The normal shape
Many measurements, such as the heights of adults or the masses of bags filled by a machine, follow the same pattern. Most values are close to the middle, fewer are further away, and the numbers fall away in the same way on both sides. Drawn as a histogram, the data has a single peak in the middle and the shape of a bell. This pattern is called a normal distribution.
A normal distribution is symmetric about its center: the left half is the mirror image of the right half. A box plot cannot show the bell shape itself, but it can show whether the data is symmetric, and a box plot that is clearly not symmetric rules the normal shape out.
A symmetric box plot
Take a box plot whose five numbers are 10, 20, 30, 40 and 50. The median, 30, sits in the center of the box. The two halves of the box match: 30 − 20 = 10 below the median and 40 − 30 = 10 above it. The two whiskers match as well: 20 − 10 = 10 on the left and 50 − 40 = 10 on the right.
When the two halves of the box are equal and the two whiskers are equal, the box plot is symmetric about the median. The box plot of a normal distribution is symmetric in this way.
The median, 30, is in the center of the box: each half of the box is 10 long, and so is each whisker.
A lopsided box plot
Eleven parcels are delivered, and the times they take, in minutes, are 10, 12, 15, 17, 18, 20, 22, 28, 35, 45 and 60. The median is the 6th value, 20. The lower half, 10, 12, 15, 17 and 18, has 15 in its middle, and the upper half, 22, 28, 35, 45 and 60, has 35 in its middle. So the five numbers are 10, 15, 20, 35 and 60.
This box plot is not symmetric. The left half of the box is 20 − 15 = 5 minutes long and the right half is 35 − 20 = 15 minutes, three times as long. The left whisker is 15 − 10 = 5 minutes and the right whisker is 60 − 35 = 25 minutes. The median sits close to the lower quartile, and the right side is stretched out.
Most deliveries took between 10 and 22 minutes, and a few took much longer. Data like this, bunched at the low end with a long tail of high values, is skewed to the right, or positively skewed. In the mirror image, with the median near the upper quartile and a long left whisker, the data is skewed to the left, or negatively skewed. The side of the long tail names the skew.
The median, 20, sits near the left end of the box. The right half of the box, 15 minutes, is three times the left half, 5 minutes, and the right whisker is the longer one.
The same 11 delivery times as a histogram, each class including its lower end: the tallest bar is at the low end, and the bars trail off to the right.
Skewed means not normal
A normal distribution is symmetric, so its box plot is symmetric too. A clearly lopsided box plot therefore rules out a normal shape: the delivery times are not normally distributed.
The reverse does not hold. A symmetric box plot shows that the data may be normal; it does not show that it is. Symmetry is one property that a normal distribution has, and other shapes have it too. It is evidence for a normal shape, not proof of one.
Real data is never perfectly symmetric, even when it comes from a normal distribution, so small differences between the two halves are expected. What rules the normal shape out is a clear difference, such as one half of the box three times the other.
Symmetric is not the same as normal
Compare two symmetric box plots, both from 10 to 50 with median 30. The first has quartiles 25 and 35, and the second has quartiles 20 and 40.
In a normal distribution, most values are close to the middle. The middle half of the data is packed into a short stretch, so the box is narrow, and the quarters at each end are spread thinly over long whiskers. The first box plot has that shape: its box is 35 − 25 = 10 wide, and each whisker is 15 long.
In the second, each quarter of the data covers the same distance, 10. The values are spread evenly from 10 to 50, with no peak in the middle at all. This box plot is perfectly symmetric, yet the data is not normal. A symmetric box plot can never prove normality on its own.
Both are symmetric about 30. The first has a narrow box and long whiskers, as a normal distribution does. The second has four equal quarters, as evenly spread data does.
median − q1 = q3 − median at every spread, so symmetry is evidence and never proof
Keep both halves of the box equal and make the four bands the same width.
Under the box, each band holds a quarter of the data, so a narrow band is drawn tall and a wide band short. It opens on the narrow box, 25 to 35, with a peak in the middle. Drag the box out to 20 to 40: the four bands become the same height, a flat shape that is still symmetric. Move the median off 30 and the box turns lopsided.
What a box plot cannot show
Fourteen people are at a children’s party. The seven children are 5, 6, 6, 7, 7, 8 and 9 years old, and the seven adults are 33, 34, 34, 35, 36, 36 and 37. With 14 values, the median is halfway between the 7th and the 8th, years. The lower half, the seven children, has 7 in its middle, so the lower quartile is 7, and the upper half, the seven adults, has 35 in its middle, so the upper quartile is 35.
The box plot is exactly symmetric: 21 − 7 = 14 years below the median and 35 − 21 = 14 above it, and whiskers of 7 − 5 = 2 years and 37 − 35 = 2 years. Yet no one at the party is anywhere near 21. The data has two peaks, one for the children and one for the adults, and nothing in between.
A box plot shows only five numbers. It does not show where the values cluster between them, so data with two peaks can give a perfectly symmetric box plot. Before calling data normal, draw a histogram or a dot plot of it as well: if it has two peaks, it is not normal, however symmetric its box plot is.
The ages at the party give a symmetric box plot, with the median, 21, in the center of a wide box.
The histogram of the same ages has two peaks, 7 children and 7 adults, and no one aged from 10 to 30.