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Test Statistic: What is it? What are its Types?

Published by at August 31st, 2021 , Revised On November 19, 2021

Having trouble understanding what a test statistic is? Confused about when to use which type of test statistic for your research?

Well, it sounds like a tedious task, especially for those who belong to non-statistical backgrounds, but we have got you covered. This guide has answers to all your questions and queries about test statistics.

First things first, let us see what exactly test statistic is, how it is calculated, and how it is significant in research and experimentation.

What is Test Statistic?

A random variable deduced from sample data and utilised in a hypothesis test is what you call a test statistic. A test statistic’s primary role is to assess and find out whether the null hypothesis is plausible or not. You can remember what a null hypothesis is, right?

For a quick recap, the null hypothesis is a statistical theory that suggests that there is no statistical relationship and importance existing in either a set of single-observed variables or between two sets of observed data.

What happens is that test statistics compares your data with what is expected in the case of a null hypothesis. The test statistic in research is used to figure out the p-value. Again, the p-value is the probability of getting results or a set of observations if the null hypothesis were true.

So, it would be right, to sum up that test statistic calculates the degree of agreement between a null hypothesis and sample data. Its value differs from one random sample to another. The sampling distribution here is called a null distribution. The magnitude of the test statistic becomes high or low depending on the alternative hypothesis when the data show great evidence against the assumptions in the null hypothesis. Because of this, the p-value becomes so low or small that it automatically rejects the null hypothesis.

Does this all make sense now?

Great! Let’s move ahead then.

Before we get into the test statistic types, there is another important thing you must know about. And that is a critical value.

What is a Critical Value?

The critical value is a point (or multiple points) on the test statistic scale that rejects the null hypothesis and is obtained from the test significance level α.

What does it tell?

Critical value gives us an idea of what the probability of two samples means in the same distribution is. Probability and critical value are inversely proportional to each other. This means, if the critical value is higher, then the probability of two sample means in the same distribution is lower.

Note: The general critical value is 1.96 for a two-tailed test. This is based on the fact that 95 percent of the area’s normal distribution is within the general critical value (1.96) standard deviations of the mean.

Now, if you are wondering in what way do critical values be utilized for hypothesis testing, here it is:

  • Measuring test statistic
  • Comparing critical values with a test statistic
  • Find out critical values depending on the significant level of alpha

It looks like we have the prerequisite to start studying test statistic types, so let’s do it.

How many Types of Test Statistics Are There? Which One Should You Use?

If you Google the types of test statistics, you will get a bunch of those in results. However, we will try to study and comprehend some of the most common ones we often use.  We will also try to learn some of the hypotheses attached to each type.

Note: Different statistical tests will have different values for measuring these test statistics. However, the underlying interpretations and hypotheses of the tests remain constant.

Having that said, we will be talking about four common test statistic types, namely:

  • T-test
  • Z-test
  • ANOVA
  • Chi-Square Test

T-test

A t-test is used to find out how significant the difference between two or more groups is. Or, we can say that it is used to compare the mean of two groups or samples. This type of test assumes a normal distribution of the samples. When population parameters, both standard and mean, are not known, the t-test is used.

The t-test can be of three different types or versions:

  1. Independent samples t-test
  2. One sample t-test
  3. Paired sample t-test

The first one compares the mean for two groups, the second one tests the mean of a single group, and the last one compares means at different times of the same group. The names pretty much sum up their objectives. The value you get from a t-test is called the t-value.

T-statistic is what you call the statistic of this hypothesis testing. The formula for it is:

t = (x1 — x2) / (σ / √n1 + σ / √n2), where

  • x1 is the mean of sample 1
  • x2 is the mean of sample 2
  • n1 is the size of sample 1
  • n2 is the size of sample 2

Null hypothesis: Means of two groups are equal

Alternative hypothesis: Means are not equal of two groups

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Z-test

Just like the t-test, the sample is thought to be normally distributed here. A z-test helps you find out a way to assess if the results from the test are repeatable and valid.

The z-value (value deduced from a z-test) here is deduced with the population mean and population standard deviation. It is utilized to confirm a hypothesis that the sample received belongs to the same population.

Null hypothesis: Means of two groups are equal

Alternative hypothesis: Means are not equal of two groups

The formula for a z-value is:

z = (x — μ) / (σ / √n), where

  • x is the sample mean
  • μ is the population mean
  • σ / √n is the population standard deviation

ANOVA

An ANOVA allows you to discover if the experiment results are significant or not. With an ANOVA test, you can tell if you need to reject the null hypothesis or accept the alternative hypothesis.

It is also commonly known as Analysis of Variance and is used to three or more than three samples with a single test.

ANOVA can be of two types:

  1. One-way ANOVA
  2. MANOVA

The first one is used for comparing the difference between multiple samples, and the second one is used to test the impact of multiple independent variables on two or more dependent variables.

Null Hypothesis: All pairs of samples are equal

Alternate Hypothesis: 1 pair of samples is at least different

Chi-Square Test

When comparison of categorical variables is needed, the Chi-Square Test is used.  There are usually two types of variables in statistics, namely numerical or countable and non-numerical or categorical variables. The statistic used to calculate the significance of a chi-square test is called a chi-square statistic. It is a number that reflects the difference between expected and observed counts if there were no relationship in the population at all.

The formula for measuring the chi-square statistic is:

Χ2 = Σ [ (Or,c — Er,c)2 / Er,c ]

In this formula,

  • Or,c is the detected frequency count at Variable A’s level r and Variable B’s level c.

While,

  • Er,c is the predicted frequency count at Variable A’s level r and Variable B’s level c.Next up are the types of chi-square tests.

There are two versions of chi-square tests. These are:

  • Goodness of fit test-shows if a sample matches the population
  • Test for independence-shows comparison between two variables to see if they are related in a contingency table

Null Hypothesis: Variable A and Variable B= independent

Alternate Hypothesis: Variable A and Variable B =not independent.

With this, the guide on test statistics ends. Hope it clears all your confusion and questions.

However, if you still have any queries, please leave a comment in the section below, and we will get back to you soon.

FAQs about Test Statistic and its Types

A random variable deduced from sample data and utilized in a hypothesis test is what you call a test statistic. A test statistic’s primary role is to assess and find out whether the null hypothesis is plausible or not.

Though there are many kinds of test statistics, these four are the most commonly used ones:
• T-test
• Z-test
ANOVA
• Chi-Square Test

A t-test is used to find out how significant the difference between two or more groups is. Or, we can say that it is used to compare the mean of two groups or samples.

The formula for a z-value is:
z = (x — μ) / (σ / √n), where
• x is the sample mean
• μ is the population mean
• σ / √n is the population standard deviation

An ANOVA allows you to discover if the experiment results are significant or not. With an ANOVA test, you can tell if you need to reject the null hypothesis or accept the alternative hypothesis. It is also commonly known as Analysis of Variance and is used to three or more than three samples with a single test.
ANOVA can be of two types:
1. One-way ANOVA
2. MANOVA
The first one is used for comparing the difference between multiple samples, and the second one is used to test the impact of multiple independent variables on two or more dependent variables.

p-value is the probability of getting results or a set of observations if the null hypothesis were true.

The critical value is a point (or multiple points) on the test statistic scale that rejects the null hypothesis and is obtained from the test significance level α.

About Owen Ingram

Ingram is a dissertation specialist. He has a master's degree in data sciences. His research work aims to compare the various types of research methods used among academicians and researchers.