Z Score Formula

Formula to Calculate Z-Score

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, it 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
Z-Score-Formula

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For eg:
Source: Z Score Formula (wallstreetmojo.com)

where

  • x = Datapoint
  • μ = Mean
  • ơ = Standard deviation

Calculation of Z Score (Step by Step)

The Equation for the 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 xi, while the total number of data points in the data set is denoted by N.
Z Score Formula Step 1
  • Step 2: Next, determine the standard deviation of the population on the basis of the population mean μ, data points xi, and the number of data points in the population N.
Step 2
  • 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.
    Step 3

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 me 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,

Example 1.1

Z = (93 – 68) / 13

Z Score will be –

Z Score Formula Example 1.2

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 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.

Solution:

Given,

  • x = 65,
  • 4th 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,

Z Score Formula 4

ơ = 13.44

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

Z = (x – x ) / s

  • Z = (65 –30) / 13.44
  • Z = -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 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 the calculation of the Z Score.

Z Score Formula Example 0.3
Z Score Formula Example 3.2

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 perspective of 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, 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 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. There are the diverse application of z-score across fields if applied properly.

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