Measures of dispersionEdexcel International A Level Maths: Revision notes

A measure of dispersion (or spread) describes how widely data vary. The range is the largest value minus the smallest, which is simple but depends on two values only. The quartiles split ordered data into four equal parts. For $n$ values, $Q_1$ is the $\frac n4$th value, the median $Q_2$ is the $\frac n2$th and $Q_3$ is the $\frac{3n}{4}$th. If the position is not a whole number, round up to the next whole position; if it is a whole number, take the mean of that value and the next one. The interquartile range is $IQR=Q_3-Q_1$, the spread of the middle 50% of the data, so it is not affected by extreme values. Example: 2, 4, 4, 4, 5, 5, 7, 9 has $n=8$, so $Q_1=\frac{4+4}{2}=4$, $Q_3=\frac{5+7}{2}=6$ and $IQR=2$.

An interpercentile range is the difference between two percentiles, such as the 10th to 90th, $P_{90}-P_{10}$, which ignores the most extreme 10% at each end. For grouped continuous data, estimate any percentile by linear interpolation assuming even spread in each class: locate the class containing the required position (for example $\frac{n}{4}$ for $Q_1$), then value $\approx L+\frac{\text{position}-F}{f}\times w$. Example: 80 waiting times with cumulative frequencies 6, 30, 58, 74, 80 at 10, 20, 30, 40, 60 minutes. $Q_1$ is the 20th value, in the 10 to 20 class: $10+\frac{20-6}{24}\times10=15.83$. $Q_3$ is the 60th value, in the 30 to 40 class: $30+\frac{60-58}{16}\times10=31.25$. So $IQR\approx15.4$.

The variance measures the mean squared distance from the mean: $\sigma^2=\frac{\sum(x-\bar x)^2}{n}=\frac{\sum x^2}{n}-\bar x^2$, and the standard deviation is its square root, $\sigma=\sqrt{\text{variance}}$. For a frequency table or grouped data (using midpoints), $\sigma^2=\frac{\sum fx^2}{\sum f}-\bar x^2$. Use the second form, which is quicker. Example: $n=20$, $\sum x=140$, $\sum x^2=1160$: $\bar x=7$, variance $=58-49=9$, $\sigma=3$. The variance is never negative; if you get a negative value you have made an error. Standard deviation has the same units as the data, whereas variance has the units squared.

If $y=ax+b$, then $\bar y=a\bar x+b$ but the standard deviation becomes $\sigma_y=|a|\sigma_x$ and the variance becomes $a^2\sigma_x^2$. Adding or subtracting a constant does not change the spread. Example: $\sigma_x=2$ and $y=3x+2$ give $\sigma_y=6$. To add or remove a value, update $n$, $\sum x$ and $\sum x^2$ and recalculate. Adding a value equal to the mean leaves the mean unchanged but reduces the standard deviation. For example, adding 7 to the sample above gives $\frac{1209}{21}-49=8.57$, a smaller variance than 9.

Compare data sets using a measure of location and a measure of spread, in context. A smaller standard deviation (or IQR) means the data are more consistent, a larger one means they are more spread out. Use the median and IQR when data are skewed or contain extreme values, and the mean and standard deviation otherwise. Example: two machines have target 500 ml. A has mean 501.5 and $\sigma=1.89$; B has mean 500.6 and $\sigma=0.8$. B is better as its mean is closer to the target and it is more consistent. A value far from the mean increases the standard deviation a lot, because deviations are squared.