Representation of data — common mistakes

When asked to compare two distributions, you must give two distinct points of comparison. One must be a measure of central tendency (e.g., 'The median time for Group A was higher than for Group B, suggesting they were slower on average'). The second must be a measure of spread (e.g., 'The IQR for Group A was smaller, suggesting their times were more consistent'). Simply stating the values is not enough; you must interpret what they mean in the context of the question.

A histogram is used for continuous data grouped into intervals, and the bars must touch. The area of each bar represents frequency. A bar chart is for discrete or categorical data, and the bars have gaps between them. The height of each bar represents frequency.

When data is grouped, we lose the original individual values. The midpoint of a class is our best estimate for the average value of all the data points within that class. By using it, we can calculate a weighted average that provides a good approximation of the true mean.

Use the median and IQR when the data is skewed or contains significant outliers. The median is a 'resistant' measure, meaning it's not heavily influenced by extreme values. Use the mean and standard deviation when the data is roughly symmetrical and free from major outliers, as they use all data points in their calculation and are often more powerful.

The cumulative frequency for a given class represents the total count of all data points up to and including that class. Therefore, this total is reached at the very end of the class interval, which is the upper class boundary. The graph starts at the lower boundary of the first class with a cumulative frequency of zero.