After finding the mean, the next question is: How much do the scores vary around that mean?
Variation tells us whether data are tightly clustered or widely spread. Two common measures are the variance and the standard deviation.
Variance and standard deviation - formal level
Variance and standard deviation - intuitive level
Variance
Variance is the average squared distance of each score from the mean.
Symbolic formula:
$$s^2 = \frac{\sum (X - \bar{X})^2}{n - 1}$$
Formula in words:
$$\text{Variance} = \frac{\text{sum of squared deviations from the mean}}{\text{number of scores} - 1}$$
Where:
- $$s^2$$ = variance
- $$X$$ = each score
- $$\bar{X}$$ = mean
- $$n$$ = number of scores
Example: Data: 6, 8, 10
- Mean = 8
- Deviations: (6–8) = –2, (8–8) = 0, (10–8) = 2
- Squared deviations: 4, 0, 4
- Sum = 8
Variance = $$\tfrac{8}{3-1} = 4$$
Standard Deviation
The standard deviation is the square root of the variance.
Symbolic formula:
$$s = \sqrt{\frac{\sum (X - \bar{X})^2}{n - 1}}$$
Formula in words:
$$\text{Standard deviation} = \sqrt{\frac{\text{sum of squared deviations from the mean}}{\text{number of scores} - 1}}$$
Example continued:
Variance = 4 → Standard deviation = $$\sqrt{4} = 2$$
So, on average, scores are about 2 units away from the mean.
Definition
- Variance: average squared distance from the mean.
- Standard Deviation: square root of variance; typical distance from the mean.
Visuals
Figure 3.1 — Variability Around the Mean. A dot plot of scores with the mean marked, vertical lines showing deviations, and shaded boxes for squared deviations.
Why This Matters
Two sets of data can have the same mean but very different spreads.
Variance and standard deviation give us the language to describe that spread.
They are the foundation for most inferential tests in statistics.
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