To get the best deal on Tutoring, call 1-855-666-7440 (Toll Free)
Top

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:
T Test Formula
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:
Formula for Standard Deviation
Where,
x = Values given
$\bar{x}$ = Mean
n = Total number of values.

Related Calculators
1 Sample T Test
 

T Test Problems

Back to Top
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:

 x
 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
1.52.25
    $\sum (x_{1}-\bar{x_{1}})^{2}$ = 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
-39
4
-11
6
11
8
39
    $\sum (x_{2}-\bar{x_{2}})^{2}$ = 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
*AP and SAT are registered trademarks of the College Board.