Four measures of spread
Dispersion tells how spread out data are. The range is largest $-$ smallest. The interquartile range is $Q_3-Q_1$, the spread of the middle half. The variance $\sigma^2$ measures average squared distance from the mean, and its square root is the standard deviation $\sigma$.
Key formula
Variance $\sigma^2=\dfrac{\sum x^2}{n}-\bar{x}^2$, standard deviation $\sigma=\sqrt{\dfrac{\sum x^2}{n}-\bar{x}^2}$, interquartile range $=Q_3-Q_1$.
Effect of changing data
If every value has a constant $k$ added, the mean rises by $k$ but the range, IQR, variance and standard deviation are unchanged (spread stays the same). If every value is multiplied by $k$, the range, IQR and standard deviation are multiplied by $|k|$, while the variance is multiplied by $k^2$.
Worked example
Data: $2, 4, 4, 6, 9$. Mean $\bar{x}=\dfrac{25}{5}=5$. $\sum x^2=4+16+16+36+81=153$. Variance $=\dfrac{153}{5}-5^2=30.6-25=5.6$. Standard deviation $=\sqrt{5.6}\approx 2.37$. Range $=9-2=7$. If each value $+3$: new mean $8$, variance still $5.6$. If each $\times 2$: standard deviation $=2\times 2.37=4.73$.
Remember
- Adding $k$: mean shifts, spread unchanged.
- Multiplying by $k$: SD and range $\times|k|$, variance $\times k^2$.
- Standard deviation is the square root of the variance.