Measures of central tendency and spreadAQA A-Level Maths: Revision notes

The mean is $\bar x=\frac{\sum x}{n}$, or $\frac{\sum fx}{\sum f}$ from a frequency table. The median is the middle value of the ordered data (the mean of the middle two if $n$ is even). The mode is the most common value (or modal class for grouped data). The mean uses every value, so extreme values pull it; the median is not affected by extremes, so it suits skewed data. The mode is the only average for categorical data. Example: for 12, 15, 15, 18, 22, 26: mean $=18$, median $=16.5$, mode $=15$.

The range is largest minus smallest value; it uses only two values, so one extreme value can distort it. The interquartile range (IQR) is $Q_3-Q_1$, the spread of the middle half of the data, which is not affected by extremes. The standard deviation uses every value and measures the typical distance of the data from the mean. A larger standard deviation means more spread. It is in the same units as the data.

$\text{variance}=\frac{\sum(x-\bar x)^2}{n}=\frac{\sum x^2}{n}-\bar x^2,\qquad \text{standard deviation}=\sqrt{\text{variance}}.$ For a frequency table, use $\frac{\sum fx^2}{\sum f}-\bar x^2$. It is the mean of the squares minus the square of the mean. The second form is quicker when you are given $\sum x$ and $\sum x^2$. Example: $n=25$, $\sum x=400$, $\sum x^2=7000$. Mean $=16$, variance $=\frac{7000}{25}-16^2=24$, standard deviation $=\sqrt{24}=4.90$. This course uses divisor $n$ (calculators may label it $\sigma$ or $\sigma_n$); a version with divisor $n-1$ gives a slightly larger value, so check which one a question requires.

Given $n$, $\sum x$ and $\sum x^2$ you can find the mean and standard deviation without the raw data. You can also correct for a mistake: remove the wrong value from the sums and add the right one, then recalculate. If a value 40 should be 20, then $\sum x$ falls by 20 and $\sum x^2$ by $1600-400=1200$. To combine two sets, add the $n$, the $\sum x$ and the $\sum x^2$ values, then use the formulae on the totals. Never average two standard deviations. Use the statistics mode on your calculator to enter data or a frequency table, but know the formulae: you may need to work from $\sum x$ and $\sum x^2$.

To compare two data sets make two comments in context: one about the average (centre) and one about the spread. Example: machine A has mean 1004 g and standard deviation 3.2 g; machine B has mean 1001 g and standard deviation 1.1 g. B is closer to the 1000 g label and its bags are more consistent. Adding a value equal to the mean leaves the mean unchanged and decreases the standard deviation, because it adds no squared deviation while increasing $n$.