The same mean can hide completely different levels of variation.
Suppose two classes both have an average score of 70. In one class, most scores may fall between 68 and 72; in the other, they may range from 40 to 100. The means are identical, but the learning gaps are not.
Spread describes how far values lie from the center. A measure of center tells you where the data sit; a measure of spread tells you how widely they are dispersed around that point.
Key question
Which number should summarize how far the observations are spread around the center?
What does spread describe?
| Measure | Formula or definition | Feature |
|---|---|---|
| Range | maximum − minimum | simple, but uses only the endpoints |
| IQR | Q3 − Q1 | width of the middle 50% |
| Variance | sum of squared deviations divided according to sample size | uses every observation |
| Standard deviation | square root of variance | same unit as the original data |
Is the range enough?
The data sets 2, 4, 6, 8, 10 and 2, 2, 6, 10, 10 both have a range of 8. Yet the first is evenly spread, while the second piles up at the endpoints. The range cannot show that difference.
Effect of an extreme value
If the maximum changes from 10 to 100, the range jumps from 8 to 98. The IQR may move much less because it focuses on the middle half of the data.
How are variance and standard deviation built?
First subtract the mean from each observation to get deviations. Raw deviations cancel because some are positive and others negative, so we square them and add them. Divide by n for a population or usually by n−1 for a sample to obtain variance, then take the square root to return to the original unit.
Sample variance and sample standard deviation
s² = Σ(xᵢ − x̄)² / (n−1), s = √s²
Why divide by n−1?
The sample mean is estimated from the same data, which uses one degree of freedom. Dividing by n−1 reduces the tendency to underestimate the population variance.
Which measure of spread should you use?
| Data situation | Recommended summary |
|---|---|
| Roughly symmetric with few extremes | Mean and standard deviation |
| Skewed or containing extremes | Median and IQR |
| Full width matters | Also report the range |
| Comparing different units | Consider a standardized measure such as the coefficient of variation |
Reading the standard deviation
A standard deviation of 10 points does not mean that every value is exactly 10 points from the mean. It summarizes the overall scale of dispersion around the mean.
Change the spread
Keep the mean fixed at 10 while moving the values farther from the center. Watch how the range, variance, and standard deviation increase even though the mean does not change.
Range
8.0
Sample variance
10.00
Standard deviation
3.16
What to notice
All measures of spread grow as observations move farther apart. Variance responds especially strongly because it squares the distances.