T Table

A T Table represents the critical values of the t distribution curve. This curve is similar to a normal distribution curve but it tells about the number of observations that may vary from the mean value. T-statistic is useful when the sample size is smaller and the standard deviation is unknown.

Read this article to gain a complete understanding of t distribution, t table, t value, and the t-tests.

Understanding T Distribution

The T distribution is also known as the Student’s T Distribution. It is a form of continuous probability distribution where the mean of a normal distribution is calculated, but the sample size is smaller. This distribution is useful when the standard deviation is not known.

The t-distribution is symmetric, bell-shaped and looks similar to a normal distribution curve. However, the tails are heavier which means more observations deviate from the mean.

The T distribution has (n-1) degree of freedom. The t-distribution becomes closer to the normal distribution (z statistic) as the degree of freedom is increased.

The Student’s T distribution is used in null hypothesis testing.

• In a null hypothesis testing, you have a null statement which is considered as true.
• There is a counter statement referred to as ‘alternate hypothesis’ to reject the hypothesis.
• If a value lies within the central region, the null hypothesis is accepted.
• If the t-score lies within tails, the null hypothesis is rejected.

The T distribution is used when the sample size is smaller i.e less than 30. For larger sample sizes, the distribution looks almost the same as the normal distribution curve.

What is T-value?

T-statistic is required because the standard deviation of the population is not known for a small sample. T-statistic allow for the use of the sample standard deviation which measures a specific sample’s variation.

T-value measure the size of difference relative to the variation in the sample data. It is the calculated difference represented in terms of standard error. As defined on Wikipedia

In statistics, the t-statistic is the ratio of the departure of the estimated value of a parameter from its hypothesized value to its standard error.

Consider a random sample of n observations with mean and standard deviation σ, from a normally distributed population with mean μ. The t-statistic is calculated as follows:

What is T Table?

The values in the t distribution table represent the critical values of the t distribution curve with df for the selected tail probabilities. The t-table gives us the probability of an absolute t value with a given degree of freedom lying above the tabulated value.

The table rows represent the upper tail area and are labeled as ‘df’ which tells us about the ‘Degrees of Freedom’. There are three column headers – cumulative probability or percent, one-sided alpha or one-tail, and two-sided alpha or two-tail. The body contains the t values.

Or

Get Z Table PDF here >>

The degrees of freedom refers to the number of observations that may vary after the sample mean of the population has been calculated.

Example

Consider a sample size of 3 and Let’s say n = 10, the df= 10-1 = 9. If significance level a is 0.10 then a/2 = 0.05. From the table we can observe that t-value = 1.833.

What is t-test?

The t-test is used to find out if the difference between the groups has occurred by chance. A greater magnitude of T means that there is a significant difference between the examined t – value and the variation in the entire population of the sample. The null hypothesis is rejected when the magnitude of t is higher.

A minimal value of t, approaching 0, represents a minute difference.

There are three types of t-test: one-tailed test, two-tailed test, and paired test.

One-tailed Test

In a one-tailed test, the critical area of distribution is one-sided. This t-test is used to find a difference between the population mean and a hypothesized value.

A one-tailed test gives the information related to one direction only i.e. a certain value is either less than or greater than the mean value. If the sample value falls within the one-sided critical area, the null hypothesis is rejected and the alternative hypothesis is accepted.

You need to choose the direction before applying the test. For example, suppose the null hypothesis states that the mean is equal to 10 and you choose ‘less than’ direction.  Using a one-tail test, you can only determine if certain value X is less than 10 or not.

Two-tailed Test

In a two-tailed test, the region of rejection lies on both sides of the data distribution. A two-tailed test is performed to know if both means are different from one another.

A two-tailed test gives more information as compared to a one-tailed test. It provides more accuracy and unbiased results. The results can be one of the possible options: one mean is greater than the other, one mean is lower than the other mean or both means are similar.

You don’t need to choose the direction before applying the test. For example, suppose the null hypothesis states that the mean is equal to 10. Using a two-tail test, you can either determine if certain value X is less than 10, equals to 10 or greater than 10.

Source: CliffsNotes

Paired T-Test

The paired sample t-test is also referred to as ‘Dependent sample t-test’. This test is used to determine whether the mean difference between the two sets is zero.

A paired t-test is used when the two groups have paired observations as each entity is measured twice. Paired t-test is used in case-control studies where test results need to be compared ‘before’ and ‘after’ a certain condition X.

For example, the condition ‘X’ can be scenarios where you want to evaluate the effectiveness of a drug, A training program or an experiment.

Watch this video to know more about t-tests.