Worksheet for how to calculate T Test

This worksheet help you to understand how to calculate the significance of observed differences between the means of two samples when there is null hypothesis. The formula used to calculate the T Test is,

where
x₁ is the mean of first data set
x₂ is the mean of first data set
S₁² is the standard deviation of first data set
S₂² is the standard deviation of first data set
N₁ is the number of elements in the first data set
N₂ is the number of elements in the first data set

Example:
Calculate the T test value whose inputs are 10, 20,30,40,50 and 1, 29,46,78,99
First calculate standard deviation & mean of the given set,

For the data set 10, 20,30,40,50
Total Inputs(N) =(10,20,30,40,50)
Total Inputs(N)=5
Mean(x_m)= (x₁+x₂+x₃...x_n)/N
Mean(x_m)= 150/5
Means(x_m)= 30
SD=sqrt(1/(N-1)*((x₁-x_m)²+(x₂-x_m)²+..+(x_n-x_m)²))
=sqrt(1/(5-1)((10-30)²+(20-30) ²+(30-30) ²+(40-30) ²+(50-30) ²))
=sqrt(1/4((-20) ²+(-10) ²+(0) ²+(10) ²+(20) ²))
=sqrt(1/4((400)+(100)+(0)+(100)+(400)))
=sqrt(250)
=15.8114
Variance=SD²
Variance=15.8114²
Variance=250

For the data set 1, 29,46,78,99
Total Inputs(N) =(1,29,46,78,99)
Total Inputs(N)=5
Mean(x_m)= (x₁+x₂+x₃...x_n)/N
Mean(x_m)= 253/5
Means(x_m)= 50.6
SD=sqrt(1/(N-1)*((x₁-x_m)²+(x₂-x_m)²+..+(x_n-x_m)²))
=sqrt(1/(5-1)((1-50.6)²+(29-50.6) ²+(46-50.6) ²+(78-50.6) ²+(99-50.6) ²))
=sqrt(1/4((-49.6) ²+(-21.6) ²+(-4.6) ²+(27.4) ²+(48.4) ²))
=sqrt(1/4((2460.16)+(466.56)+(21.16)+(750.76)+(2342.56)))
=sqrt(1510.3)
=38.8626
Variance=SD²
Variance=38.8626²
Variance=1510.3

To find T Test:
From above we know that,
x₁ = 30
x₂ = 50.6
S₁² = 250
S₂² = 1510.3
N₁ = 5
N₂ = 5
Substitute these values in the above formula,

T = (30 - 50.6)/√((250/5) + (1510.3/5))
= -1.0979

When you try such calculations on your own, this t-test calculator can be used to verify your results of calculations.