Dispersion from a frequency table
For grouped data, use each class midpoint $x$ and its frequency $f$. The mean is $\bar{x} = \dfrac{\sum fx}{\sum f}$. The variance and standard deviation then measure how spread out the values are around the mean.
Key formula
$\text{Variance } \sigma^2 = \dfrac{\sum fx^2}{\sum f} - \bar{x}^2$ and standard deviation $\sigma = \sqrt{\sigma^2}$. A larger $\sigma$ means the data are more spread out (less consistent).
Worked example
Worked example
Midpoints $5, 15, 25$ with frequencies $2, 5, 3$: $\sum f = 10$, $\sum fx = 10 + 75 + 75 = 160$, so $\bar{x} = 16$. $\sum fx^2 = 50 + 1125 + 1875 = 3050$, so $\sigma^2 = \dfrac{3050}{10} - 16^2 = 305 - 256 = 49$ and $\sigma = \sqrt{49} = 7$.
Comparing two data sets
When two sets have the same mean, the set with the smaller standard deviation is more consistent (its values cluster more tightly). This is how dispersion lets us compare, for example, the marks of two classes.
Remember
- Use midpoints for grouped data.
- Variance is in squared units; standard deviation is in the original units.
- Adding a constant to every value leaves the standard deviation unchanged.