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**Table of Content**

Try the test statistic calculator to estimate the t-value of a given dataset within seconds by providing a couple of simple inputs.

This online tool takes sample mean, population mean, standard deviation, and sample size into account to calculate t statistics precisely.

A particular statistical calculation that figures out the relationship between the sample and its population is known as the test statistics.

It’s a score that is used in the hypothesis test and informs about how likely the results are under the assumption that the null hypothesis is true.

Moreover, a student’s t-test is used to evaluate hypotheses about the population mean. It tells whether you should support or reject the null hypotheses.

The test statistic formula depends on the population standard deviation (σ) whether it’s known or unknown. Let’s take a look at the following general test statistic equations:

**For UnKnown Standard Deviation:**

t = \(\frac{\overline{x} – μ_0}{\frac{s}{\sqrt{n}}}\)

**For Known Standard Deviation:**

t = \(\frac{\overline{x} – μ_0}{\frac{σ}{\sqrt{n}}}\)

Where

- \( \overline{x} \) indicates the sample mean
- μ shows the population mean
- n is the representative of the sample size
- s is the standard deviation

There are different standard cases that you must know to perform test statistics. Below we have four standard cases for which t value formulas differ. Let’s see them:

Calculating one population mean, if the variable is numerical and one population or one group is being studied.

For instance, you believe that the students of O level spend $20 a day, here the variable is numerical and the population is the students of O level. In this case, it’s good to use one population mean.

Test Statistic for One Population Mean = \(\frac{\overline{x} – μ_0}{\frac{σ}{\sqrt{n}}}\)

This test is good when the variable is numerical and you have to compare two populations or groups. For this, you must choose two separate random samples from each population.

Test Statistic Comparing Two Means = \(\frac{\overline{x} – \overline{y}}{\sqrt{\frac{σ^2_x}{n_1} + \frac{σ^2_y}{n_2}}}\)

Perform this calculation, when the variable is categorical such as genders, workers/ unemployed and one population has to be considered.

Test Statistic for a Single Population Proportion = \(\frac{\stackrel{\text{^}}{p} – \ p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}}\)

Use it when the variable is categorical, and you have to consider the proportion of two individuals having specific characteristics of two different categories such as male/female.

Test Statistic for Two Population Proportions = \( \frac{\stackrel{\text{^}}{p_1}-\stackrel{\text{^}}{p_2}}{\sqrt{\stackrel{\text{^}}{p}(1-\stackrel{\text{^}}{p})(\frac{1}{n_1} + \frac{1}{n_2})}} \)

Let us make a supposition for a cricket series in which Jack has an average score of 66 or consecutive 16 matches. Now as you better know an average batting average for a player is 40 (maximum). Keeping in view the deviation in scoring that is 4 in the case, what are the performance stats of Jack? Now how to find the test statistics?

\(\overline{x}=66\)

\(n=16\)

\(μ=40\)

\(σ=4\)

\(\text{Test Statistic}=\frac{\overline{x} – μ_0}{\frac{σ}{\sqrt{n}}}\)

\(\text{Test Statistic}=\frac{66 – 40}{\frac{4}{\sqrt{16}}}\)

\(\text{Test Statistic}=\frac{26}{\frac{4}{4}}\)

\(\text{Test Statistic}=\frac{26}{1}\)

\(\text{Test Statistic}=26\)

Now as the computed value is 26, find a Standardized test statistic calculator to verify it. With it, you will have the correct results in fractions of seconds.

Suppose the standard significance level is 5% and compare the results with it. Now if you look for the critical value for the normal threshold of 5%, it is 1.645. It means that the performance for 16 matches is considerably better than average.

Our test statistic formula calculator allows you to perform four types of tests by considering certain types of parameters that are:

- Hypothesis Mean
- Standard Deviation
- Sample Mean
- Sample Size

It provides you with a numerical value that sums up the information of a given data set and allows you to make inferences about a population, comparing various groups and help in:

**Hypothesis Testing:**It helps the analysts to determine whether they should support or reject the hypothesis.**Statistical Significance:**Allows to quantify the difference between the observed data and the expectation under the null hypothesis.**Decision-Making:**Helps in decision-making by providing the effectiveness of a product, idea, or change in a process.**Research Validation**: With it, researchers can ensure the validation of the studies.

When finding the Z-score, we assume that the population standard deviation is given but while finding the T-score, we need to estimate the population standard deviation on our own. It is not wrong to say that both T-score and Z-score are used to make comparisons.

Gosset was a talented statistician who proposed the theory of student’s t-distribution in the year 1908.

A z score is a value that shows a standard deviation above or below the mean. Meanwhile, it represents the relationship of value with the mean of a group.

This is a statistical test that is done to compare the mean of two groups. Mostly, it is performed in hypothesis testing. It helps in determining whether a process or treatment has an effect on the population or not. You can easily use a test statistics calculator to know that within seconds.

From the source of Wikipedia: Common test statistics

From the source of Khan Academy: two-sample t test, Hypotheses, conclusions about the difference of means

From the source of Lumen Learning: Random Variables, Properties