What it is
A measure of centre is a number that shows where the values in a list lie, their middle or their typical value. A measure of spread is a number that shows how spread out the values are. Both are needed, because the centre says nothing about the spread: the lists 4, 5, 6 and 0, 5, 10 both have mean 5 and median 5, but their ranges are 2 and 10.
| Measure | Kind | What it gives | The list 2, 4, 4, 4, 5, 5, 7, 9 |
|---|---|---|---|
| Mean | centre | the sum divided by the count | 5 |
| Median | centre | the middle value in order of size | 4.5 |
| Mode | centre | the most frequent value | 4 |
| Range | spread | largest minus smallest value | 7 |
| Interquartile range | spread | Q₃ − Q₁, the width of the middle half | 2 |
| Standard deviation σ | spread | how far from the mean the values tend to lie | 2 |
The interquartile range is built on the quartiles and drawn as the box of a box plot. Percentiles divide more finely: Q₁, the median and Q₃ are the 25th, 50th and 75th percentiles. The variance is the standard deviation squared.
Why it matters
Which measure fits depends on the data and on what the measure is for.
| Data | Centre | Spread |
|---|---|---|
| Roughly symmetric values without outliers | mean | standard deviation |
| A skewed distribution or outliers, such as salaries, house prices and waiting times | median | interquartile range (box plot) |
| Categories, such as colours or clothing brands | mode | – |
| Values to be used further, in totals, z-scores or confidence intervals | mean | standard deviation |
The mean and the standard deviation belong together, because both are computed from the values themselves. The median and the interquartile range belong together, because both are computed from the order of the values.
An outlier. Replace 9 with 100 in the list 2, 4, 4, 4, 5, 5, 7, 9. The mean goes from 5 to about 16.4, the range from 7 to 98 and the standard deviation from 2 to about 31.6. The median 4.5, the mode 4 and the interquartile range 2 are unchanged. The measures built on order only notice that 100 is the largest; those built on the values notice how large it is.
A skewed distribution. Seven waiting times in a phone queue, in minutes: 3, 4, 4, 5, 6, 7, 42. The mean is 71/7 ≈ 10.1 and the median 5. The mean is larger than six of the seven waiting times, while the median shows a typical wait.
When the mean is right. Together the waiting times come to 71 minutes, which is 7 times the mean. The median cannot be multiplied back into a sum. If you need the total, or will calculate further with the values, you need the mean.
Other means. “Mean” usually refers to the arithmetic mean. When groups carry different weight, the weighted mean is used; for growth rates the geometric mean; and for speeds over equal distances the harmonic mean. The spread of quantities of different sizes is compared with the coefficient of variation, the standard deviation divided by the mean.
The statistics calculator takes a pasted list, works out the measures above and draws the box plot.
Common misconceptions
- “The mean is the typical value.” The mean need not be a value that occurs, or even a possible one: no family has 1.8 children, but an average can be 1.8. In a skewed distribution it can also lie far from most of the values, as with the waiting times above.
- “The median is better than the mean.” The median withstands outliers, but that also means it ignores how large the largest and smallest values are, and it cannot be turned back into a sum. For symmetric data without outliers, the mean carries more information.
- “The standard deviation is the average distance from the mean.” In the list 2, 4, 4, 4, 5, 5, 7, 9 the distances are 3, 1, 1, 1, 0, 0, 2 and 4, with mean 12/8 = 1.5, while the standard deviation is 2. Squaring gives large distances more weight, and the standard deviation is never smaller than the average distance.
- “The range is enough as a measure of spread.” It uses only the two most extreme values. A single outlier moved it from 7 to 98 above.
- “The same mean means similar groups.” The lists 5, 6, 6, 7 and 1, 3, 9, 11 both have mean 6, but their standard deviations are about 0.71 and 4.12.
Common questions
What is the difference between a measure of centre and a measure of spread?
A measure of centre gives where the values lie, a measure of spread how spread out they are. A centre without a
spread gives an incomplete picture, because groups with the same mean can have completely different spreads.
Mean or median – which should I choose?
The median when a few values lie far from the rest or the distribution is skewed. The mean when the values are
evenly spread, or when you need the total or will calculate further. Sort or plot the values first and see how
they lie.
Why does a spreadsheet give different quartiles from the textbook?
There are several definitions of a quartile, and spreadsheets often use a different one from the one usually
taught in school. The quartiles page shows the list above with
both methods.
Are there more measures of centre and spread?
Yes. The weighted, geometric and harmonic means are measures of centre, and the mean absolute deviation, the
variance and the coefficient of variation are measures of spread.