Formula to Calculate Z Test in Statistics

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where x = any value from the population

  • μ = population meanơ = population standard deviation

In the case of a sample, the formula for Z-test statistics of value calculates deducting the sample mean from the X-value. Then the result is divided by the sample standard deviation. Mathematically, it represents as

where

  • x = any value from the samplex_mean = sample means = sample standard deviation

Z Test Calculation (Step by Step)

Examples

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. But, first, determine the Z-test score of David, who scored 90 on the test.

  • Firstly, calculate the population means and standard deviation based on the observation captured in the population mean, and each observation is denoted by xi. Then, the total number of observations in the population is denoted by N. Population mean,Population standard deviation, Finally, the Z-test statistics computes deducting the population mean from the variable. Then, the result is divided by the population standard deviation, as shown below. Z = (x – μ) / ơThe formula for Z-test statistics for a sample derives from the following steps: Firstly, calculate the sample mean and standard deviation the same as above. Here, the total number of observations in the sample is denoted by n such that n < N.Sample mean, Sample standard deviation, Finally, the Z-test statistics are calculated by deducting the sample mean from the X-value. Then, the result is divided by the sample standard deviation, as shown below. Z = (x – x_mean) / s

Population mean,Population standard deviation,

Z = (x – μ) / ơThe formula for Z-test statistics for a sample derives from the following steps:

Sample standard deviation,

Z = (x – x_mean) / s

Given

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

Therefore, one can calculate the Z-test statistics 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 the z-scoreZ-scoreThe Z-score of raw data refers to the score generated by measuring how many standard deviations above or below the population mean the data, which helps test 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.read more table, 84.13% of students score less than David.

Example #2

Take the example of 30 students selected as a part of a sample team surveyed to see how many pencils used in a week. Determine the Z-test score for the 3rd 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 3rd student’s response, is 5Sample 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

One can calculate the sample standard deviation using the above formula.

ơ = 1.90

Therefore, one calculates the Z-test score for the 3rd student as,

Z = (x – x ) / s

  • Z = (5 –17) / 1.90Z = 0.44

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

Example #3

Take the example of 30 students selected as a part of a sample team surveyed to see how many pencils were used in a week. Determine the Z-test score for the 3rd 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 the calculation of Z-test statistics.

You can refer to the Excel sheet below for the detailed calculation of Z-test statistics.

Relevance and Uses

It is essential to understand the concept of Z-test statistics because it usually uses whenever it is arguable whether or not a test statistic follows a normal distributionNormal DistributionNormal 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 under the concerned null hypothesis. However, one should remember that one can use the Z-test only when the sample size exceeds 30. Otherwise, one may use the t-testT-testA T-test is a method to identify whether the means of two groups differ from one another significantly. It is an inferential statistics approach that facilitates the hypothesis testing.read more.

This article is a guide to the Z-Test Statistics Formula. Here, we learn how to calculate Z-test in statistics using its formula, examples, and a downloadable Excel template. You can learn more about financial analysis from the following articles: –

  • Z Test vs. T-TestZ Test Vs. T-TestThe Z Test is a statistical hypothesis that is used to determine whether the calculated means of two samples differ when the standard deviation is available and the sample size is large. The T test, on the other hand, is used to determine how the averages of different data sets differ from one another when the standard deviation or variance is unknown.read moreSample Size FormulaSample Size FormulaThe sample size formula depicts the relevant population range on which an experiment or survey is conducted. It is measured using the population size, the critical value of normal distribution at the required confidence level, sample proportion and margin of error.read moreF-Test FormulaF-Test FormulaF-test formula is used in order to perform the statistical test that helps the person conducting the test in finding that whether the two population sets that are having the normal distribution of the data points of them have the same standard deviation or not.read moreSkewness Formula