Formula to Calculate Student’s T Distribution

The formula to calculate T-distribution (also popularly known as student’s T-distribution) is shown as subtracting the population mean (mean of the second sample) from the sample mean ( mean of the first sample) that is [ x̄ – μ ] which is then divided by the standard deviation of means. It is initially divided by the square root of n, the number of units in that sample[ s ÷ √(n)].

The T-distribution is a kind of distribution that looks almost like the normal distribution curve or bell curve but with a bit fatter and shorter tail. When the sample size is small, it will use this distribution instead of the normal distribution.

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Where,

  • x̄ is the sample meanμ is the population means is the standard deviationn is the size of the given sample

Calculation of T Distribution

The student’s T-distribution calculation is quite simple, but the values are required. For example, one needs the population means, which is the universe means, which is nothing but the average population. Whereas the sample mean is required to test the authenticity of the population mean, whether the statement claimed based on population is true, and sample, if any, taken will represent the same statement. So, the T-distribution formula subtracts the sample mean from the population mean, divides it by standard deviation, and multiples it by the square rootSquare RootThe Square Root function is an arithmetic function built into Excel that is used to determine the square root of a given number. To use this function, type the term =SQRT and hit the tab key, which will bring up the SQRT function. Moreover, this function accepts a single argument.read more of the sample size to standardize the value.

However, since there is no range for T-distribution calculation, the value can go weird, and we won’t be able to calculate probability as the student’s T-distribution has limitations in arriving at a value. Hence, it is only useful for a smaller sample size. Also, one needs to find that value from the student’s T-distribution table to calculate probability after arriving at a score.

Examples

Example #1

Consider the following variables given to you:

  • Population mean = 310Standard deviation = 50Size of the sample = 16Sample mean = 290

Calculate the T-distribution value.

Solution:

Use the following data for the calculation of the T-distribution.

So, the calculation of the T-distribution can be as follows:

Here, given all the values. Then, we need to incorporate the values.

We can use the T-distribution formula:

Value of t = (290 – 310) / (50 / √16)

T Value = -1.60

Example #2

SRH company claims that its employees at the analyst level earn an average of $500 per hour. A sample of 30 employees at the analyst level was selected. Their average hourly earnings were $450, with a sample deviation of $30. And assuming their claim to be true, calculate the T-distribution value, which shall use to find the probability for T–distribution.

Here, we have all the values. So, we need to incorporate the values.

Value of t = (450 – 500) / (30 / √30)

T Value = -9.13

Hence, the value for T-score is -9.13.

Example #3

Universal college board administered an IQ level test to 50 randomly selected professors. And the result they found was that the average IQ level score was 120 with a variance of 121. Assume that the t score is 2.407. What is the population mean for this test, which would justify the T-score value as 2.407?

Here, all the values are given along with the T-value. This time, we need to calculate the population mean instead of the T-value.

Again, we will use the available data and calculate the population means by inserting the values in the formula below.

The sample mean is 120, the population means is unknown, the sample standard deviation will be the square root of the variance, which would be 11, and the sample size is 50.

So, the population mean(μ) calculation   can be as follows:

Value of t = (120 – μ ) / (11 / √50)

2.407 = (120 – μ ) / (11 / √50)

-μ = -2.407 * (11/√50)-120

Population Mean (μ) will be:

μ = 116.26

Hence, the value for the population mean will be 116.26.

Relevance and Use

The T-distribution (and those associated T-score values) is used in 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 when determining if one should reject or accept the null hypothesis.

In the above graph, the central region will be the acceptance area, and the tail region will be the rejection region. In this graph, which is a two-tailed test, the blue shaded will be the rejection region. One can describe the tail region’s area with either the T-scores or the z-scoresThe Z-scoresThe Z-score of raw data refers to the score generated by measuring how many standard deviations above or below the population mean the data, which helps test the hypothesis under consideration. In other words, it is the distance of a data point from the population mean that is expressed as a multiple of the standard deviation.read more. For example, the image on the left will depict an area in the tails of five percent (which is 2.5% on both sides). The Z-score should be 1.96 (taking the value from the Z-table), representing that 1.96 standard deviations from the average or the mean. One can reject the 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 if the value of the Z-score is less than the value of -1.96 or the value of the Z-score is greater than 1.96.

This distribution shall describe earlier when one has a smaller sample size (mostly under 30) or if one doesn’t know the population variance or standard deviation. It would always be the case for practical purposes (that is, in the real world). On the other hand, if the sample size is large enough, then the two distributions will be practically similar.

This article is a guide to T-Distribution Formula. Here, we learn how to calculate students’ T-distribution value and population mean (μ), along with practical examples in Excel and a downloadable Excel template. You can learn more about Excel modeling from the following articles: –

  • Calculate Sample SizeCalculate Sample 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 moreBell Curve FormulaBell Curve FormulaBell 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 moreFormula of Binomial Distribution Skewness Formula