Skewness and outliersEdexcel International A Level Maths: Revision notes

Skewness describes the symmetry of a distribution. If the upper tail is longer the distribution has positive skew; if the lower tail is longer it has negative skew. A symmetrical distribution has no skew. In a histogram or box plot, positive skew has a longer upper whisker or tail. Using quartiles: if $Q_3-Q_2>Q_2-Q_1$ the skew is positive; if $Q_3-Q_2<Q_2-Q_1$ it is negative; if they are equal, the data are symmetrical. Example: $Q_1=24$, $Q_2=28$, $Q_3=40$ gives gaps of 4 and 12, so positive skew.

For a positively skewed distribution, mean $>$ median $>$ mode: extreme high values pull the mean up most. For a negatively skewed distribution, mean $<$ median $<$ mode. For a symmetrical distribution the three are equal (or very close). Example: mode 8, median 10 and mean 12 indicates positive skew. This also explains why the median is often preferred for skewed data such as incomes: it is not distorted by a few very large values.

An outlier is a value that is unusually far from the rest of the data. Outliers may be genuine extreme values or recording errors, so investigate before removing them. In an exam the rule for identifying outliers is always given in the question. Two common rules are:

• more than $1.5\times IQR$ above $Q_3$ or below $Q_1$, so outliers lie outside $Q_1-1.5(Q_3-Q_1)$ and $Q_3+1.5(Q_3-Q_1)$;
• more than $k$ standard deviations from the mean, so outliers lie outside $\bar x\pm k\sigma$. Example: $Q_1=2$, $Q_3=11$ give $IQR=9$ and an upper limit of $11+13.5=24.5$, so 40 is an outlier and 19 is not.

A box plot shows the minimum, $Q_1$, median, $Q_3$ and maximum. The box runs from $Q_1$ to $Q_3$ with a line at the median. When there are outliers, each whisker stops at the most extreme value that is not an outlier, and each outlier is marked separately, usually with a cross. If there is no outlier at an end, the whisker goes to the smallest or largest value. Example: $Q_1=2$, median 5, $Q_3=11$, smallest 0, largest 40, next largest 19, with 40 an outlier. The box is 2 to 11, the lower whisker goes to 0, the upper whisker goes to 19 and 40 is marked with a cross.

Outliers have a large effect on the mean and especially on the standard deviation, because deviations are squared. They have little effect on the median and the IQR. When an outlier is removed, update $n$, $\sum x$ and $\sum x^2$ and recalculate. Example: $n=20$, $\sum x=1240$, $\sum x^2=78\,880$ (mean 62, standard deviation 10). Removing the marks 38 and 85 gives $n=18$, $\sum x=1117$, $\sum x^2=70\,211$, so the mean is 62.06 and the standard deviation falls to 7.05. The mean hardly changed because the outliers lie on opposite sides, but the standard deviation fell a lot.