Mean and standard deviation of grouped data, and combining groups

Last time

Before we start: last time, in Cumulative frequency, you learned these points. Here they are again on one card. Keep them in mind, because today builds on them.

How spread out?

The standard deviation measures how far values typically lie from the mean. This lesson covers the mean and standard deviation of grouped data, and how to combine the means and standard deviations of two groups.

Grouped data

For grouped data the exact values are unknown, so each class is represented by its mid-point, x. The mean is Σfx divided by Σf, and it is only an estimate. Grouping makes a large data set easier to handle and display, but the original values are lost.

Variance and standard deviation

The variance is Σfx² divided by Σf, minus the mean squared. The standard deviation is the square root of the variance. A larger standard deviation means the values are more spread out; a smaller one, more consistent.

A worked table

Columns for f times x and f times x squared, with their totals.

Working it out

The mean is 160 over 10, which is 16. The variance is 3,050 over 10, which is 305, minus 16 squared, 256: so 49. The standard deviation is the square root of 49, which is 7.

Combining means

The combined mean is the total of all the values divided by the total number: n one m one plus n two m two, over n one plus n two. 20 students with a mean of 60 and 30 with a mean of 70: 1,200 plus 2,100 is 3,300, and 3,300 over 50 is 66. The combined mean is not simply halfway between 60 and 70, because the groups differ in size.

Combining standard deviations

For each group, Σx² is n times the standard deviation squared plus the mean squared. With standard deviations 5 and 10: 20 times 3,625 is 72,500, and 30 times 5,000 is 150,000. Together, 222,500. The variance is 222,500 over 50, which is 4,450, minus 66 squared, 4,356: so 94. The standard deviation is about 9.70.

Comparing with mean and SD

A higher mean shows larger values on average. A smaller standard deviation shows more consistent values. Always say what this means in the context: for example, one machine fills bags more consistently than the other.

Mistakes that lose marks

Mistake one: using a class boundary or the class width instead of the mid-point as x. Mistake two: squaring Σfx instead of adding up f times x squared. Mistake three: averaging two means, or two standard deviations, without allowing for the group sizes. The fix: a table with fx and fx², and Σx² to combine.

The full-marks tip

The full-marks tip: set out columns for x, f, fx and fx², and show their totals. To combine groups, rebuild each group's Σx and Σx², then use the formulas on the totals.

On one card

So, on one card: mean and standard deviation. Keep it for revision.