Spread tells you how much values differ from each other. Two groups can share the same mean yet look completely different.
Why it matters
A mean without spread hides variability — for example a drug that helps some patients and harms others.
Key ideas
- Range = max − min (very sensitive to extremes)
- Variance = average squared distance from the mean (sample: divide by n − 1)
- SD = √variance, in the same unit as the data
- IQR = Q3 − Q1, the spread of the middle 50%
- SD pairs with the mean; IQR pairs with the median
Real-life research example
Two clinics both have a mean waiting time of 30 minutes. Clinic A has SD 4 min, Clinic B has SD 20 min. Patients at B face far less predictable waits.
When to use
- SD: roughly symmetric data
- IQR: skewed data or outliers
When NOT to use
- Don't rely on the range alone
- Don't mix the population formula (÷ n) with the sample formula (÷ n − 1)
Mini example
Data: 4, 6, 8, 10, 12 → mean = 8 Deviations: −4, −2, 0, 2, 4 → squares: 16, 4, 0, 4, 16 = 40 Variance = 40 ÷ 4 = 10, SD = √10 = 3.16
Quick takeaway
Never report a mean alone. Pair it with a measure of spread.
Try it yourself
Enter two different sets of numbers with the same mean (e.g. 8, 8, 8, 8, 8 and 2, 5, 8, 11, 14) one after the other. Compare their SD and IQR.
Model answer
The means match, but the second set has a much larger SD and IQR. Spread reveals differences that the mean hides.