# Hypothesis Testing- Parametric and Non-Parametric Tests in Statistics

*This article was published as a part of the Data Science Blogathon*

## Introduction

Hypothesis testing is one of the most important concepts in Statistics which is heavily used by **Statisticians**, **Machine Learning Engineers**, and **Data Scientists**.

In hypothesis testing, Statistical tests are used to check whether the **null hypothesis** is rejected or not rejected. These Statistical tests assume a null hypothesis of no relationship or no difference between groups.

So, In this article, we will be discussing the statistical test for hypothesis testing including both parametric and non-parametric tests.

**Table of Contents**

**1.** What are Parametric Tests?

**2.** What are Non-parametric Tests?

**3.** Parametric Tests for Hypothesis testing

- T-test
- Z-test
- F-test
- ANOVA

**4.** Non-parametric Tests for Hypothesis testing

- Chi-square
- Mann-Whitney U-test
- Kruskal-Wallis H-test

**Let’s get started,**

**Parametric Tests**

The basic principle behind the parametric tests is that we have a fixed set of parameters that are used to determine a probabilistic model that may be used in Machine Learning as well.

Parametric tests are those tests for which we have prior knowledge of the population distribution (i.e, normal), or if not then we can easily approximate it to a normal distribution which is possible with the help of the Central Limit Theorem.

Parameters for using the normal distribution is –

- Mean
- Standard Deviation

Eventually, the classification of a test to be parametric is completely dependent on the population assumptions. There are many parametric tests available from which some of them are as follows:

- To find the confidence interval for the population means with the help of known standard deviation.
- To determine the confidence interval for population means along with the unknown standard deviation.
- To find the confidence interval for the population variance.
- To find the confidence interval for the difference of two means, with an unknown value of standard deviation.

** **

**Non-parametric Tests**

In Non-Parametric tests, we don’t make any assumption about the parameters for the given population or the population we are studying. In fact, these tests don’t depend on the population.

Hence, there is no fixed set of parameters is available, and also there is no distribution (normal distribution, etc.) of any kind is available for use.

This is also the reason that nonparametric tests are also referred to as** distribution-free tests**.

In modern days, Non-parametric tests are gaining popularity and an impact of influence some reasons behind this fame is –

- The main reason is that there is no need to be mannered while using parametric tests.
- The second reason is that we do not require to make assumptions about the population given (or taken) on which we are doing the analysis.
- Most of the nonparametric tests available are very easy to apply and to understand also i.e. the complexity is very low.

** **

** Image Source: Google Images**

**T-Test**

**1.** It is a parametric test of hypothesis testing based on **Student’s T distribution**.

**2.** It is essentially, testing the significance of the difference of the mean values when the sample size is small (i.e, less than 30) and when the population standard deviation is not available.

**3.** Assumptions of this test:

- Population distribution is normal, and
- Samples are random and independent
- The sample size is small.
- Population standard deviation is not known.

**4.** Mann-Whitney ‘U’ test is a non-parametric counterpart of the T-test.

**A T-test can be a:**

**One Sample T-test: **To compare a sample mean with that of the population mean.

**where,**

**x̄** is the sample mean

**s** is the sample standard deviation

**n** is the sample size

**μ** is the population mean

**Two-Sample T-test:** To compare the means of two different samples.

where,

**x̄ _{1}** is the sample mean of the first group

**x̄ _{2}** is the sample mean of the second group

**S _{1}** is the sample-1 standard deviation

**S _{2} **is the sample-2 standard deviation

**n** is the sample size

__Conclusion:__

- If the value of the test statistic is greater than the table value ->
**Rejects the null hypothesis**. - If the value of the test statistic is less than the table value ->
**Do not reject the null hypothesis**.

**Z-Test**

**1.** It is a parametric test of hypothesis testing.

**2.** It is used to determine whether the means are different when the population variance is known and the sample size is large (i.e, greater than 30).

**3. **Assumptions of this test:

- Population distribution is normal
- Samples are random and independent.
- The sample size is large.
- Population standard deviation is known.

**A Z-test can be:**

**One Sample Z-test: **To compare a sample mean with that of the population mean.

** Image Source: Google Images**

**Two Sample Z-test:** To compare the means of two different samples.

** **

**where,**

**x̄**_{1} is the sample mean of 1st group

**x̄**_{2} is the sample mean of 2nd group

**σ _{1}** is the population-1 standard deviation

**σ _{2} **is the population-2 standard deviation

**n** is the sample size

** **

**F-Test**

**1.** It is a parametric test of hypothesis testing based on **Snedecor F-distribution**.

**2.** It is a test for the null hypothesis that two normal populations have the same variance.

**3.** An F-test is regarded as a comparison of equality of sample variances.

**4.** F-statistic is simply a ratio of two variances.

**5.** It is calculated as:

F = s_{1}^{2}/s_{2}^{2}

**6.** By changing the variance in the ratio, F-test has become a very flexible test. It can then be used to:

- Test the overall significance for a regression model.
- To compare the fits of different models and
- To test the equality of means.

**7.** Assumptions of this test:

- Population distribution is normal, and
- Samples are drawn randomly and independently.

** **

**ANOVA **

**1.** Also called as **Analysis of variance**, it is a parametric test of hypothesis testing.

**2.** It is an extension of the T-Test and Z-test.

**3.** It is used to test the significance of the differences in the mean values among more than two sample groups.

**4.** It uses F-test to statistically test the equality of means and the relative variance between them.

**5.** Assumptions of this test:

- Population distribution is normal, and
- Samples are random and independent.
- Homogeneity of sample variance.

**6.** One-way ANOVA and Two-way ANOVA are is types.

**7. F-statistic = variance between the sample means/variance within the sample**

**Chi-Square Test**

**1.** It is a non-parametric test of hypothesis testing.

**2.** As a non-parametric test, chi-square can be used:

- test of goodness of fit.
- as a test of independence of two variables.

**3.** It helps in assessing the goodness of fit between a set of observed and those expected theoretically.

**4.** It makes a comparison between the expected frequencies and the observed frequencies.

**5.** Greater the difference, the greater is the value of chi-square.

**6.** If there is no difference between the expected and observed frequencies, then the value of chi-square is equal to zero.

**7.** It is also known as the **“Goodness of fit test”** which determines whether a particular distribution fits the observed data or not.

**8.** It is calculated as:

**9.** Chi-square is also used to test the independence of two variables.

**10.** Conditions for chi-square test:

- Randomly collect and record the Observations.
- In the sample, all the entities must be independent.
- No one of the groups should contain very few items, say less than 10.
- The reasonably large overall number of items. Normally, it should be at least 50, however small the number of groups may be.

**11. **Chi-square as a parametric test is used as a test for population variance based on sample variance.

**12.** If we take each one of a collection of sample variances, divide them by the known population variance and multiply these quotients by (n-1), where n means the number of items in the sample, we get the values of chi-square.

**13.** It is calculated as:

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**Mann-Whitney U-Test**

**1.** It is a non-parametric test of hypothesis testing.

**2.** This test is used to investigate whether two independent samples were selected from a population having the same distribution.

**3.** It is a true non-parametric counterpart of the T-test and gives the most accurate estimates of significance especially when sample sizes are small and the population is not normally distributed.

**4.** It is based on the comparison of every observation in the first sample with every observation in the other sample.

**5.** The test statistic used here is “U”.

**6.** Maximum value of “U” is ‘n_{1}*n_{2}‘ and the minimum value is zero.

**7.** It is also known as:

- Mann-Whitney Wilcoxon Test.
- Mann-Whitney Wilcoxon Rank Test.

**8.** Mathematically, U is given by:

**U _{1} = R_{1} – n_{1}(n_{1}+1)/2**

where n_{1} is the sample size for sample 1, and R_{1} is the sum of ranks in Sample 1.

**U _{2} = R_{2} – n_{2}(n_{2}+1)/2**

When consulting the significance tables, the smaller values of U_{1} and U_{2} are used. The sum of two values is given by,

**U _{1} + U_{2} = { R_{1} – n_{1}(n_{1}+1)/2 } + { R_{2} – n_{2}(n_{2}+1)/2 } **

Knowing that R_{1}+R_{2} = N(N+1)/2 and N=n_{1}+n_{2}, and doing some algebra, we find that the sum is:

**U _{1 }+ U_{2} = n_{1}*n_{2}**

** **

**Kruskal-Wallis H-test**

**1.** It is a non-parametric test of hypothesis testing.

**2.** This test is used for comparing two or more independent samples of equal or different sample sizes.

**3.** It extends the Mann-Whitney-U-Test which is used to comparing only two groups.

**4.** One-Way ANOVA is the parametric equivalent of this test. And that’s why it is also known as ‘**One-Way ANOVA on ranks**.

**5.** It uses ranks instead of actual data.

**6.** It does not assume the population to be normally distributed.

**7.** The test statistic used here is “H”.

**This completes today’s discussion!**

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**Endnotes**

*Thanks for reading!*

I hope you enjoyed the article and increased your knowledge about Statistical Tests for Hypothesis Testing in Statistics.

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__About the Author__

__About the Author__

**Aashi Goyal**

Currently, I am pursuing my Bachelor of Technology (B.Tech) in Electronics and Communication Engineering from **Guru Jambheshwar University(GJU), Hisar. **I am very enthusiastic about Statistics, Machine Learning and Deep Learning.

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