Standard scores and z-scores

Standardising is used when normally distributed data is transformed into a new set of units to display the number of standard deviations each data value lies from the mean. This is particularly useful for making comparisons with data that have different means and/or standard deviations. This process is called standardising and the transformed data values are called standard, or z-scores.

In order to calculate the standard (z) scores, the following method is used:

standard score = (data value – mean) / standard deviation or
z = \dfrac{x-\bar{x}}{s}

It can be inferred from this that a positive z-value would mean the data value is above the mean, whereas a negative value would indicate the data value is below the mean.

Application

The marks of a student across two subjects was recorded. Further Math = 61, Maths Methods = 60. The mean and standard deviation for Further Maths was 70 and 15. The mean and standard deviation for Maths Methods was 61 and 8.

In which subject did the student perform better in?

Firstly, calculate the z-scores for both subjects

Further Math
z = \frac{61-70}{15} = -0.6

Maths Methods
z = \frac{60-61}{8} = -0.125

And then compare the z-scores

The z-score in Maths Methods was higher and hence demonstrated that the student did better on this in this subject.

See also: