**Introduction**

An Irish statistician, W.S. Gossett in 1908, applied this test for test for testing the significance of difference between the means of two different samples of small size. It was named ‘t-test’. The pen name of Gossette was student, hence this test is called Student’s ‘t’ test. Student t-test is also described as t-distribution or t-ratio.

Student’s t-test is applied to small samples only. It is used only for two or less groups and the observation is less than 30.

**I) Student’s t-test for a Single Population**

Student’s t-test represents the ratio of the difference of population parameter and corresponding statistic and the standard error of the statistic with n-1 degrees of freedom if the sample size is n. This can be represented as follows with (n-1) degrees of freedom if the sample is represented by n. The degree of freedom for t = size of sample – 1. If the size of sample is represented by n then the degree of freedom will be n – 1.

If a sample of size n is drawn from a normal population with mean µ and variance σ^{2}, the student’s t-test will be:

Where ** X̅** = sample mean

** µ** = population mean

** n** = sample size and

** S** = standard deviation of the sample

**II) Student’s t-test for Two Samples**

The student t-test for two samples or for two variables is the ratio of difference between the means of samples and the standard error of difference between two means. The t-ratio is obtained by using the following formula:

Here **X̅**_{1}= Mean of one variable or one sample

**X̅2**= Mean of second variable or second sample

** S.E. _{D}** = Standard error of difference between two means

**III) How t-Distribution different from Normal Distribution **

The graph of t-distribution is similar to normal distribution except for the following differences:

a) The normal distribution curve is higher in the middle than t-distribution curve.

b) The largest differences between normal distribution and t distributions occur in the tails which is the most important area in statistical tests.

c) t-distribution has a greater spread sideways than the normal distribution curve. It means there is more area in the tails of t-distribution.

#### **IV) Properties of t-Distribution**

a) t-distribution is a symmetrical distribution with mean as zero.

b) The graph of t-distribution is similar to normal distribution except for the following two differences:

i) The normal distribution curve is higher in the middle than t-distribution curve.

ii) t-distribution has a greater spread sideways than the normal distribution curve. It means there is more area in the tails of t-distribution.

c) The t-distribution curve is asymptotic to X-axis, i.e. it extends to infinity on either side.

d) The shape of t-distribution curve varies with the degree of freedom.

e) The larger is the number of degree of freedom the more closely t-distribution resembles standard normal distribution.

f) Sampling distribution of t does not depend on population parameter. It depends on degree of freedom (n-1).

**Degree Of Freedom**

In **statistics**, the number of **degrees of freedom** is the number of values in the final calculation of a statistic that are free to vary. The number of independent ways by which a dynamic system can move, without violating any constraint imposed on it, is called number of **degrees of freedom**. It is denoted by df.

a. When data is given in the form of a series of variables in a row or column, then:

where, n is the number of variables in the series in a row or column.

b. When the number of frequencies are put in cells of a contingency table, the degree of freedom will be the product of number of rows less one and the number of columns less one, i.e.,

Here, R is the number of rows and C is the number of columns.

**Types of t-Tests**

** I) One Sample Test**

This function gives a single sample t-Test with a confidence interval for the mean difference.

The single sample t method tests a null hypothesis that the population mean is equal to a specified value. If this value is zero (or not entered) then the confidence interval for the sample mean is given.

The test statistic is calculated as:

Where x̅ is the sample mean, s^{2} is the sample variance, n is the sample size, µ is the specified population mean and t is a Student t quantile with (n-1) degree of freedom.

**II) Two Sample Test**

** a) Paired t-Test: **This function gives a paired t-Test, confidence intervals for the difference between a pair of means and, optionally, limits of agreement for a pair of samples.

The paired t-Test provides an hypothesis test of the difference between population means for a pair of random samples whose differences are approximately normally distributed. Please note that a pair of samples, each of which are not from normal a distribution, often yields differences that are normally distributed.

The test statistic is calculated as:

Where d̅ is the mean difference, s^{2} is the sample variance, n is the sample size and t is a Student t quantile with (n-1) degree of freedom.

** b) Unpaired t-Test: **This function gives an unpaired two sample t-Test with a confidence interval for the difference between the means.

The unpaired t-Tests the null hypothesis that the population means related to two independent, random samples from an approximately normal distribution are equal.

**References**

i) http://www.statsdirect.com/help/parametric_methods/single_sample_t.htm

ii) http://www.stat.columbia.edu/~martin/W2024/R2.pdf

### About Suman Maity:

Suman Maity is a B.Tech(Electrical Engineering) . Currently he is working as an Analyst Intern with NikhilGuru Consulting Analytics Service LLP, Bangalore. He has prior worked for around 1+ year with T&M Services Consulting Pvt Ltd and HR Chamber Outsourcing Pvt Ltd.

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