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Introduction to t-test theory for surveys

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Table of Contents

01

What is a T-test?

The T-test is an inferential statistic used to test differences between the means of two groups. The T-test is generally used when the data sets follow a normal distribution and may have unknown variances. In the t-test, the data for grouping variables is categorical, while for the dependent variable is an interval scale. 

Used as a hypothesis testing tool, a t-test allows you to test assumptions made about a population. It helps you to understand if there is any effect of the process on the population. 

In order to establish the statistical significance, when calculating the t-test, you need to note the t-statistics, t-distribution, and the degree of freedom.

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02

When can you use the T-test?

A T-test is used when you want to compare the means between two groups. It cannot be of use when there are more than two groups. In such cases, you should use ANOVA or post-hoc tests. 

As a parametric test of difference, a T-test also makes similar assumptions as another parametric test. T-test assumes that:

  • The scale of measurement which is applied to the collected data is continuous, i.e., an ordinal scale. 
  • The data collected is a randomly selected segment of the population, i.e., a representative.
  • When the data is plotted, it shows a normal distribution, i.e., a bell-shaped distribution curve.
  • Equal variance exists when the standard deviation of samples is also approximately equal, i.e., homogeneity of variance.
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03

How to determine which T-test to use?

  • The groups which are being compared, do they come from a single population or from two different populations?
  • Do you want to test the difference between the groups in a specific direction?

These are two things you need to consider before selecting the t-test that you should use. 

One-sample T-test

    • When you have to compare one group against a standard value, you should use a one-sample t-test.

Two-sample T-test

    • When you have groups that are derived from two different populations, you need to use a two-sample t-test. This t-test is also called Independent T-test. 

Paired T-test

    • This t-test is used when you have the groups from a single population. 

One-tailed T-test

    • Use this T-test when you want to determine if the mean of one population is greater or less than the other population means. 

Two-tailed T-test

    • This T-test should be used when you want to determine if two populations are different from one another.
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04

Calculating T-test

In order to calculate T-test, you need three data values: 

Mean difference, i.e., a difference of the means from each set of data

The standard deviation of each group

The number of data values from each group

T-test Formula 

The formula of the Two-sample T-test is as follows:

Ttest

T: T-value

x1 and x2: the difference between means of the two groups 

S2: pooled standard error of the two groups

n1 and n2: the number of data values in each group

In the calculation of the T-test, the outcome you get as a result is called T-value. This T-value is compared to a critical value table, called the T-Distribution Table. The comparison is made to determine the effect of chance on the difference and determine if the difference is outside that chance range. 

The T-test is used to question if the difference between the groups represents a true difference or it is a meaningless random difference.

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05

T-distribution Tables

A T-distribution table is available in the following formats:

  • One-tail: used to assess a case that has a fixed value or range and a clear direction, positive or negative.
  • Two-tail: used for range-bound analysis, for instance, questioning if the coordinates fall between -1 and +1.

You can use software that supports fundamental statistical functions like MS Excel to calculate.

06

T-values and Degrees of Freedom

These are the outputs from the T-test. 

T-value

  • The difference between the mean of two sets/variation within the sets.
  • The numerator of the ratio, the difference between the means of the sets, is easy to calculate. However, the denominator, variation within the sets, is complex as it depends on the type of value involved. 
  • The T-value is also called T-score. 
    • When T-value is higher, it indicates a significant difference between the two sets. 
    • When you have a smaller T-value, it suggests that there is a similarity between the groups. 

Degrees of Freedom

  • Is a term that refers to the values that have the freedom to vary. This value is vital to assess the importance and validity of the null hypothesis. Calculation of the degrees of freedom is based on the number of data records available. 
  • By calculating T-value against the T-Distribution Table, you can determine if T-value is greater than expected by chance. If the t-value is greater, you can reject the null hypothesis, concluding that the two groups are different.

06

FAQs

The T-test is used to measure the difference between the means of groups divided by the pooled Standard Error of the means of two groups. 

The calculation gives us an at-value that represents the magnitude of the difference between the means of two groups. Also, it determines if the difference exists purely by chance, i.e., p-value.

In case you have a big sample size, you can say that your T-value is significant if the value (absolute) is higher or equal to 1.96.

P-value: Probability Value, this tells you the likelihood of your data existing under the null hypothesis. It tells you the likelihood of seeing a test statistic as extreme or more extreme as the one calculated under a statistical test in case the null hypothesis is true. 

For example, if your p-value is 0.5, this implies that 5% of the time, you may see a test statistic as extreme as the one found in case the null hypothesis is true.

The three types of T-tests are:

  • One-sample T-test
  • Two-sample T-test
  • Paired-sample T-test

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