**Formula of Z Test Statistics (Table of Contents)**

## Z Test Formula in Statistics

The term z-test in statistics refers to the hypothesis test in which it is measured how many standard deviations above or below the mean a raw data is. A z-test score is also known as standard score and 99.9% of the value fall in the range from -3 standard deviations (extreme left of the normal distribution curve) up to +3 standard deviations (extreme right of the normal distribution curve). The formula for z-test statistics of value is calculated by deducting population mean from the x-value and then the result is divided by the population standard deviation.

Mathematically, z-test statistics formula is represented as,

** Z = (x – ****μ) / ****ơ **

where x = any value from the population

- μ = population mean
- ơ = population standard deviation

In case of a sample, the formula for z-test statistics of value is calculated by deducting sample mean from the x-value and then the result is divided by the sample standard deviation. Mathematically, it is represented as,

**Z = (x – x_mean****) / s**

where

- x = any value from the sample
**x_mean**= sample mean- s = sample standard deviation

### Explanation of the Z Test Statistics Formula

The formula for z-test statistics for a population is derived by using the following steps:

**Step 1:** Firstly, calculate the population mean and population standard deviation based on the observation captured in the population mean and each observation is denoted by x_{i}. The total number of observations in the population is denoted by N.

Population mean,

Population standard deviation,

**Step 2:** Finally, the formula for z-test statistics is computed by deducting population mean from the variable and then the result is divided by the population standard deviation as shown below.

**Z = (x – ****μ) / ****ơ**

The formula for z-test statistics for a sample is derived by using the following steps:

**Step 1:** Firstly, calculate the sample mean and sample standard deviation the same as above. Here, the total number of observations in the sample is denoted by n such that n < N.

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Sample mean,

Sample standard deviation,

**Step 2:** Finally, the formula for z-test statistics is calculated by deducting sample mean from the x-value and then the result is divided by the sample standard deviation as shown below.

**Z = (x – x_mean****) / s**

### Examples of Z Test Statistics Formula

Let’s see some simple to advanced examples of Z-Test Statistics Formula to understand it better.

#### Example #1

**Let us assume a population of students in a school who appeared for a class test. The mean score in the test is 75 and the standard deviation is 15. Determine the z-test score of David who scored 90 in the test.**

Given,

- The population mean, μ= 75
- Population standard deviation, ơ = 15

Therefore, the z-test statistics can be calculated as,

Z = (90 – 75) / 15

**Z Test Statistics will be –**

**Z = 1**

Therefore, David’s test score is one standard deviation above the mean score of the population i.e. as per z-score table, 84.13% students less score than David.

#### Example #2

**Let us take the example of 30 students who were selected as a part of a sample team to be surveyed to see how many pencils were being used in a week. Determine the z-test score for the 3 ^{rd} student of based on the given responses: 3, 2, 5, 6, 4, 7, 4, 3, 3, 8, 3, 1, 3, 6, 5, 2, 4, 3, 6, 4, 5, 2, 2, 4, 4, 2, 8, 3, 6, 7.**

Given,

- x = 5, since the 3
^{rd}students response is 5 - Sample size, n = 30

Sample mean, = (3 + 2 + 5 + 6 + 4 + 7 + 4 + 3 + 3 + 8 + 3 + 1 + 3 + 6 + 5 + 2 + 4 + 3 + 6 + 4 + 5 + 2 + 2 + 4 + 4 + 2 + 8 + 3 + 6 + 7) / 30

**Mean = 4.17**

Now, the sample standard deviation can be calculated by using the above formula.

**ơ = 1.90**

Therefore, the z-test score for the 3^{rd} student can be calculated as,

Z = (x – x ) / s

- Z = (5 –17) / 1.90
- Z = 0.44

Therefore, the 3^{rd} student’s usage is 0.44 times the standard deviation above the mean usage of the sample i.e. as per z- score table, 67% students use fewer pencils than the 3^{rd} student.

### Z Test Statistics Formula (with Excel Template)

**Let us take the example of 30 students who were selected as a part of a sample team to be surveyed to see how many pencils were being used in a week. Determine the z-test score for the 3 ^{rd} student of based on the given responses: 3, 2, 5, 6, 4, 7, 4, 3, 3, 8, 3, 1, 3, 6, 5, 2, 4, 3, 6, 4, 5, 2, 2, 4, 4, 2, 8, 3, 6, 7.**

Below is given data for calculation of Z Test Statistics

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

### Relevance and Uses

It is very important to understand the concept of z-test statistics because it is usually used whenever it is arguable whether or not a test statistic follows a normal distribution under the concerned null hypothesis. However, it should be kept in mind that a z-test is used only when the sample size is greater than 30, otherwise, the t-test is used.

You can download this Z Test Formula Excel Template from here – Z-Test Statistics Formula Excel Template

### Recommended Articles

This has been a guide to Z Test Statistics Formula. Here we discuss the calculation of Z Test Statistics along with examples and downloadable excel template. You can learn more about financial analysis from the following articles –

- Z Test vs T Test – Compare
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