Z Score Formula | Step by Step Calculation of Z-Score with Examples

Formula to Calculate Z-Score

Z Score Formula is used for the purpose of calculation of the Z score when the mean and the standard deviation is known and large and according to the formula mean of the population is subtracted from the Mean of Sample and the resultant is divided by the standard deviation of the population.

The term “z-score” of raw data refers to the score generated by measuring how many standard deviations above or below the population mean is the data, which helps in testing the hypothesis under consideration. In other words, a z-score is the distance of a data point from the population mean that is expressed as a multiple of the standard deviation.

The z-scores vary in the range of -3 times the standard deviation (far left of the normal distribution) to +3 times the standard deviation (far right of the normal distribution).
The z-scores have a mean of 0 and a standard deviation of 1.

The equation for z-score of a data point is calculated by subtracting the population mean from the data point (referred to as x) and then the result is divided by the population standard deviation. Mathematically, it is represented as,

Z Score = (x – μ) / ơ

where

x = Datapoint
μ = Mean
ơ = Standard deviation

Calculation of Z Score (Step by Step)

The Equation for z-score of a data point can be derived by using the following steps:

Step 1: Firstly, determine the mean of the data set based on the data points or observations which are denoted by x_i, while the total number of data points in the data set is denoted by N.

Step 2: Next, determine the standard deviation of the population on the basis of the population mean μ, data points x_i and the number of data points in the population N.

Step 3: Finally, the z-score is derived by subtracting the mean from the data point and then the result is divided by the standard deviation as shown below.

Examples

You can download this Z Score Formula Excel Template here – Z Score Formula Excel Template

Example #1

Let us take the example of a class of 50 students who have written the science test last week. Today is the result day and the class teacher told that John scored 93 in the test while the average score of the class was 68. Determine the z-score for John’s test mark if the standard deviation is 13.

Solution:

Given,

John’s test score, x = 93
Mean, μ= 68
Standard deviation, ơ = 13

Therefore, the z-score for John’s test score can be calculated using the above formula as,

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Z = (93 – 68) / 13

Z Score will be –

Z Score = 1.92

Therefore, John’s Ztest score is 1.92 standard deviation above the average score of the class, which means 97.26% of the class (49 students) scored less than John.

Example#2

Let us take another detailed example of 30 students (as z-test is not appropriate for less than 30 data points) who appeared for a class test. Determine the z-test score for the 4^th student of based on the marks scored by the students out of 100 – 55, 67, 84, 65, 59, 68, 77, 95, 88, 78, 53, 81, 73, 66, 65, 52, 54, 83, 86, 94, 85, 72, 62, 64, 74, 82, 58, 57, 51, 91.

Solution:

Given,

x = 65,
4^th student scored = 65,
Number of data points, N = 30.

Mean = (55 + 67 + 84 + 65 + 59 + 68 + 77 + 95 + 88 + 78 + 53 + 81 + 73 + 66 + 65 + 52 + 54 + 83 + 86 + 94 + 85 + 72 + 62 + 64 + 74 + 82 + 58 + 57 + 51 + 91) / 30

Mean = 71.30

Now, the standard deviation can be calculated by using the formula as shown below,

ơ = 13.44

Therefore, Z-score of the 4^th student can be calculated using the above formula as,

Z = (x – x ) / s

Z = (65 –30) / 13.44
Z = -0.47

Therefore, the 4^th student’s score is 0.47 standard deviation below the average score of the class, which means that 31.92% of the class (10 students) scored less than the 4^th student as per z- score table.

Z Score in Excel (with Excel Template)

Now, let us take the case mentioned in example 2 to illustrate the concept of z-score in the excel template below.

Below is given data for calculation of Z Score

You can refer to the given excel sheet below for the detailed calculation of Z Score Test Statistics.

Relevance and Uses

From the perspective of hypothesis testing, z-score is a very important concept to understand because it is employed to test whether or not a test statistic falls in the acceptable range of value. The z-score is also used to standardize data prior to analysis, calculate the probability of a score or comparison of two or more data points which are from different normal distributions. There are the diverse application of z-score across fields if applied properly.