Changing every value, standardised scores and scaling

Last time

Before we start: last time, in Mean, median, mode and range, and comparing rates, you learned these points. Here they are again on one card. Keep them in mind, because today builds on them.

Changing every value

Sometimes every value in a data set is changed in the same way: a constant is added, or every value is multiplied. This lesson covers what that does to the mean and the standard deviation, how to scale marks to a chosen mean and standard deviation, and how scaling compares performances fairly.

Adding a constant

If the same number is added to every value, the mean increases by that number. The spread does not change: the standard deviation, the range and the interquartile range stay the same. If 5 marks are added to every score, a mean of 20 becomes 25, and a standard deviation of 3 stays 3.

Multiplying by a constant

If every value is multiplied by k, the mean is multiplied by k. The standard deviation is also multiplied by k, for positive k, and the variance by k squared. For y equals a plus b x: the new mean is a plus b times the old mean, and the new standard deviation is b times the old one.

The effects, side by side

What each change does to the mean and the standard deviation.

Standardised scores

A standardised score, z, is the value minus the mean, divided by the standard deviation. It says how many standard deviations a value lies above or below the mean. A mark of 52, when the mean is 40 and the standard deviation 8, has z equal to 12 over 8, which is 1.5.

Scaling to a new mean and SD

To scale to a new mean M and standard deviation S, keep each value's standardised score: the new value is M plus S times z. Scaling to a mean of 50 and a standard deviation of 10, the mark 52 becomes 50 plus 10 times 1.5, which is 65. Or find a and b in y equals a plus b x: b is S over s, 1.25, and a is M minus b times m, which here is 0.

Comparing by scaling

Standardised scores compare performances in different tests or events, where the means and spreads differ. A student scores 62 in Physics, where the mean is 50 and the standard deviation 6: z is 2. In History she scores 70, where the mean is 58 and the standard deviation 8: z is 1.5. So Physics is her better performance, despite the lower mark.

When scaling helps

Scaling makes marks from different tests comparable before they are combined, such as a written test and a practical test. It keeps every student's position in the order unchanged. For times, where lower is better, a negative standardised score is the good result.

Mistakes that lose marks

Mistake one: adding the constant to the standard deviation as well as to the mean. Mistake two: multiplying the variance by k instead of k squared. Mistake three: comparing raw marks from tests with different means and spreads. The fix: shifts leave the spread alone; compare standardised scores.

The full-marks tip

The full-marks tip: work out the standardised score of each value first, then scale or compare. Say clearly which performance is better, and why, using the z values.

On one card

So, on one card: transformations of data. Keep it for revision.