T-Test: What It Is With Multiple Formulas and When To Use Them

What Is a T-Test?

A t-test is an inferential statistic used to determine if there is a significant difference between the means of two groups and how they are related. T-tests are used when the data sets follow a normal distribution and have unknown variances, like the data set recorded from flipping a coin 100 times.

The t-test is a test used for hypothesis testing in statistics and uses the t-statistic, the t-distribution values, and the degrees of freedom to determine statistical significance.

Understanding the T-Test

A t-test compares the average values of two data sets and determines if they came from the same population. For example, the grades of students from a physics class and those of a different group of students from a writing class would not likely have the same mean and standard deviation. Similarly, samples taken from the placebo-fed control group of a drug test and those taken from the drug-prescribed group should have a slightly different mean and standard deviation.

Four assumptions are made while using a t-test:

1. The data collected must follow a continuous or ordinal scale, such as the scores for an IQ test.
2. The data is collected from a randomly selected portion of the total population
3. The data will result in a normal distribution of a bell-shaped curve.
4. Equal or homogenous variance exists when the standard variations are equal.

Mathematically, the t-test takes a sample from each of the two sets and establishes the problem statement. It assumes a null hypothesis, which means that it assumes the two means are equal.

Using the t-test formulas, values are calculated and compared against the standard values. This comparison helps to determine the effect of chance on the difference, and whether the difference is outside that chance range. The t-test questions whether the difference between the groups represents a true difference in the study or merely a random difference.

Based on the results, the assumed null hypothesis is accepted or rejected. If the null hypothesis is rejected, it indicates that data readings are strong and are probably not due to chance.

• Null hypothesis rejected: Differences are statistically significant
• Null hypothesis accepted: Differences are not statistically significant
  

The t-test is just one of many tests used for this purpose. Others may be more appropriate depending on the number of variables or the size of the sample. For example, statisticians use a z-test for data sets with a large sample size. Other testing options include the chi-square test and the f-test.

Example of When a T-Test Would Be Used

Imagine that a drug manufacturer tests a new medicine. Following standard procedure, the drug is given to one group of patients, and a placebo is given to another group called the control group. The placebo is a substance with no therapeutic value and serves as a benchmark to measure how the other group, administered the actual drug, responds.

After the drug trial, the members of the control group reported an increase in average life expectancy of three years. Members of the group that was prescribed the new drug reported an increase in average life expectancy of four years.

Initial observation indicates that the drug is working. However, it is also possible that the observation may be due to chance. A t-test can be used to determine if the results are significant and applicable to the entire population, or whether they are random and not due to the drug intervention.

Using the T-Test

Calculating a t-test requires three fundamental data values:

1. The difference between the mean values from each data set, also known as the mean difference
2. The standard deviation of each group
3. The number of data values of each group
   

The t-test produces two values as its output: t-value and degrees of freedom. The t-value, or t-score, is a ratio of the difference between the mean of the two sample sets and the variation that exists within the sample sets.

The numerator value is the difference between the mean of the two sample sets. The denominator is the variation that exists within the sample sets and is a measurement of the dispersion or variability.

This calculated t-value is then compared against a value obtained from a critical value table called the T-distribution table. Higher values of the t-score indicate that a large difference exists between the two sample sets. The smaller the t-value, the more similarity exists between the two sample sets.

Degrees of freedom refer to the values in a study that has the freedom to vary and are essential for assessing the importance and the validity of the null hypothesis. Computation of these values usually depends upon the number of data records available in the sample set.

Paired Sample T-Test Formula

The correlated t-test, or paired t-test, is a dependent type of test and is performed when the samples consist of matched pairs of similar units, or when there are cases of repeated measures. For example, there may be instances where the same patients are repeatedly tested before and after receiving a particular treatment. Each patient is being used as a control sample against themselves.

This method also applies to cases where the samples are related or have matching characteristics, like a comparative analysis involving children, parents, or siblings.

The formula for computing the t-value and degrees of freedom for a paired t-test is:

$\begin{aligned}&T=\frac{\textit{mean}1 - \textit{mean}2}{\frac{s(\text{diff})}{\sqrt{(n)}}}\\&\textbf{where:}\\&\textit{mean}1\text{ and }\textit{mean}2=\text{The average values of each of the sample sets}\\&s(\text{diff})=\text{The standard deviation of the differences of the paired data values}\\&n=\text{The sample size (the number of paired differences)}\\&n-1=\text{The degrees of freedom}\end{aligned}$​T=(n)​s(diff)​mean1−mean2​where:mean1 and mean2=The average values of each of the sample setss(diff)=The standard deviation of the differences of the paired data valuesn=The sample size (the number of paired differences)n−1=The degrees of freedom​

Equal Variance or Pooled T-Test Formula

The equal variance t-test is an independent t-test and is used when the number of samples in each group is the same, or the variance of the two data sets is similar.

The formula used for calculating t-value and degrees of freedom for equal variance t-test is:

$\begin{aligned}&\text{T-value}=\frac{\textit{mean}1-\textit{mean}2}{\sqrt{\frac{(n1-1)\times\textit{var}1^2+(n2-1)\times\textit{var}2^2}{n1+n2-2}\times\frac{1}{n1}+\frac{1}{n2}}}\\&\textbf{where:}\\&\textit{mean}1 \text{ and } \textit{mean}2=\text{Average values of each}\\&\text{of the sample sets}\\&\textit{var}1\text{ and }\textit{var}2=\text{Variance of each of the sample sets}\\&n1\text{ and }n2=\text{Number of records in each sample set}\end{aligned}$​T-value=n1+n2−2(n1−1)×var12+(n2−1)×var22​×n11​+n21​​mean1−mean2​where:mean1 and mean2=Average values of eachof the sample setsvar1 and var2=Variance of each of the sample setsn1 and n2=Number of records in each sample set​

and,

$\begin{aligned} &\text{Degrees of Freedom} = n1 + n2 - 2 \\ &\textbf{where:}\\ &n1 \text{ and } n2 = \text{Number of records in each sample set} \\ \end{aligned}$​Degrees of Freedom=n1+n2−2where:n1 and n2=Number of records in each sample set​

Unequal Variance T-Test Formula

The unequal variance t-test is an independent t-test and is used when the number of samples in each group is different, and the variance of the two data sets is also different. This test is also called Welch's t-test.

The formula used for calculating t-value and degrees of freedom for an unequal variance t-test is:

$\begin{aligned}&\text{T-value}=\frac{mean1-mean2}{\sqrt{\bigg(\frac{var1}{n1}{+\frac{var2}{n2}\bigg)}}}\\&\textbf{where:}\\&mean1 \text{ and } mean2 = \text{Average values of each} \\&\text{of the sample sets} \\&var1 \text{ and } var2 = \text{Variance of each of the sample sets} \\&n1 \text{ and } n2 = \text{Number of records in each sample set} \end{aligned}$​T-value=(n1var1​+n2var2​)​mean1−mean2​where:mean1 and mean2=Average values of eachof the sample setsvar1 and var2=Variance of each of the sample setsn1 and n2=Number of records in each sample set​

and,

$\begin{aligned} &\text{Degrees of Freedom} = \frac{ \left ( \frac{ var1^2 }{ n1 } + \frac{ var2^2 }{ n2 } \right )^2 }{ \frac{ \left ( \frac{ var1^2 }{ n1 } \right )^2 }{ n1 - 1 } + \frac{ \left ( \frac{ var2^2 }{ n2 } \right )^2 }{ n2 - 1}} \\ &\textbf{where:}\\ &var1 \text{ and } var2 = \text{Variance of each of the sample sets} \\ &n1 \text{ and } n2 = \text{Number of records in each sample set} \\ \end{aligned}$​Degrees of Freedom=n1−1(n1var12​)2​+n2−1(n2var22​)2​(n1var12​+n2var22​)2​where:var1 and var2=Variance of each of the sample setsn1 and n2=Number of records in each sample set​

Which T-Test to Use

The following flowchart can be used to determine which t-test to use based on the characteristics of the sample sets. The key items to consider include:

• The similarity of the sample records
• The number of data records in each sample set
• The variance of each sample set

Example of an Unequal Variance T-Test

Assume that the diagonal measurement of paintings received in an art gallery is taken. One group of samples includes 10 paintings, while the other includes 20 paintings. The data sets, with the corresponding mean and variance values, are as follows:

Is the difference from 19.4 to 21.6 due to chance alone, or do differences exist in the overall populations of all the paintings received in the art gallery? We establish the problem by assuming the null hypothesis that the mean is the same between the two sample sets and conduct a t-test to test if the hypothesis is plausible.

Since the number of data records is different (n1 = 10 and n2 = 20) and the variance is also different, the t-value and degrees of freedom are computed for the above data set using the formula for the Unequal Variance T-Test.

The t-value is -2.24787. Since the minus sign can be ignored when comparing the two t-values, the computed value is 2.24787.

The degrees of freedom value is 24.38 and is reduced to 24 (the formula definition requires rounding down the value to the least possible integer value).

One can specify a level of probability (alpha level, level of significance, p) as a criterion for acceptance. In most cases, a 5% value can be assumed.

Using the degree of freedom value as 24 and a 5% level of significance, a look at the t-value distribution table gives a value of 2.064. Comparing this value against the computed value of 2.247 indicates that the calculated t-value is greater than the table value at a significance level of 5%. Therefore, it is safe to reject the null hypothesis that there is no difference between means.

Rejecting the null hypothesis means the population set has intrinsic differences, and they are not by chance.

The Bottom Line

A t-test is an inferential statistic used to determine if there is a statistically significant difference between the means of two population samples. It is used in statistics for hypothesis testing and can indicate whether differences between two populations are meaningful or random.

The t-test calculation uses three data: the difference between the mean values from each data set, the standard deviation of each group, and the number of data values. There are different variations of the t-test formula. Which one to use depends on factors such as the similarity of the sample records, the size of each data set, and the variance of each set. However, each variation of the t-test is used to investigate the same statistical question.