Measures of locationEdexcel International A Level Maths: Revision notes

A measure of location is a single value that describes the centre of a data set. For $n$ values $x_1,\dots,x_n$: the mean is $\bar x=\frac{\sum x}{n}$; the median is the middle value when the data are in order (the $\frac{n+1}{2}$th value), or the mean of the two middle values when $n$ is even; the mode is the most frequent value. Example: 0, 1, 1, 2, 2, 2, 3, 3, 4, 6 has $\sum x=24$, so $\bar x=2.4$. The 5th and 6th values are both 2, so the median is 2. The mode is also 2. Data can be discrete (counts, such as goals) or continuous (measurements, such as time and mass).

When values are repeated, data are given in a frequency table. Then $\bar x=\frac{\sum fx}{\sum f}$, where $\sum f=n$. Do not divide by the number of rows. For the median, use cumulative frequency to locate the $\frac{n+1}{2}$th value. The mode is the value with the highest frequency, not the frequency itself. Example: $x=1,2,3$ with $f=4,5,1$ gives $\sum fx=4+10+3=17$ and $\sum f=10$, so $\bar x=1.7$. The mode is 2 and the median is 2 (the 5th and 6th values are both 2).

For grouped data the individual values are lost, so only estimates are possible. The modal class is the class with the highest frequency (when class widths differ, the one with the highest frequency density, frequency $\div$ class width). The mean is estimated using class midpoints: $\bar x\approx\frac{\sum fx}{\sum f}$ with $x$ the midpoint. The median is estimated by linear interpolation, assuming data are evenly spread within each class: find the class containing the $\frac n2$th value, then $\text{median}\approx L+\frac{\frac n2-F}{f}\times w$, where $L$ is the lower boundary, $F$ the cumulative frequency before the class, $f$ its frequency and $w$ its width. Example: classes 0 to under 10, 10 to under 20, 20 to under 30, 30 to under 50 with $f=4,10,16,10$. Midpoints 5, 15, 25, 40 give $\bar x\approx\frac{970}{40}=24.25$. The median is $20+\frac{20-14}{16}\times10=23.75$.

Coding simplifies arithmetic by transforming each value, for example $y=\frac{x-a}{b}$. The mean and median obey the same transformation: if $y=\frac{x-a}{b}$ then $\bar y=\frac{\bar x-a}{b}$, so $\bar x=b\bar y+a$. The mode and the median transform in the same way, because the coding is linear and (for $b>0$) preserves order. Example: $m$ coded as $y=\frac{m-500}{10}$ with $\sum y=36$ over 20 values gives $\bar y=1.8$ and $\bar m=10(1.8)+500=518$. To combine two samples, use totals: $\bar x=\frac{n_1\bar x_1+n_2\bar x_2}{n_1+n_2}$. Never average the two means directly unless the samples are equal in size.

Choose the measure that suits the data. The mean uses every value but is distorted by extreme values (outliers). The median is not affected by extreme values, so is better for skewed data such as salaries. The mode is the only measure that works for non-numerical data and is useful for the most popular choice, but there may be no mode or several. In a context question, name the measure, give a reason from the data, and state the conclusion in the context of the question. When asked to compare two data sets, comment on a measure of location (a typical value) and a measure of spread (how consistent the values are), for example: the mean is higher, but the standard deviation is also larger, so the results are less consistent.