Mean and standard deviation
The mean is a typical value. The standard deviation tells you how far points usually sit from that typical value.
For a list of numbers $x_1,\ldots,x_N$, the mean is the ordinary average. The standard deviation (SD) is a root-mean-square distance from the mean. Variance is SD squared.
$$\mu_x = \frac{1}{N}\sum_{i=1}^{N} x_i$$
$$\sigma_x = \sqrt{\frac{1}{N}\sum_{i=1}^{N}(x_i-\mu_x)^2}$$
Same mean, different spread
Let $x = \{{1,8,4,10,2\}}$ and $y = \{{7,5,4,3,6\}}$.
- Both have mean $\mu = 5$.
- But $\sigma_x \approx 3.46$ while $\sigma_y \approx 1.41$.
Same centre, very different dispersion. If these were model errors, $y$ would look more consistent.
Exam tip
Never report only a mean when two groups might differ in variability.
Key takeaways
- Mean = average (centre).
- SD = typical distance from the mean (spread).
- Equal means do not imply similar data.