Formula to Calculate Z-Score

  • The Z-scores vary from -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 the Z-score of a data point calculated by subtracting the population mean from the data point (referred to as x). Then the result is divided by the population standard deviation. Mathematically, it represents as,

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where

  • x = Datapointμ = Meanơ = Standard deviation

Calculation of Z Score (Step by Step)

The equation for the Z-score of a data point derives 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 xi, while the total number of data points in the data set denotes by N.

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

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

Examples

Example #1

Let us take the example of a class of 50 students who wrote the science test last week. Today is the result day, and the class teacher told me that John scored 93 on the test while the average class score was 68. So first, determine the Z-score for John’s test mark if the standard deviation is 13.

Solution:

Given,

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

Therefore, one can calculate the Z-score for John’s test score using the above formula,

Z = (93 – 68) / 13

Z Score will be –

Z  Score = 1.92

Therefore, John’s Z-test 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 the Z-test is not appropriate for less than 30 data points) who appeared for a class test. Determine the Z-test score for the 4th 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.

  • x = 65,4th student scored = 65,The 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, one can calculate the standard deviation by using the formula as shown below,

ơ = 13.44

Therefore, one can calculate the Z-score of the 4th student using the above formula,

Z = (x – x ) / s

  • Z = (65 –30) / 13.44Z = -0.47

Therefore, the 4th 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 4th student as per the Z- score table.

Z Score in Excel (with Excel Template)

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 the calculation of the Z-score.

You can refer to the given excel sheet below for the detailed calculation of the Z Score FormulaZ Score FormulaZ-test formula is applied hypothesis testing for data with a large sample size. It denotes the value acquired by dividing the population standard deviation from the difference between the sample mean, and the population mean.read more Test Statistics.

Relevance and Uses

From the hypothesis testingHypothesis TestingHypothesis Testing is the statistical tool that helps measure the probability of the correctness of the hypothesis result derived after performing the hypothesis on the sample data. It confirms whether the primary hypothesis results derived were correct.read more perspective, the Z-score is very important to understand because it is employed to test whether or not a test statistic falls in the acceptable range. The Z-score is also used to standardize data before analysis and calculate the probability of a score or comparison of two or more data points from different normal distributionsNormal DistributionsNormal Distribution is a bell-shaped frequency distribution curve which helps describe all the possible values a random variable can take within a given range with most of the distribution area is in the middle and few are in the tails, at the extremes. This distribution has two key parameters: the mean (µ) and the standard deviation (σ) which plays a key role in assets return calculation and in risk management strategy.read more. Finally, there are diverse applications of Z-score across fields if applied properly.

This article has been a guide to the Z-Score Formula. Here, we provide the formula to calculate Z-score, practical examples, and a downloadable Excel template. You can learn more about financial analysis from the following articles: –

  • Altman Z Score FormulaStandard Normal Distribution CalculationBell Curve FormulaFormula of Sampling Distribution Formula Of Sampling Distribution A sampling distribution is the probability-based distribution of detailed statistics. It helps calculate means, range, standard deviation, and variance for the undertaken sample. For a sample size of more than 30, the formula is: µ͞x =µ and σ͞x =σ / √n
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