In plain English
Statistics is the craft of saying something honest about a lot of numbers using only a few. Two questions cover most of it: where is the middle, and how spread out is it? The middle is the average (or the median if a few extreme values would drag the average around). The spread is the standard deviation: roughly how far a typical observation sits from the middle.
The advanced view
A dataset is a realisation of an underlying data-generating process. Descriptive statistics are estimators of that process's moments: the sample mean estimates the first moment, the sample variance the second (with n − 1 in the denominator so the estimator is unbiased), skewness the third and kurtosis the fourth. Every claim you make afterwards — a test, a regression, a risk number — inherits the sampling error of these estimators.
Start with the distinction that everything else rests on. The population is every unit you care about: all customers, all trading days, all invoices. The sample is the part you actually measured. Statistics exists because you almost never have the population, and it works only when the sample was drawn in a way that does not systematically favour some units over others. A biased sample of a million is worse than a random sample of a hundred, because size shrinks noise but never touches bias.
Know your data type before you compute anything. Categorical data (industry, region, yes/no) can be counted and shared as proportions, but has no meaningful average. Ordinal data (satisfaction 1–5, credit rating) is ordered but the gaps are not equal, so a median is safer than a mean. Numeric data (revenue, price, days) supports the full toolkit. Half the bad charts in a case interview are a mean computed on something that has no middle.
For the middle: the mean is the balance point and uses every observation, which is also why one outlier moves it. The median is the middle observation and ignores how extreme the extremes are. The mode is the most common value, useful for categories. If mean and median differ a lot, the distribution is skewed — and in business data it usually is, because revenue per customer, deal size and firm size all have a long right tail.
For the spread: the range is fragile, the interquartile range is robust, and the standard deviation is the one everything downstream uses. Variance is the average squared distance from the mean; standard deviation is its square root, which puts it back into the units of the data. The coefficient of variation (sd ÷ mean) makes spread comparable across things measured on different scales — the correct way to say whether a 2% move in a bond is bigger news than a 2% move in a small-cap stock.
The descriptive toolkit
Mean x̄ = Σx / n Variance s² = Σ(x − x̄)² / (n − 1) Standard deviation s = √s² Coefficient of variation = s / x̄ z-score z = (x − x̄) / s Weighted mean = Σ(wᵢ xᵢ) / Σwᵢ Empirical rule (normal-ish data): ≈68% within ±1s, ≈95% within ±2s, ≈99.7% within ±3s
Essential vocabulary
- Population vs. sample
- Everything you care about vs. the part you measured. Population parameters are μ and σ; sample estimates are x̄ and s.
- Sampling bias
- A selection rule that makes the sample systematically unrepresentative. Survivorship bias in fund returns is the classic finance case.
- Skewness
- Asymmetry. Positive skew means a long right tail — most business distributions, from deal size to customer value.
- Kurtosis
- Tail weight. Fat tails mean extreme observations are far more likely than a normal distribution admits. Returns are fat-tailed.
- Percentile / quartile
- The value below which a given share of observations falls. P50 is the median; Q1–Q3 span the middle half.
- Outlier
- An observation far from the rest. Investigate before deleting — in finance the outliers are often the whole story.
- Standardisation (z-score)
- Recentring on 0 and rescaling by the standard deviation, so different variables can be compared.
Common pitfalls
- ×Quoting a mean for a heavily skewed distribution. Average revenue per customer with three enterprise accounts in the sample describes nobody.
- ×Averaging averages. The mean of five country margins is not the group margin unless the countries are the same size — use a weighted mean.
- ×Treating a bigger sample as a fix for a biased one. More data only shrinks noise.
- ×Reporting spread without scale. A standard deviation of 4 means nothing until you know whether the mean is 5 or 5,000.
- ×Deleting outliers to make a chart tidy, then presenting the result as the typical case.
- ×Confusing precision with accuracy. Three decimal places on a badly sampled number is still the wrong number.
Worked example — five stores, one honest summary
Step 1 of 11
- 1Monthly profit (k): 20, 24, 26, 30, 150
In the interview
When an exhibit hands you five numbers, say the median and the shape before the average. "The typical store makes 26k; one store makes 150k and carries the region" is the sentence that shows you read the data rather than summed it.
Deeper
Deeper: standard deviation vs. standard error
Standard deviation describes the spread of the data. Standard error describes the uncertainty of a statistic computed from it: SE = σ ÷ √n. They are different questions — 'how variable are customers?' versus 'how sure am I about the average customer?' — and confusing them is the single most common statistical error in business analysis.
Because SE falls with √n, precision is expensive: halving the error needs four times the data. That is also why sub-segments in a survey are so unreliable — a 1,000-person survey cut into eight segments has 125 per cell and roughly three times the error of the headline.
Must know cold
- ✓Mean is pulled by outliers; median is not. Skewed data → quote the median.
- ✓SD describes data spread; SE = SD ÷ √n describes estimate precision.
- ✓z = (x − μ) ÷ σ puts any value on a common scale.
- ✓Roughly 68 / 95 / 99.7% of a normal sits within 1 / 2 / 3 SD.
Exercises
Try each one on paper before revealing the worked solution.
Exercise 1
Order values: 40, 45, 50, 55, 610. Compute mean and median, and say which you would quote.
Exercise 2
A survey of 400 customers gives mean spend 500 with SD 200. How precise is the mean?
About 68% of outcomes fall within one standard deviation and 95% within 1.96. The shaded tails are the 2.5% on each side that drive confidence intervals, VaR and significance tests. Financial returns have fatter tails than this, which is exactly why the picture is worth memorising as a baseline to argue against.