**T-Tests, what are they?**

Two-Sample T-tests are statistical tests that are used for the comparison of the means of two populations. These are also simply called Student’s t-tests. The output generated from the T test helps us to know if there exists a significant difference between the mean of two samples which is unlikely going to happen because of sampling error or random chance of any kind.

T-tests can be further broken down into two categories, paired t-tests, and unpaired t-tests. Their application is common to see in the domains of biology, business, and psychology.

**What is a paired T-Test?**

A paired T–test is also called a dependent or correlated test. itis a statistical test that compares the averages/means and standard deviations of two related groups to determine if there is a significant difference between the two groups. A significant difference occurs only when the differences between the groups are unlikely to be due to sampling error or chance. The groups can be related by being the same group of people, the same item, or being subjected to the same conditions. Paired t-tests are considered more powerful than unpaired t-tests because using the same participants or item eliminates variation between the samples that could be caused by anything other than what’s being tested.

**What are the hypotheses of a paired T-Test?**

There are two hypotheses that can be developed in a paired T-test

The first one is the null hypothesis (H0) which states that there is no significant difference between the means of the two groups. The second hypothesis is the alternate hypothesis (H1) states that there is a significant difference between the two population means and that there is a significant difference between the two population means and this difference is most likely not caused by sampling error or chance.

**There are some assumptions of paired t-tests. These are the key assumptions:**

• The dependent variable in the study has a normal distribution

• The observations are sampled independently

• The measurement of the dependent variable will be on the basis of ratios or intervals

• The independent variable must have two related groups or matched pairs.

**Application of paired t-test:**

The application of paired T-test is when the same item is tested twice and it is called a repeated measures t-test. Some of the examples where we can see the application of paired t-test appropriate are:

• The before and after effects of any therapy or treatment on the user or patients

• The body weight on a digital and non-digital weighing scale on the same set of people

• Results of a competitive exam of students after a study preparation course

**The concept of unpaired t-test:**

An Independent T-test or unpaired t-test is different from paired t-test on the grounds of the sample it works on. It is a statistical procedure that compares the means of independent or unrelated groups to understand if there is a significant difference between the two.

Hypotheses of unpaired t-test:

There is no difference between the hypotheses of unpaired and paired t-tests. So, in this case also the hypotheses will go as under:

• The null hypothesis (H0) states that there is no significant difference between the means of the two groups

• The alternate hypothesis (H1) states that there is a significant difference between the two population means and this difference is not caused by any sampling error or chance.

The assumptions of unpaired T-test:

There are certain assumptions that are associated with unpaired T Test and they are different and some are opposed to the assumptions of a paired T-test

They go like this:

• The dependent variable is normally distributed

• The observations are sampled independently

• The dependent variable is measured on an incremental level, which could be ratios or intervals

• The groups have the same standard deviation, which means that the variance of the data is the same between the groups

• The independent variables must consist of two independent groups

**Application of an unpaired T-test**

The application of an unpaired t-test is seen when we have to compare the mean between two independent groups. They can be used when there is a comparison between two groups with equal variance

Some of the examples of the application of unpaired T-test can be:

• Research the benefit of some therapy or treatment where half the group is administered one kind of treatment and the other half is administered another kind of treatment or the control group.

• Research where some difference is evaluated between men and women. It could be related to emotional quotient, bone density, or any other variable where the gender comparison is done to study the difference

In a situation where the variance between the two groups is unequal, we use Welch’s test.

** Comparison of Paired and Unpaired T-Test:**

Though both tests are undertaken with the same objectives, there are some key differences in the prerequisites of the two tests. Some of the main differences are:

• A paired T-test is designed to compare the means of the same group or the items but the scenario is different. The independent T-test compares the means of two independent or unrelated groups.

• In the case of an unpaired t-test the variance is assumed to be equal whereas in a paired t-test the variance is not assumed to be equal.

**Table of Comparison between Paired and Unpaired T-Test**

Whenever we have to compare the means of two or more population groups, ANOVA and t-test are the two best practices that are preferred. There is a thin line of difference between them, which is:

A t-test is taken up when we have to compare the means between two groups whereas when we have to compare the means of three or more groups then the ANOVA test is the one to be taken up. Both these techniques being popular statistical methods of hypotheses testing for comparing means, they have common assumptions, which are:

• The sample drawn from the population has a normal distribution

• There is homogeneity in the variance

• There is random data sampling

• The observations are independent of each other

• The dependent variable is measured in interval or ratio variables

Because of the common assumptions of the T-test and ANOVA, most people tend to confuse both. Despite all the similarities, both these tests have their own share of unique characteristics and situations in which they can apply them. Let us understand this here

**Difference between t-test and ANOVA**

Let us understand the difference between t-test and ANOVA with an overview of the differences between the two so that researchers when they dive deep into analysis, can understand the difference in application of one from the other clearly and explicitly.

**Comparison Criteria**

**Let us understand this difference between T-test and ANOVA more clearly with some theoretical description**

T test as a method of data analysis examines how greatly the population means of two samples differ from each other.

The best use of the t-test is to test a hypothesis. The data analysis method helps determine if a process has any effect on the target population. The method should be used when you want to compare the means of two groups.

**How to perform t-test and ANOVA**

The T-test is a statistical hypothesis testing method that examines whether the sample population means of two groups largely differ from one another.

It uses the t-distribution method when the standard deviation is unknown and small sample size. The t Test is based on t statistics which assumes the normal distribution of variables and a known mean. Population variance is then calculated from the sample.

• Null hypothesis H0: µ(x) = µ(y) against

• Alternative hypothesis H1: µ(x) ≠ µ(y)

**Where µ(x) and µ(y) represent the population means.**

The degree of freedom of the t-test is n1 + n2 – 2

**ANOVA**

Analysis of Variance is used when there is a comparison between more than two population means. It assumes that the sample is drawn from a normally distributed population and that its variances are equal. The total amount of variation is split into two: the amount assigned to chance, and the amount assigned to causes. So, ANOVA proceeds to test the variance in population means by evaluating the variance among the group items which is proportional to the amount of variance in the groups. This variance occurs due to an unexplained disturbance because of different treatments.

It tests the hypothesis:

• **Null hypothesis H0: **All population means are the same

• **Alternative hypothesis H1:** At least one population mean is different

**One-Way and Two-Way ANOVA:**

ANOVA is of two types, one-way and two-way ANOVA, It is also called unidirectional or two-way ANOVA. There are further variations in ANOVA< example MANOVA. MANOVA, which is multivariate ANOVA differs from ANOVA as the former tests for multiple dependent variables simultaneously while the latter assesses only one dependent variable at a time. One-way or two-way refers to the number of independent variables in your analysis of the variance test. A one-way ANOVA evaluates the impact of a sole factor on a sole response variable. It determines whether all the samples are the same. The one-way ANOVA is used to determine whether there are any statistically significant differences between the means of three or more independent (unrelated) groups.

A two-way ANOVA is an extension of the one-way ANOVA. With a one-way, you have one independent variable affecting a dependent variable. With a two-way ANOVA, there are two independents. For example, a two-way ANOVA allows a company to compare worker productivity based on two independent variables, such as salary and skill set. It is utilized to observe the interaction between the two factors and tests the effect of two factors at the same time. ANCOVA(Analysis of Covariance) is another variation of ANOVA and it combines ANOVA and regression. It can be useful for understanding within-group variance that ANOVA tests do not explain.

ANOVA is a good way to compare more than two groups to identify relationships between them. The technique can be used in scholarly settings to analyze research or in the world of finance to try to predict future movements in stock prices. Understanding how ANOVA works and when it may be a useful tool can be helpful for advanced investors.

**Using Post Hoc with ANOVA**

Post hoc tests are an important part of ANOVA. When you use ANOVA to test the equality of at least three group means, statistically significant results indicate that not all the group means are equal. However, ANOVA results do not identify which differences between pairs of means are significant. Use post hoc tests to explore differences between multiple groups means while controlling the experiment-wise error rate.

**Conclusion**

After studying the above differences, we can safely say that the t-test is a special type of Analysis of Variance that is used when we only have two population means to compare. Hence, to avoid an increase in error while using a t-test to compare more than two population groups, we use ANOVA. There is a lot of online software available now, that can help you to eliminate the problem of human error and make you apply t-test and ANOVA seamlessly.