Statistics From Zero · Lesson 26 of 31
Histograms and the Shape of Data: Skew, Peaks and Tails
In one sentence: A histogram counts how many values fall into each range, and its shape (symmetric, skewed, two-humped) tells you which average to trust before you calculate anything.
The idea
A list of numbers is hard to read. A histogram turns it into a picture: chop the range of values into equal slices, called bins, count how many values land in each one, and draw a bar for each count. Tall bars show where most values live; short bars show where they're rare.
The point isn't the bars themselves. It's the shape they make. The shape tells you whether the average is a fair summary, whether there are oddballs, and whether you might be mixing two different groups. All before a single formula.
Uncle Gary times the pizza
The pizza place on the corner promises "delivery in about 30 minutes". Uncle Gary, who has a stopwatch and opinions, timed his last 20 orders. "Average: 30 minutes," he admits. "So they're honest."
Aunt Linda remembers it differently: usually it's quicker than that, and once, in the snow, it took over an hour.
Both are right. Here are Gary's 20 times, in minutes: 18, 20, 21, 22, 22, 23, 24, 24, 25, 25, 26, 27, 28, 29, 30, 32, 35, 41, 52, 75. Count them into 10-minute bins:
| Delivery time | Orders |
|---|---|
| 10 to 19 minutes | 1 |
| 20 to 29 minutes | 13 |
| 30 to 39 minutes | 3 |
| 40 to 49 minutes | 1 |
| 50 to 59 minutes | 1 |
| 60 to 69 minutes | 0 |
| 70 to 79 minutes | 1 |
Draw that as bars and you get a tall block on the left and a long, thin tail stretching to the right. Most deliveries take 20 to 29 minutes; 75% arrive within 30. But the snowy 75 minutes drags the mean up to 30 (29.95, to be exact), while the median (Lesson 2) sits at 25.5. Take the 75 out and the mean drops to 27.6, while the median hardly moves (25). Linda's "usually quicker" is the median; Gary's "30" is the mean, propped up by one bad night.
Histogram of 20 pizza delivery times (10-minute bins)
The four shapes worth knowing
- Symmetric: one hump in the middle, both sides alike, like heights or the bell curve (Lesson 5). Mean and median are close; either works.
- Right-skewed (positive skew): a hump on the left and a long tail to the right. Waiting times, incomes, house prices, Gary's pizza. The mean is pulled towards the tail, above the median.
- Left-skewed (negative skew): the mirror image, with a tail to the left. Scores on an easy exam: most people near the top, a few far below. The mean sits below the median.
- Two humps (bimodal): usually a sign that two different groups are mixed together, like lunchtime and evening customers, or two delivery drivers, one fast and one lost. Summarising it with one average describes nobody.
A handy rule: the tail points to where the mean has gone. If the mean is well above the median, expect a right tail; well below, a left tail.
Bin width changes the picture
The same 20 numbers look different with different bins. With 5-minute bins, the hump splits into 20 to 24 minutes (7 orders) and 25 to 29 (6), and the tail becomes a scatter of single bars with gaps between them. With 30-minute bins, almost everything falls into one bar and the tail disappears. Too few bins hide the shape; too many turn it into noise. There's no single right answer, so try a couple and be suspicious of anyone who shows you only one.
Where the shape matters
- Choosing the average. For skewed data, the median usually describes the typical case better (Lesson 2).
- Spotting oddballs. A bar far away from the rest is worth checking: a real extreme, or a typing error (Lesson 19).
- Before a test. Many tests, including the t-test (Lesson 14), assume roughly bell-shaped data or large samples. A histogram is the quickest check.
- Markets. Daily stock returns look roughly bell-shaped in the middle but have fatter tails than the bell curve allows: big moves happen far more often than it predicts.
Try it yourself
A company reports that the average salary of its 50 employees is $95,000. The median is $52,000.
1. What shape would a histogram of the salaries most likely have? 2. Which number better describes a typical employee, and why?
Show the answer
1. Right-skewed: most salaries bunched lower down, with a long tail of a few very high ones (probably the managers). The mean is far above the median, and the tail points to where the mean has gone.
2. The median, $52,000: half the employees earn less, half more. The mean is pulled up by a handful of large salaries and describes almost nobody. A histogram would show this at a glance.
The words your lecturer uses
| Lecturer says | It means |
|---|---|
| Histogram | Bars showing how many values fall into each equal-width range |
| Bin (class interval) | One of those ranges, such as 20 to 29 minutes |
| Frequency (relative frequency) | The count in a bin (or that count as a share of all values) |
| Distribution | The overall pattern of where the values fall |
| Positive (right) skew | A long tail to the right; mean above median |
| Negative (left) skew | A long tail to the left; mean below median |
| Unimodal, bimodal | One hump, two humps |
| Tails | The thin ends of the distribution, where the rare values live |
Three things to remember
- Look at the histogram before you average anything. The shape decides which summary is fair.
- The tail points to where the mean has gone: right skew pulls the mean above the median.
- Two humps usually mean two groups. Split them before you summarise.
Part of Statistics From Zero, a free beginner course by WrongStat. Not a substitute for your lecturer, but hopefully a good translator.