T Test Meaning Types Examples Formula Calculation

T-Test Meaning

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The type of T-test to be conducted is decided by whether the samples to be analyzed are from the same category or distinct categories. The inference obtained in the process indicates the probability of the mean differences to have happened by chance. The test is useful when comparing population age, length of crops from two different species, student grades, etc.

Key Takeaways

A T-test is a statistical method of comparing the means or proportions of two samples gathered from either the same group or different categories. It is aimed at hypothesis testing, which is used to test a hypothesis pertaining to a given population. It is the difference between population means and a hypothesized value. One-sample, two-sample, paired, equal, and unequal variance are the types of T-tests users can use for mean comparisons.

T-Test Explained

A T-test studies a set of data gathered from two similar or different groups to determine the probability of the difference in the result than what is usually obtained. The accuracy of the test depends on various factors, including the distribution patterns used and the variants influencing the collected samples. Depending on the parameters, the test is conducted, and a T-value is obtained as the statistical inference of the probability of the usual resultant being driven by chance.

For example, if one wishes to figure out if the mean of the length of petals of a flower belonging to two different species is the same, a T-test can be done. The user can select petals randomly from two other species of that flower and come to a standard conclusion. The final T-test interpretation could be obtained in either of the two ways:

A null hypothesisNull HypothesisNull hypothesis presumes that the sampled data and the population data have no difference or in simple words, it presumes that the claim made by the person on the data or population is the absolute truth and is always right. So, even if a sample is taken from the population, the result received from the study of the sample will come the same as the assumption.read more signifies that the difference between the means is zero and where both the means are shown as equal.An alternate hypothesis implies the difference between the means is different from zero. This hypothesis rejects the null hypothesis, indicating that the data set is quite accurate and not by chance.

This T-test, however, is only valid and should be done when the mean or average of only two categories or groups needs to be compared. As soon as the number of comparisons to be made is more than two, conducting this is not recommended.

Assumptions

The test runs on a set of assumptions, which are as follows:

The measurement scale used for such hypothesis testingHypothesis TestingHypothesis Testing is the statistical tool that helps measure the probability of the correctness of the hypothesis result derived after performing the hypothesis on the sample data. It confirms whether the primary hypothesis results derived were correct.read more follows a set of continuous or ordinal patterns. The accounted parameters and variants influencing the samples and surrounding the groups are based on the standard consideration.The tests are completely based on random sampling. As no individuality is maintained in the samples, the reliability is often questioned. When the data is plotted with respect to the T-test distribution, it should follow a normal distributionNormal DistributionNormal Distribution is a bell-shaped frequency distribution curve which helps describe all the possible values a random variable can take within a given range with most of the distribution area is in the middle and few are in the tails, at the extremes. This distribution has two key parameters: the mean (µ) and the standard deviation (σ) which plays a key role in assets return calculation and in risk management strategy.read more and bring about a bell-curved graph.For a clearer bell curveBell CurveBell Curve graph portrays a normal distribution which is a type of continuous probability. It gets its name from the shape of the graph which resembles to a bell. read more, the sample sizeSample SizeThe sample size formula depicts the relevant population range on which an experiment or survey is conducted. It is measured using the population size, the critical value of normal distribution at the required confidence level, sample proportion and margin of error.read more needs to be bigger.The variance should be such that the standard deviationsStandard DeviationsStandard deviation (SD) is a popular statistical tool represented by the Greek letter ‘σ’ to measure the variation or dispersion of a set of data values relative to its mean (average), thus interpreting the data’s reliability.read more of the samples are almost equal.

Types

Some of the widely used T-test types are as follows:

#1 – One-Sample T-Test

While performing this test, the mean or average of one group is compared against the set average, which is either the theoretical value or means of the population. For example, a teacher wishes to figure out the average height of the students of class 5 and compare the same against a set value of more than 45 kgs.

The teacher first randomly selects a group of students and records individual weights to achieve this. Next, she finds out the mean weight for that group and checks if it meets the standard set value of 45+. The formula used to obtain one-sample t-test results is:

Where,

T = t-statisticm = mean of the group= theoretical mean value of the populations = standard deviation of the groupn = sample size

#2 – Independent Two-Sample T-Test

This is the test conducted when samples from two different groups, species, or populations are studied and compared. It is also known as an independent T-test. For example, if a teacher wants to compare the height of male students and female students in class 5, she would use the independent two-sample test.

The T-test formula used to calculate this is:

mA – mB = means of samples from two different groups or populationsnA – nB = respective sample sizess2 = standard deviation or common variance of two samples

#3 – Paired Sample T-Test

This hypothesis testing is conducted when two groups belong to the same population or group. The groups are studied either at two different times or under two varied conditions. The formula used to obtain the t-value is:

T = t-statisticm = mean of the group = theoretical mean value of the populations = standard deviation of the groupn = sample size

#4 – Equal Variance T-Test

This test is conducted when the sample size in each group or population is the same or the variance of the two data sets is similar. It is also referred to as pooled T-test. The formula applied here is as follows:

Mean1 and mean2 = average value of each set of samplesvar1 and var2 = variance of each set of samplesn1 and n2 = number of records in each set

#5 – Unequal Variance T-Test

The unequal variance testing is used when the variance and the number of samples in each group are different. It is often referred to as Welch’s test, and the formula is:

mean1 and mean2 = Average value of each set of samplesvar1 and var2 = Variance of each set of samplesn1 and n2 = number of records in each set

Example With Calculation

Let us consider the scores for each subject in the examination held in two phases.

Step 1: Subtract the marks scored in both the phases

Step 2: Add up all the differences, i.e., -55

Step 3: Square up the differences

Step 4: Add up all the squares of difference, i.e., 983

Step 5: Usage of formula to calculate the T value

= -9.16/√ {983-(-55)2/6)}/ (6-1) *6= -9.16/√15.96= -9.16/3.99

T Value = -2.29

Now, get the degrees of freedomDegrees Of FreedomDegrees of freedom (df) refers to the number of independent values (variable) in a data sample used to find the missing piece of information (fixed) without violating any constraints imposed in a dynamic system. These nominal values have the freedom to vary, making it easier for users to find the unknown or missing value in a dataset.read more. To obtain this, subtract 1 from the sample size (6 – 1 =5). The next thing is to find out the p-valueP-valueP-Value, or Probability Value, is the deciding factor on the null hypothesis for the probability of an assumed result to be true, being accepted or rejected, & acceptance of an alternative result in case of the assumed results rejection. read more, which, if smaller in value, supports the null hypothesis result. For example, if the p-value is something around 0.9, i.e., 90%, it indicates that the T-value obtained has the probability of being a random observation. On the other hand, if the p-value is around 0.025, i.e., 2.5%, the result or t-value obtained is significant.

T-Test Meaning#

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T-Test Explained#

Assumptions#

Types#

#1 – One-Sample T-Test#

#2 – Independent Two-Sample T-Test#

#3 – Paired Sample T-Test#

#4 – Equal Variance T-Test#

#5 – Unequal Variance T-Test#

Example With Calculation#

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