# Z Score Formula

Z-score is a numerical measurement used to describe a value's relationship to the mean of a group of values. Z-score is measured in terms of standard deviations of values from their mean. If a Z-score is 0, it means that the data point's score is identical to the mean score, while a Z-score of 1.0 would indicate a value that is one standard deviation from the mean. We will learn about the formula to find the z score with the help of a few solved examples ahead.

## What Is the Formula to Find Z Score?

Z-scores may be positive or negative, where a positive value indicates the score is above the mean and a negative score indicates that it is below the mean. The formula to calculate the z- score is given as:

\( \dfrac{x - \overline{x}}{\sigma} \)

where,

- x = Random standardised variable
- \(\overline{x}\) = Mean
- \(\sigma\) = Standard deviation

**Break down tough concepts through simple visuals.**

## Solved Examples Using Formula to Find Z Score

**Example 1:** Jake scored 70 marks out of 100 on a test. The mean marks scored by a class of 15 students were 60. Calculate the z- score for the marks secured by Jake using the Z score formula.

**Solution:**

To find: z- score for marks secured by Jake

Given:

Marks secured by Jake, x = 70

Total marks = 100

Total students(standard deviation), \(\sigma\) = 15

Mean marks, \(\overline{x}\) = 60

Using z - score formula,

z- score for secured marks = \(\dfrac{70 - 60}{15}\)

= 10/15

= 0.6667

**Answer: **z- score for Jake's marks = 0.6667

**Example 2:** A student appeared for two tests. He secured 80 out of 100 in the first and 75 in the second. The mean and deviation for first were 70 and 15 respectively, while 64 and 12 for the second respectively. What conclusion can you make on comparing student's performance for both exams? Solve using the Z score formula.

**Solution:**

To find: z- score for student's marks for both tests.

Given:

**First test:**

Marks secured, \(x_1\) = 80

Total marks = 100

Standard deviation, \(\sigma_1\) = 15

Mean, \(\overline{x}_1\) = 15

**Second test:**

Marks secured, \(x_2\) = 75

Total marks = 100

Standard deviation, \(\sigma_2\) = 15

Mean, \(\overline{x}_2\) = 15

Using z- score formula,

Z- score for first test = \( \dfrac{80 - 70}{15} \)

= 0.6667

Z- score for second test = \( \dfrac{75 - 64}{12} \)

= 0.9167

Since the z- score is more for the second test, the student performed better in the second test.

**Answer: **The student performed better in the second test.