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T Score – Formula and Examples


A t score is a test statistic which is used in the testing of equality of sample means in one or two samples (of small sizes). Suppose you are given the value of a particular random variable which follows the t distribution. The t score tell us how many standard deviations away from the mean that particular value lies.

The formula to calculate the t score of a variable X following the t distribution is:

t score= (X-µ)/σ

where, X is the value taken by the random variable, µ is the mean of the distribution and σ is the standard deviation of the distribution.

So the procedure to calculate the t score is to simply subtract the mean and divide by the value of the standard deviation.

T score for sample mean:

Suppose we draw a sample of size ‘n’ from the t distribution then the sample mean also follows the t distribution with mean µ and standard deviation σ/√n.

In this case the formula for t score of sample mean is,

t score for sample mean= (x̄-µ)/(σ/√n)

If the population variance σ is unknown we use the sample variance ‘s’ instead.

When we apply the T test for testing equality of means in one or two samples we use the above t score as our test statistic and test the null hypothesis by comparing the statistic with the critical t value.

Example of a t score calculation and one sample t test:

The lifetimes (in ‘000 hours) of a random sample of 10 bulbs from a company are as follows:

4.2, 4.6, 3.9, 4.1, 5.2, 3.8, 3.9, 4.3, 4.4, 5.6

Can we accept the hypothesis that average lifetime of bulbs is 4000 hours.

Solution: The null hypothesis is that the bulb has an average life of 4000 hours.

We calculate the mean as x̄=44/10=4.4

The sample standard deviation is calculated using the formula  s=  √{∑ (xi-x̄)2/n-1}= 0.589

Then t score for sample mean= (x̄-µ)/(s/√n) = 2.147

The critical t table value for 9 degrees of freedom at 5% level of significance is 2.262.

Since the calculated test statistic is lesser than the table value we accept the null hypothesis and conclude that the mean life of bulb is indeed 4000 hours.

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