The standard score (more commonly referred to as a z-score) is a very useful statistic because it (a) allows us to calculate the probability of a score occurring within our normal distribution and (b) enables us to compare two scores that are from different normal distributions. The standard score does this by converting (in other words, standardizing) scores in a normal distribution to z-scores in what becomes a standard normal distribution. To explain what this means in simple terms, let's use an example (if needed, see our statistical guide, Normal Distribution Calculations, for background information on normal distribution calculations).
A tutor sets a piece of English Literature coursework for the 50 students in his class. We make the assumption that when the scores are presented on a histogram, the data is found to be normally distributed. The mean score is 60 out of 100 and the standard deviation (in other words, the variation in the scores) is 15 marks (see our statistical guides, Measures of Central Tendency and Standard Deviation, for more information about the mean and standard deviation).
Having looked at the performance of the tutor's class, one student, Sarah, has asked the tutor if, by scoring 70 out of 100, she has done well. Bearing in mind that the mean score was 60 out of 100 and that Sarah scored 70, then at first sight it may appear that since Sarah has scored 10 marks above the 'average' mark, she has achieved one of the best marks. However, this does not take into consideration the variation in scores amongst the 50 students (in other words, the standard deviation). After all, if the standard deviation is 15, then there is a reasonable amount of variation amongst the scores when compared with the mean.
Whilst Sarah has still scored much higher than the mean score, she has not necessarily achieved one of the best marks in her class. The question arises: How well did Sarah perform in her English Literature coursework compared to the other 50 students? Before answering this question, let us look at another problem.
The tutor has a dilemma. In the next academic year, he must choose which of his students have performed well enough to be entered into an advanced English Literature class. He decides to use the coursework scores as an indicator of the performance of his students. As such, he feels that only those students that are in the top 10% of the class should be entered into the advanced English Literature class. The question arises: Which students came in the top 10% of the class?
Therefore, we are left with two questions to answer. First, how well did Sarah perform in her English Literature coursework compared to the other 50 students? Second, which students came in the top 10% of the class?
Whilst it is possible to calculate the answer to both of these questions using the existing mean score and standard deviation, this is very complex. Therefore, statisticians have come up with probability distributions, which are ways of calculating the probability of a score occurring for a number of common distributions, such as the normal distribution. In our case, we make the assumption that the students' scores are normally distributed. As such, we can use something called the standard normal distribution and its related z-scores to answer these questions much more easily.
When a frequency distribution is normally distributed, we can find out the probability of a score occurring by standardising the scores, known as standard scores (or z scores). The standard normal distribution simply converts the group of data in our frequency distribution such that the mean is 0 and the standard deviation is 1 (see below).