Measures of location and spread, histograms, box plots and cumulative frequency, identifying outliers, scatter diagrams, correlation and the use of regression lines.

What this dot point is asking

AQA wants you to calculate and interpret measures of location and spread, draw and read histograms, box plots and cumulative frequency graphs, identify outliers using standard rules, describe correlation in scatter diagrams, and use a regression line to make and judge predictions. These skills are applied to the large data set on Paper 3, where interpretation in context earns as many marks as the calculation.

Location and spread

The median is the middle value of ordered data, and the mode is the most common value. The mean uses every data point but is sensitive to outliers, whereas the median and IQR are more robust. Comparing the mean with the median is a quick way to detect skew: mean above median suggests positive skew.

Displaying data

Histograms plot frequency density (frequency divided by class width) so that the area of each bar represents frequency. This is what makes them valid for unequal class widths, where bar height alone would mislead. Box plots display the minimum, lower quartile, median, upper quartile and maximum, and make comparisons of two distributions easy. Cumulative frequency graphs plot the running total against the upper class boundary, letting you read off the median and quartiles for grouped data.

Outliers

Correlation and regression

Correlation measures how closely points follow a straight line. Positive correlation means the variables rise together; negative means one falls as the other rises. The regression line of $y$ on $x$ is used to predict $y$ from $x$: its gradient gives the predicted change in $y$ per unit change in $x$. Predicting within the data range (interpolation) is reliable; predicting far outside it (extrapolation) is not, because the linear pattern may not continue.

Choosing and comparing summaries

A recurring exam skill is justifying which average or measure of spread to use. For skewed data or data with outliers, the median and interquartile range are preferred because they are not distorted by extreme values; for roughly symmetric data the mean and standard deviation use all the information and are usually reported. When comparing two data sets you should compare both a measure of location and a measure of spread, and you must do so in context: "the second class had a higher median mark and a smaller interquartile range, so on average they scored more and were more consistent."

Coding (linear transformation) sometimes simplifies calculation: if $y = \frac{x - a}{b}$, then $\bar{x} = a + b\bar{y}$ and the standard deviation of $x$ is $b$ times that of $y$. This lets you work with smaller numbers and scale back at the end, and AQA occasionally tests the effect of coding on the mean and standard deviation directly.