A Full Guide On Z Test: Formula & Example


What Is a Z Test?

Z test is basically a statistical method which is used to analyse data having its application in science, business, and numerous further disciplines.

Z test definition tells us that it is used to calculate the mean of the population and to analyse that the mean of two populations are different, where we know the variance as well as there is large sample size. Though, while analysing the data we assume that it has a distribution, and standard deviation is given or known.

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How Z-Tests Work

Z test make use of normal distribution to resolve the complications involving too large samples. Though, Z test and t test are closely related to each other, but t test is performed having a smaller sample size., and in t test the standard deviation is not known to us. Whereas, in the Z test the sample size is large and the standard deviation is known to us.

Requirements of Z Test:

While analysing and calculating the Z test, you require the following:

  • Running a Z test on your data requires five steps:
  • You need to demonstrate up the null hypothesis as well as the alternate hypothesis.
  • The alpha level should be selected.
  • You need to look up for the critical value of z in the table of z.
  • You will need to Calculate up the z test statistic
  • You will then Compare up the test statistic with the critical z value and then take up the decision to accept or to reject the given null hypothesis.

Essential condition to Z Test:

You need the following conditions to do or run a Z test, which are as follows:

  • The data should be independent from each other.
  • The sample size of the data should be greater than 30. If not, t test is recommended for smaller sample size.
  • The sample should be equal.
  • There should be normal distributed data for the sample.
  • The data should be selected arbitrarily.

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Z Test Formula

According to the Z test formula we subtract the mean of population from the mean of sample and divide the result by the result of the Standard Deviation of the Population divided by the number of the observations square roots. Which states that,

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Formula 1:

Z = (x – μ) / ơ

where in the above formula,

  • x = slightly value from the population
  • μ = mean of the population
    • ơ = populations standard deviation

Formula 2:

Z = (x – x_mean) / s

Where in the above Z test formula,

  • x = somewhat value from the sample
  • x mean = mean of the sample
  • s = standard deviation of the sample

Z test Examples:

Example 1: We assume that the mean score of the company who have given upgradation test is given as 75 and the standard deviation is given to us as 15. Now, we need to calculate the z-test score of ABC who has scored 90 in the upgradation test of the company.

So, we have the following data with us as:

  • The mean of the population i.e. μ= 75
  • Standard deviation of the Population i.e. ơ = 15

Therefore, by using the Z test formula, we calculate Z test as:

Z = (x – μ) / ơ

Z = (90 – 75) / 15

Z = 1

Here, we will consider the Z table and find the value and see to it that 84.13{367c01af22dc6c3a8611ff25983b0f0a247ed9fc1c45fd9103ad49b47a0c5f39} of the colleges scored less than ABC.

Calculation of Z test in excel:

Now, the above example is calculated in excel as follows:

mean of the population 75
Standard deviation of the Population 15
Any value from the population 90
Z test result =(B4-B2)/B3

Applying or using the formula =(B4-B2)/B3 for calculating the Z test we get the following result as:

A B</b
mean of the population 75
Standard deviation of the Population 15
Any value from the population 90
Z test result 1

Example 2: Taking up or assuming the values as follows:

Population Mean (μ): 20
Population Variance (σ2): 10
Sample Mean (M): 4
Sample Size (N): 35

Calculate the value of z and p, for the single Z sample.

Z Score or Z test Calculation using the Z test formula

Z = (M – μ) / √(σ2 / n)

Z = (2 – 4) / √(3 / 40)

Z = -2 / 0.27386

Z = -7.30297

Result: The value obtained of z = -7.30297. The value obtained of p = < .00001. Therefore, the noteworthy result at p < .05.

One Proportion Z-Test

A One Proportion Z-Test or one sample z test takes up only one sample at a time and calculate the hypothesis as well as interpret the result. It is used to analyse whether the population is different from the other parameters of the hypothesis or not.

Here, you should do the following:

  • Analyse the Z test hypotheses.
  • Consider the following table to analyse the hypothesis.
  • You can choose the value lying between 0 and 1.
  • Calculate test statistic by using the Z test formula as: Z = (x – μ) / ơ
  • Calculate the Z test and the P-value.
  • Assess as well as analyse the null hypothesis.

Two Proportion Z-Test

A two Proportion Z-Test or two sample z-test takes up two sample at a time and calculate the hypothesis as well as interpret the result. Here, you compare the two proportions of Z test to see if they are alike or not.

Z test calculator:

As we know the Z test is calculated for the one sample as well as for the two samples. Thus, for calculating it we use calculator. So, calculate the Z test for one population and the Z test for 2 population.

When to use z test?

The Z test should be used in the following conditions:

  • When there is need to compare two data or sample.
  • The size of sample is larger than 30.
  • The normal distribution of data is there or exists.
  • The standard deviation is known to us.


It should be noted that Z- test should only be used when the sample size is large or greater than 30, otherwise you should opt for a t test. And one should also use it with the normal distribution and when the standard deviation is known to us.