Standard score (z-score)

There is only one problem with this z-score; that is, it is based on a value of 0.8997 rather than the 0.9 value we are interested in. This is one of the difficulties of refer to the standard normal distribution table because it cannot give every possible z-score value (that we require a quite enormous table!). Therefore, you can either take the closest two values, 0.8997 and 0.9015, to your desired value, 0.9, which reflect the z-scores of 1.28 and 1.29, and then calculate the exact value of “z” for 0.9, or you can use a z-score calculator. If we use a z-score calculator, our value of 0.9 corresponds with a z-score of 1.282. In other words, P ( z > 1.282 ) = 0.1.



Now that we have the key information (that is, the mean score, μ, the standard deviation, σ , and z-score, z) we can answer our question directly, namely: What mark would a student have to achieve to be in the top 10% of the class and qualify for the advanced English Literature class? First, let us reiterate the facts:

Score	Mean	Standard Deviation	z-score
(X)	µ	σ	z
?	60	15	1.282

To find out the relevant score, we apply the following formula:



Therefore, students that scored above 79.23 marks out of 100 came in the top 10% of the English Literature class, qualifying for the advanced English Literature class as a result.

Setting the Scene: Part II

Clearly, the z-score statistic is helpful in highlighting how Sarah performed in her English Literature coursework and what mark a student would have to achieve to be in the top 10% of the class and qualify for the advanced English Literature class. However, we have only been talking about one distribution here, namely the distribution of scores amongst 50 students that completed a piece of English Literature coursework. What if Sarah wanted to compare how well she performed in her Maths coursework compared with her English Literature coursework?

In this case, Sarah achieved a higher mark in her Maths coursework, 72 out of 100. However, as we have already learnt, just because her Maths score (72) is higher than her English Literature score (70), we shouldn’t assume that she performed better in her Maths coursework compared to her English Literature coursework. The question therefore arises: How well did Sarah perform in her Maths coursework compared to her English Literature coursework?