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# T Test Formula

T Test is often called Student's T test in the name of its founder "Student". T test is used to compare two different set of values. It is generally performed on a small set of data. T test is generally applied to normal distribution which has a small set of values. This test compares the mean of two samples. T test uses means and standard deviations of two samples to make a comparison. The formula for T test is given below:
Where,
$\bar{x_{1}}$ = Mean of first set of values
$\bar{x_{2}}$ = Mean of second set of values
S1 = Standard deviation of first set of values
S2 = Standard deviation of second set of values
n1 = Total number of values in first set
n2 = Total number of values in second set.

The formula for standard deviation is given by:
Where,
x = Values given
$\bar{x}$ = Mean
n = Total number of values.

 Related Calculators 1 Sample T Test

## T Test Problems

Few problems based on T test are given below:

### Solved Examples

Question 1: Find the t-test value for the following two sets of values:
7, 2, 9, 8 and 1, 2, 3, 4?

Solution:

Formula for mean:
$\bar{x}$ = $\frac{\sum x}{n}$
Formula for standard deviation:
$S=\sqrt{\frac{\sum (x-\bar{x})^{2}}{n-1}}$
Calculation for first set:
Number of terms in first set:
n1 = 4
Mean for first set of data:
$\bar{x_{1}}$ = 6.5
Construct the following table for standard deviation:

 x1 $x_{1}-\bar{x_{1}}$ $(x_{1}-\bar{x_{1}})^{2}$ 7 0.5 0.25 2 -4.5 20.25 9 2.5 6.25 8 1.5 2.25 $\sum (x_{1}-\bar{x_{1}})^{2}$ = 29

Standard deviation for first set of data:
S1 = 3.11
Calculation for second set:
Number of terms in second set:
n2 = 4
Mean for second set of data:
$\bar{x_{2}}$ = 2.5
Construct the following table for standard deviation:

 x2 $x_{2}-\bar{x_{2}}$ $(x_{2}-\bar{x_{2}})^{2}$ 1 -1.5 2.25 2 -0.5 0.25 3 0.5 0.25 4 1.5 2.25 $\sum (x_{2}-\bar{x_{2}})^{2}$ = 5

Standard deviation for first set of data:
S2 = 1.29
Formula for t-test value:
$t$ = $\frac{\bar{x_{1}}-\bar{x_{2}}}{\sqrt{\frac{S_{1}^{2}}{n_{1}}+\frac{S_{2}^{2}}{n_{2}}}}$
$t$ = $\frac{6.5-2.5}{\sqrt{\frac{9.667}{4}+\frac{1.667}{4}}}$
t = 2.3764 = 2.38 (approx)

Question 2: Find the t-test value for the following two sets of data:

 x1 9 10 11 12 x2 2 4 6 8

Solution:

Formula for mean:
$\bar{x}$ = $\frac{\sum x}{n}$
Formula for standard deviation:
$S$ = $\sqrt{\frac{\sum (x-\bar{x})^{2}}{n-1}}$
Calculation for first set:
Number of terms in first set:
n1 = 4
Mean for first set of data:
$\bar{x_{1}}$ = 10.5
Construct the following table for standard deviation:

 x1 $x_{1}-\bar{x_{1}}$ $(x_{1}-\bar{x_{1}})^{2}$ 9 -1.5 2.25 10 -0.5 0.25 11 0.5 0.25 12 1.5 2.25 $\sum (x_{1}-\bar{x_{1}})^{2}$ = 5

Standard deviation for first set of data:
S1 = 1.291
Calculation for second set:
Number of terms in second set:
n2 = 4
Mean for second set of data:
$\bar{x_{2}}$ = 5
Construct the following table for standard deviation:

 x2 $x_{2}-\bar{x_{2}}$ $(x_{2}-\bar{x_{2}})^{2}$ 2 -3 9 4 -1 1 6 1 1 8 3 9 $\sum (x_{2}-\bar{x_{2}})^{2}$ = 20

Standard deviation for first set of data:
S2 = 2.582
Formula for t-test value:
$t$ = $\frac{\bar{x_{1}}-\bar{x_{2}}}{\sqrt{\frac{S_{1}^{2}}{n_{1}}+\frac{S_{2}^{2}}{n_{2}}}}$
$t$ = $\frac{10.5-5}{\sqrt{\frac{1.667}{4}+\frac{6.667}{4}}}$
t = 3.8105 = 3.81 (approx)

 More topics in T Test Formula T-Distribution Formula
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