## The Z-score

The Z-Score

The Z-score also referred to as standardized raw scores is a useful statistic because not only permits to compute the probability (chances or likelihood) of raw score (occurring within normal distribution) but also it helps to compare two raw scores from different normal distributions. The Z-score is a dimensionless measure since it is derived by subtracting the population mean from an individual raw score and then this difference is divided by the population standard deviation. This computational procedure is called standardizing raw score, which is often used in the Z-test of testing of hypothesis.

Any raw score can be converted to a Z-score by

$Z Score=\frac{raw score - mean}{\sigma}$

Example 1:

If the mean = 100 and standard deviation = 10, what would be the Z-score of the following raw score

 Raw Score Z-Score 90 $\frac{90-100}{10}=-1$ 110 $\frac{110-100}{10}=1$ 70 $\frac{70-100}{10}=-3$ 100 $\frac{100-100}{10}=0$

Note that:

• If Z-score have a zero value then it means that raw score is equal to the population mean.
• If Z-score has positive value then it means that raw score is above the population mean.
• If Z-score has negative value then it means that raw score is below the population mean.

Example 2:

Suppose you got 80 marks in Exam of a class and 70 marks in another exam of that class. You are interested in finding that in which exam you have performed better. Also suppose that the mean and standard deviation of exam1 are 90 and 10 and in exam2 60 and 5 respectively. Converting both exam marks (raw scores) into standard score (Z-Score), we get

$Z_1=\frac{80-90}{10} = -1$

The Z-score results ($Z_1=-1$) shows that 80 marks are one standard deviation below the class mean.

$Z_2=\frac{70-60}{5}=2$

The Z-score results ($Z_2=2$) shows that 70 marks are two standard deviation above the mean.

From Z1 and Z2 means that in second exam student performed well as compared to the first exam. Another way to interpret the Z-score of $-1$ is that about 34.13% of the students got marks below the class average. Similarly the Z-score of 2 implies that 47.42% of the students got marks above the class average.