 118(1). — The landmark study on rising inequality.
3. Taleb, N.N. (2007). The Black Swan. Random House. — Brilliant book on fat tails and why we underestimate rare events.
4. Anscombe, F.J. (1973). Graphs in statistical analysis. The American Statistician, 27(1). — Shows four datasets with identical means and SDs but completely different shapes.
5. ONS (2023). Household income inequality, UK. Office for National Statistics.
6. Wooldridge, J.M. (2019). Introductory Econometrics (7th ed.). Cengage.