Measures of location and spreadEdexcel 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 mode is the most frequent value (or modal class for grouped data). The mean uses every value but is affected by outliers; the median is not affected by extreme values; the mode is the only measure for non-numerical data. For grouped data use class midpoints, so the mean is an estimate.

The range is the largest minus the smallest value. A percentile splits ordered data: the $p$th percentile has $p\%$ of values below it. The interquartile range is $Q_3-Q_1$, and an interpercentile range such as the 10th to 90th percentile covers the middle $80\%$. For grouped data, estimate a percentile by linear interpolation: find the class containing position $\frac{p}{100}n$ and use $L+\frac{\text{position}-F_{\text{below}}}{f}\times w.$ Ranges based on percentiles ignore extreme values; the range does not.

The standard deviation measures the typical distance of values from the mean. Define $S_{xx}=\sum x^2-\frac{(\sum x)^2}{n}.$ Then variance $=\frac{S_{xx}}{n}=\frac{\sum x^2}{n}-\bar x^2$ and standard deviation $=\sqrt{\frac{S_{xx}}{n}}$. The version with $\sqrt{\frac{S_{xx}}{n-1}}$ is also accepted. For frequency tables, use $\sum fx^2$ and $\sum f$ in place of $\sum x^2$ and $n$. Example: $\sum x=240$, $\sum x^2=6100$, $n=10$ gives $S_{xx}=340$ and SD $=\sqrt{34}=5.83$.

From a grouped frequency table, estimate the mean using midpoints: $\bar x\approx\frac{\sum fx}{\sum f}$, and the standard deviation from $\sum fx^2$. Both are estimates because the real values within classes are unknown. Example: classes with midpoints $5,15,25,40$ and frequencies $4,12,20,14$ give $\sum fx=1260$, $\bar x=25.2$; $\sum fx^2=37700$, variance $=754-635.04=118.96$, SD $=10.9$.

Coding simplifies data: $y=\frac{x-a}{b}$, so $x=by+a$. Then

• $\bar x=b\bar y+a$ (the mean is multiplied by $b$ and shifted by $a$),
• standard deviation of $x=b\times$ standard deviation of $y$ (shifting does not change spread),
• variance of $x=b^2\times$ variance of $y$. Example: $y=\frac{x-400}{10}$ with $\bar y=3$ and variance $7.25$ gives $\bar x=430$ and variance $725$.

Pearson's large data set gives weather data. Questions may use its terminology (daily mean temperature, rainfall) but need no knowledge of the actual data. To compare two data sets, make one comment on a measure of location (mean or median) and one on a measure of spread (SD, IQR), each in context: 'a higher mean means warmer on average' and 'a smaller standard deviation means more consistent temperatures'.