Measures of Dispersion for Grouped Data

When data is grouped into class intervals, you cannot use each value directly. This chapter shows how to find the mean, variance and standard deviation from a frequency table, and how to draw and read a histogram, frequency polygon and ogive (cumulative frequency curve).

Paper 2 gives a frequency table and asks for the mean and standard deviation, or provides data to plot an ogive and then read the median and interquartile range from it. A carefully built table and a smooth, accurate curve are where the marks are.

Grouped-data questions punish students who mix up the three x-values a class can supply. The midpoint, the average of a class's lower and upper limits, stands in for every value in the class and is the x you put into the mean and standard deviation calculations.

The class boundaries, found by closing the half-unit gaps between classes, set the width of each bar in a histogram and are what you use to identify and calculate the modal class. The upper boundary of each class is the x-value you plot the cumulative frequency against on an ogive, because everything up to that boundary has already been counted.

Match the right x to the right graph and the marks follow.

An ogive turns a frequency table into a smooth cumulative curve, and its main job in Paper 2 is to hand you the median and the quartiles by reading, not by formula. With N values in total, the median sits at the N/2 mark: find N/2 on the vertical cumulative-frequency axis, run a horizontal line across to the curve, then drop straight down to read the value on the horizontal axis.

The first quartile Q1 uses N/4 and the third quartile Q3 uses 3N/4, read the same way. Note that for grouped data from an ogive you use N/4, N/2 and 3N/4, not the (N + 1) positions used for a short ungrouped list.

The interquartile range is then simply Q3 − Q1.

A common final part asks you to compare two sets, two classes, two machines, two months, using the figures you have just calculated, and full marks need both numbers read together. The mean tells you the centre: which group is higher on average.

The standard deviation tells you the consistency: a smaller standard deviation means the values cluster tightly around the mean, so the group is more uniform and predictable, while a larger one means the values are more spread out. So a class with the same mean but a smaller standard deviation is the more consistent performer, and a machine with a larger standard deviation is the less reliable one.

Always state what each number says about the situation, not just which is bigger.

A frequency table with clean class midpoints, taken through to the mean and standard deviation.