Dispersion: Range, Variance and Standard Deviation
Two cafés both average 48 cold coffees a day. One sells between 45 and 51 every day; the other swings from 10 to 120. Same average, completely different businesses to run. The average tells you the centre. Dispersion tells you how far to trust it.
The week again, with its mean of 48: 40, 38, 36, 43, 46, 95, 38.
The rule
Spread is distance from the centre: square it, average it, root it.
- Range = max − min = 95 − 36 = 59. Quick, but it depends only on the two most extreme values.
- Variance is the average squared distance from the mean. Squaring makes every distance positive and weighs big gaps heavily.
- Standard deviation (SD) is the square root of the variance. It brings the answer back into the data's own units, so it reads as "the typical distance from the mean".
Working it out for the week:
| day | cups | minus 48 | squared |
|---|---|---|---|
| Mon | 40 | −8 | 64 |
| Tue | 38 | −10 | 100 |
| Wed | 36 | −12 | 144 |
| Thu | 43 | −5 | 25 |
| Fri | 46 | −2 | 4 |
| Sat | 95 | 47 | 2,209 |
| Sun | 38 | −10 | 100 |
| sum | 2,646 |
Now divide. Two versions exist:
- Sample variance divides by n − 1: 2,646 ÷ 6 = 441, so SD = √441 = 21 cups. Use this when your data is a sample standing in for a bigger picture, such as one week representing a typical week. This is
=VAR.Sand=STDEV.Sin Excel. - Population variance divides by n: 2,646 ÷ 7 = 378, SD ≈ 19.4. Use this only when you have every value you care about. This is
=VAR.Pand=STDEV.P.
Notice Saturday alone contributes 2,209 of the 2,646. Squaring makes the far-out values dominate.
The shortcut
Read a standard deviation next to its mean: "about mean ± SD" is where a typical value falls. For the week, 48 ± 21 spans 27 to 69. If that band feels too wide to be useful, something unusual is in the data.
Distances
Subtract the mean, 48, from each day: −8, −10, −12, −5, −2, 47, −10.
The trap: forgetting that variance is in squared units. "441 cups squared" is not something anyone can picture; always report the standard deviation.
Drills
Answer fast — trust the rule.
What is the range of the five weekdays, Monday to Friday (40, 38, 36, 43, 46)?
Why do we square the distances before averaging them?
The variance of the week's sales, treated as a sample, is 441. What is the standard deviation?
Café A has mean 48 and SD 3. Café B has mean 48 and SD 21. Which is true?
Which Excel function gives the sample standard deviation?
Remember
- Spread is distance from the centre: square it, average it, root it.
- Shortcut: read "mean ± SD" as the typical band; sample divides by n − 1.
- Trap: variance is in squared units; report the standard deviation.