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T Test Calculator
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T Test Calculator (Critical Value Calculator or T Critical Value Calculator or T Distribution Calculator) performs as a hypothesis of statistics test on which the statistic test follows the t distribution of the student if it is supported by null hypothesis.

It is the widely and most commonly used in the statistic test and it follows a normal distributions if the values of the scale term statistics (within certain condition) follow the t distribution of the student.

T Test Formula

t = $\frac{\bar{x_{1}} - \bar{x_{2}}}{\sqrt{\frac{s_{1}^{2}}{n_{1}} + \frac{s_{2}^{2}}{n_{2}}}}$

where,
$\bar{x_{1}}$ = The mean of first set of values
$\bar{x_{2}}$ = The mean of second set of values
s12 = The standard deviation of first set of values
s22 = The standard deviation of second set of values
n1 = Total number of elements in the first set of values
n2 = Total number of elements in the second set of values.

## T Test Steps

Step 1 :

Observe the given two set of values.

Step 2 :

Calculate the mean and standard deviation for each and individual set of values.

Step 3 :

Apply the formula to calculate the T-Test value
t = $\frac{\bar{x_{1}} - \bar{x_{2}}}{\sqrt{\frac{s_{1}^{2}}{n_{1}} + \frac{s_{2}^{2}}{n_{2}}}}$
where,
$\bar{x_{1}}$ = The mean of first set of values
$\bar{x_{2}}$ = The mean of second set of values
s12 = The standard deviation of first set of values
s22 = The standard deviation of second set of values
n1 = Total number of elements in the first set of values
n2 = Total number of elements in the second set of values.

## T Test Example

1. ### Calculate the t-test for the below two set of values:First set: 5,20,40,80,100Second set: 1,29,46,78,99

Step 1 :

Given two set of values:
First set: 5,20,40,80,100
Second set: 1,29,46,78,99

Step 2 :

Let first set of values = x1
second set of values = x2
to find mean for x1 = 5,20,40,80,100
$\bar{x_{1}}$ = $\frac{5 + 20 + 40 + 80 + 100}{5}$ = $\frac{245}{5}$ = 49
to find mean for x2 = 1,29,46,78,99
$\bar{x_{2}}$ = $\frac{1 + 29 + 46 + 78 + 99}{5}$ = $\frac{253}{5}$ = 50.6

 x1 (x1 - $\bar{x_{1}}$)2 x2 (x2 - $\bar{x_{2}}$)2 5 (5 - 49)2 = 1936 1 (1 - 50.6)2 = 2460.16 20 (20 - 49)2 = 841 29 (29 - 50.6)2 = 466.56 40 (40 - 49)2 = 81 46 (46 - 50.6)2  = 21.16 80 (80 - 49)2 = 961 78 (78 - 50.6)2 = 750.76 100 (100 - 49)2 = 2601 99 (99 - 50.6)2 = 2342.56

So, $\sum{(x_{1} - \bar{x_{1}})^2}$ = 1936+841+81+961+2601 = 6420

$\sum{(x_{2} - \bar{x_{2}})^2}$ = 2460.16 + 466.56 + 21.16 + 750.76 + 2342.56 = 6041.2

Standard deviation for first data(s1)2 = $\frac{\sum{(x_{1} - \bar{x_{1}})^2}}{n_{1}}$ = $\frac{6420}{5}$ = 1284

Standard deviation for first data(s2)2 = $\frac{\sum{(x_{2} - \bar{x_{2}})^2}}{n_{2}}$ = $\frac{6041.2}{5}$ = 1208.24

Step 3 :

T-Test value is given by
t = $\frac{\bar{x_{1}} - \bar{x_{2}}}{\sqrt{\frac{s_{1}^{2}}{n_{1}} + \frac{s_{2}^{2}}{n_{2}}}}$

t = $\frac{49 - 50.6}{\sqrt{\frac{1284}{5} + \frac{1208.24}{5}}}$

t = -0.0716654866

t = -0.072

t = -0.072

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