Descriptive Statistics

Imagine you're handed a spreadsheet with 10,000 exam scores, salaries, or blood-pressure readings. Your eyes glaze over — no human can "see" ten thousand numbers at once. Descriptive statistics is the art of shrinking that flood of data down to a handful of honest, meaningful numbers and pictures: a typical value, a sense of how spread out things are, and a shape you can grasp in a glance. It is the first thing any scientist, doctor, economist, or data analyst does with a dataset, and doing it badly is one of the easiest ways to mislead yourself and others.

The key word is honest. A single summary number always throws information away, and different summaries throw away different things. The skill this page teaches is not memorizing formulas — it's knowing which summary tells the truth for which data, and spotting when a number is quietly lying to you.

Learning Objectives

By the end of this page, you should be able to:

• Compute and interpret the three measures of center: mean, median, and mode.
• Compute and interpret measures of spread: range, variance, and standard deviation.
• Choose the appropriate measure of center and spread for a given dataset.
• Explain how outliers and skew distort the mean and inflate the standard deviation.
• Read and extract information from histograms and box plots, including identifying skew and outliers.
• Distinguish between a population and a sample, and know why their variance formulas differ.

Quick Answer

Descriptive statistics summarize a dataset with measures of center and measures of spread. The three centers are the mean (arithmetic average, $\bar{x} = \frac{1}{n}\sum x_i$), the median (middle value when sorted), and the mode (most frequent value). The main spreads are the range (max minus min), the variance (average squared distance from the mean), and its square root, the standard deviation, which is in the same units as the data. The mean and standard deviation work beautifully for roughly symmetric data but are dragged around by outliers and skew; in those cases the median and interquartile range are far more trustworthy. A histogram shows the full shape of the data, and a box plot compresses the five-number summary (min, Q1, median, Q3, max) into one picture that flags outliers instantly.

Where It Came From

The word statistics shares a root with state, and that is no accident. For most of history, "statistics" literally meant numerical facts a government needed to run a country: how many people it could tax, how many young men it could conscript, how much grain a province produced. Ancient Egypt, Babylon, China, and Rome all took censuses. But counting is not yet understanding — for centuries these were just ledgers of raw totals with no attempt to find pattern or meaning in them.

The decisive leap came in 1662, in plague-ravaged London. A haberdasher named John Graunt got hold of the Bills of Mortality — weekly parish records of who had died and, roughly, of what. Nobody had thought to treat these grim lists as data. Graunt did. In his short book Natural and Political Observations Made upon the Bills of Mortality, he tallied causes of death, noticed that some diseases held steady year to year while others (like plague) spiked, observed that more boys were born than girls, and built the first crude "life table" estimating how many people survived to each age. In doing so he discovered something profound: large numbers of individually unpredictable events produce stable, regular patterns. That single insight — that aggregates behave lawfully even when individuals don't — is the seed of all of statistics and actuarial science.

Over the next two centuries the toolkit grew. Belgian astronomer-turned-social-scientist Adolphe Quetelet (1830s–40s) popularized the "average man" and imported the bell curve from astronomy into human data. Francis Galton and Karl Pearson (late 1800s) formalized the standard deviation, correlation, and the modern histogram. The motivation never changed from Graunt's day: humans face more numbers than they can hold in their heads, and we need faithful ways to compress them without lying.

Measures of Center: Mean, Median, and Mode

A measure of center answers "what is a typical value?" — but "typical" has three legitimate meanings.

The mean is the arithmetic average. For data $x_1, x_2, \dots, x_n$:

$\bar{x} = \frac{1}{n}\sum_{i=1}^{n} x_i$

Think of it as the balance point: if the data values sat on a seesaw, the mean is where it balances. Every value pulls on it in proportion to its distance.

The median is the middle value once the data is sorted. If $n$ is odd it's the single middle number; if $n$ is even it's the average of the two middle numbers. Think of it as the 50th percentile — half the data lies below, half above. It cares only about order, not magnitude, which is exactly why a billionaire walking into a room barely moves the median income but wrecks the mean.

The mode is the most frequently occurring value. It's the only center that works on categorical data (the modal eye colour, the best-selling shirt size) and a dataset can have one mode, several, or none.

Worked Example: All Three Centers

Seven friends report their monthly streaming spend, in dollars:

$8,\ 10,\ 12,\ 12,\ 15,\ 20,\ 60$

Mean: sum $= 8+10+12+12+15+20+60 = 137$. So $\bar{x} = 137/7 \approx 19.57$ dollars.

Median: the data is already sorted; $n = 7$ is odd, so the middle (4th) value is $12$ dollars.

Mode: $12$ appears twice, everything else once, so the mode is $12$ dollars.

Notice the story these tell. The median and mode both say "a typical person spends about $12," which matches six of the seven friends. The mean says "$19.57" — a value higher than 6 of the 7 people actually spend. That one $60 subscriber (an outlier) dragged the balance point far to the right. This is the single most important lesson in the whole topic: when data is skewed or has outliers, the median describes the typical person better than the mean.

Measures of Spread: Range, Variance, and Standard Deviation

Center alone is dangerously incomplete. The old joke: a statistician who can't swim drowns crossing a river with an average depth of one metre. Two datasets can share the same mean yet be wildly different — $\{50, 50, 50\}$ and $\{0, 50, 100\}$ both average 50. Spread measures how much the data varies.

The range is simply $\text{max} - \text{min}$. Quick to compute, but it depends entirely on the two most extreme points, so a single freak value distorts it completely.

The variance measures the typical squared distance from the mean. For a population of size $N$ with mean $\mu$:

$\sigma^2 = \frac{1}{N}\sum_{i=1}^{N}(x_i - \mu)^2$

For a sample meant to estimate a larger population, we divide by $n-1$ instead of $n$ (this is Bessel's correction, which corrects a bias that would otherwise make samples underestimate the true spread):

$s^2 = \frac{1}{n-1}\sum_{i=1}^{n}(x_i - \bar{x})^2$

Why square the distances? Because raw deviations $(x_i - \bar{x})$ always sum to exactly zero — the positives and negatives cancel. Squaring makes everything positive and, as a bonus, penalizes big misses more heavily.

The problem with variance is its units: if the data is in dollars, the variance is in "dollars squared," which is meaningless to interpret. The fix is the standard deviation — just the square root of the variance:

$\sigma = \sqrt{\sigma^2}, \qquad s = \sqrt{s^2}$

This lands us back in the original units, and it's the workhorse spread measure of all of statistics. Roughly, it's "the typical amount a data point differs from the mean."

Worked Example: Variance and Standard Deviation

Five daily temperatures (°C): $18,\ 20,\ 21,\ 23,\ 18$. Treat these as the whole population.

Step 1 — mean: $\mu = (18+20+21+23+18)/5 = 100/5 = 20$.

Step 2 — deviations and their squares:

$x_i$	$x_i - \mu$	$(x_i - \mu)^2$
18	$-2$	4
20	0	0
21	1	1
23	3	9
18	$-2$	4

Step 3 — variance: sum of squares $= 4+0+1+9+4 = 18$. Population variance $\sigma^2 = 18/5 = 3.6$.

Step 4 — standard deviation: $\sigma = \sqrt{3.6} \approx 1.90$ °C.

So on a typical day the temperature sits about 1.9 °C away from the 20 °C average. If instead we had treated these five days as a sample of a longer season, we would divide by $n - 1 = 4$: $s^2 = 18/4 = 4.5$, giving $s \approx 2.12$ °C. Same data, slightly larger number, because the sample formula deliberately hedges against underestimating the unknown true spread.

The Interquartile Range: A Robust Alternative

Just as the median resists outliers, so does the interquartile range (IQR). Sort the data, find the median (Q2), then the median of the lower half (Q1, the 25th percentile) and the median of the upper half (Q3, the 75th percentile). Then:

$\text{IQR} = Q3 - Q1$

The IQR is the spread of the middle 50% of the data and completely ignores the extreme tails, making it the spread partner to the median for skewed data.

Outliers, Skew, and How to Read the Pictures

An outlier is a value far from the rest of the data. A distribution is skewed when it has a long tail on one side. In a right-skewed (positive) distribution — incomes, house prices, hospital stays — a few very large values stretch the right tail, pulling the mean above the median. In a left-skewed distribution the reverse happens. A handy rule of thumb:

• Symmetric: mean $\approx$ median
• Right-skewed: mean $>$ median
• Left-skewed: mean $<$ median

A histogram groups data into bins and draws a bar for the count in each bin, showing the distribution's whole shape: where it peaks, whether it's symmetric or skewed, whether it has one hump or several. It's the single most informative first plot you can make.

A box plot (box-and-whisker) draws the five-number summary: a box from Q1 to Q3 with a line at the median, and whiskers extending to the most extreme points within $1.5 \times \text{IQR}$ of the box. Points beyond the whiskers are drawn individually as candidate outliers. The standard fence rule flags a point as an outlier if it lies below $Q1 - 1.5\,\text{IQR}$ or above $Q3 + 1.5\,\text{IQR}$.

Worked Example: Five-Number Summary and Outlier Test

Data (already sorted), nine values: $3,\ 5,\ 6,\ 7,\ 8,\ 9,\ 10,\ 12,\ 40$.

Median (Q2): with $n = 9$, the middle is the 5th value $= 8$.

Q1: median of the lower four values $\{3,5,6,7\}$ is $(5+6)/2 = 5.5$.

Q3: median of the upper four values $\{9,10,12,40\}$ is $(10+12)/2 = 11$.

IQR $= 11 - 5.5 = 5.5$.

Upper fence $= Q3 + 1.5 \times \text{IQR} = 11 + 1.5(5.5) = 11 + 8.25 = 19.25$.

Since $40 > 19.25$, the value 40 is a statistical outlier and would appear as a lone point beyond the right whisker. Note how the box plot flagged it automatically without us eyeballing anything — that's its whole point.

Real-World Applications

• Medicine: Clinical reference ranges (e.g. "normal" cholesterol) are built from the distribution of healthy people, typically the central 95%. Doctors track a patient's value in standard deviations from the mean.
• Economics and public policy: Median household income is reported instead of the mean precisely because income is heavily right-skewed; the mean would be inflated by a handful of billionaires and misrepresent the typical family.
• Quality control: Factories monitor the standard deviation of a product's dimension. Six-sigma manufacturing literally aims to keep defects more than six standard deviations from the target.
• Education: Standardized tests report percentiles (medians and quartiles) so a score can be understood relative to everyone else's, and grading "on a curve" uses the mean and standard deviation.
• Weather and climate: "Average" temperatures are means, but climatologists watch the standard deviation too — a rising variance signals more extreme events even if the average barely moves.
• Sports analytics: Comparing players by median performance resists being fooled by one spectacular or disastrous game.

Common Mistakes

Mistake 1: Using the mean for skewed data. Reporting the mean salary at a startup where the founder earns $2M makes the "average employee" look rich. Why it's wrong: the mean is pulled toward extreme values. Correction: for skewed data or data with outliers, report the median (and the IQR for spread).

Mistake 2: Confusing variance with standard deviation. Students quote a "variance of 3.6 °C" as if it were a spread in degrees. Why it's wrong: variance is in squared units and isn't directly interpretable. Correction: take the square root to get the standard deviation, which is in the data's own units.

Mistake 3: Forgetting the $n-1$ for samples. Dividing a sample's squared deviations by $n$ instead of $n-1$. Why it's wrong: the sample mean is itself estimated from the data, which artificially shrinks the deviations, so dividing by $n$ systematically underestimates the true variance. Correction: use $n-1$ for a sample; use $n$ only when you genuinely have the entire population.

Mistake 4: Reporting a center with no spread. Saying "the average is 50" and stopping. Why it's wrong: without spread, $\{50,50,50\}$ and $\{0,50,100\}$ look identical though one is constant and the other wildly variable. Correction: always pair a center with a spread — mean with SD, or median with IQR.

Comparison and Connections

The two center–spread pairings are the heart of the choice you make on every dataset:

Situation	Best center	Best spread	Why
Symmetric, no outliers	Mean	Standard deviation	Uses all data efficiently; feeds later inference
Skewed or has outliers	Median	IQR	Resistant to extreme values
Categorical data	Mode	(counts / proportions)	Only mode is even definable

The mean, variance, and standard deviation are called non-resistant (or non-robust) because a single extreme value can move them arbitrarily far. The median, quartiles, and IQR are resistant. Descriptive statistics also sets the stage for Distributions, where the mean and standard deviation become the two parameters $\mu$ and $\sigma$ of the normal curve, and for Statistical Inference, where sample statistics are used to reason about unknown population values.

Practice Questions

Recall

State the formula for the population standard deviation and explain in one sentence what it measures. Answer: $\sigma = \sqrt{\frac{1}{N}\sum (x_i - \mu)^2}$. It measures the typical distance of a data point from the mean, in the original units of the data.

Understanding

A dataset of house prices has a mean of $420,000 and a median of $310,000. Is it skewed, and which way? Answer: Because the mean is well above the median, the data is right-skewed (positively skewed) — a few very expensive houses stretch the upper tail and pull the mean up. The median ($310,000) better represents a typical home.

Application

Compute the sample variance and sample standard deviation of $4,\ 8,\ 6,\ 5,\ 3$. Answer: Mean $= (4+8+6+5+3)/5 = 26/5 = 5.2$. Squared deviations: $(4-5.2)^2=1.44$, $(8-5.2)^2=7.84$, $(6-5.2)^2=0.64$, $(5-5.2)^2=0.04$, $(3-5.2)^2=4.84$. Sum $= 14.8$. Sample variance $s^2 = 14.8/(5-1) = 3.7$. Sample SD $s = \sqrt{3.7} \approx 1.92$.

Analysis

For the data $2,\ 4,\ 4,\ 5,\ 6,\ 7,\ 30$, decide whether 30 is an outlier by the $1.5 \times \text{IQR}$ rule, and comment on which measure of center you'd report. Answer: Median (4th of 7) $= 5$. Lower half $\{2,4,4\}$ gives Q1 $= 4$; upper half $\{6,7,30\}$ gives Q3 $= 7$. IQR $= 3$. Upper fence $= 7 + 1.5(3) = 11.5$. Since $30 > 11.5$, 30 is an outlier. With an outlier present, report the median ($5$) rather than the mean (which is $\approx 8.3$, larger than six of the seven values).

FAQ

Q: When exactly should I use the mean versus the median? Use the mean when the data is roughly symmetric with no serious outliers — it uses every value and is the foundation for later statistical methods. Switch to the median whenever the data is skewed or contaminated by outliers, because it reports the true "middle" without being dragged around.

Q: Why do we square the deviations instead of just taking absolute values? Squaring guarantees positivity (so deviations don't cancel), penalizes large errors more, and has clean mathematical properties — the mean is the value that minimizes the sum of squared deviations, and variance adds nicely across independent quantities. There is a mean-absolute-deviation measure, but squared-based variance dominates because of these properties.

Q: What's the real difference between dividing by $n$ and $n-1$? Divide by $n$ when your data is the entire population. Divide by $n-1$ when it's a sample used to estimate a bigger population; the correction compensates for the fact that using the sample's own mean makes deviations look slightly too small.

Q: Can a dataset have more than one mode? Yes. If two values tie for most frequent it's bimodal; more than two, multimodal. A bimodal histogram often signals two distinct groups mixed together (e.g. heights of a mixed-gender crowd), which is itself a valuable finding.

Q: How many standard deviations from the mean is "unusual"? As a rough guide, for bell-shaped data about 68% of values lie within one SD of the mean, 95% within two, and 99.7% within three (the "empirical rule"). A value more than 2–3 SDs out is worth a second look. This is developed fully in the Distributions topic.

Quick Revision

• Mean $\bar{x} = \frac{1}{n}\sum x_i$ — the balance point; sensitive to outliers.
• Median — the middle sorted value (50th percentile); resistant to outliers.
• Mode — most frequent value; the only center for categorical data.
• Range $= \max - \min$; IQR $= Q3 - Q1$ (spread of middle 50%).
• Population variance $\sigma^2 = \frac{1}{N}\sum (x_i-\mu)^2$; sample variance $s^2 = \frac{1}{n-1}\sum (x_i-\bar{x})^2$.
• Standard deviation $= \sqrt{\text{variance}}$ — spread in the data's own units.
• Skew rule: right-skewed $\Rightarrow$ mean $>$ median; left-skewed $\Rightarrow$ mean $<$ median.
• Outlier fence: below $Q1 - 1.5\,\text{IQR}$ or above $Q3 + 1.5\,\text{IQR}$.
• Pairings: symmetric data $\to$ mean + SD; skewed data $\to$ median + IQR.
• Histogram shows shape; box plot shows the five-number summary and flags outliers.

Related Topics

Prerequisites

• Statistics and Probability overview
• Variables and Expressions — summation notation and basic algebra.

Related Topics

• Probability Basics
• Distributions

Next Topics

• Statistical Inference
• Conditional Probability and Bayes