**Z-Score Definition:** The Z-Score also referred to as standardized raw scores is a useful statistic because not only permits to computation of the probability (chances or likelihood) of the raw score (occurring within normal distribution) but also 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 formula by

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

### Z-Score Real Life Examples

**Example 1:** If the mean = 100 and standard deviation = 10, what would be the Z-score of the following raw score

Raw Score | Z Scores |
---|---|

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,

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

**Example 2: **Suppose you got 80 marks in an 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 exam-1 are 90 and 10 and in the exam-2 60 and 5 respectively. Converting both exam marks (raw scores) into the standard score, we get

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

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

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

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

From $Z_1$ and $Z_2$ means that in the second exam, students 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.