CK-12 Probability and Statistics - Advanced

11.2. The One-Way ANOVA Test http://www.ck12.org

k=the number of groups

When simplified, the formula becomes:

SSB=

k

∑

k= 1

Tk^2

nk

−

T^2

N

where

Tk=sum of the observations in groupK

T=sum of all observations.

Once we calculate this value, we divide by the number of degrees of freedom(K− 1 )to arrive at theMSB.

MSB=

SSB

K− 1

2. Calculating the mean squares within groups(MSW). The mean squares within groups calculation is also
   called thepooled estimate of the population variance. Remember that when we square the standard deviation of a
   sample, we are estimating population variance. Therefore, to calculate this figure, we sum of the squared deviations
   within each group and then divide by the sum of the degrees of freedom for each group.

To calculate theMSWwe first find theSSW, which is calculated using the formula:

∑(Xi 1 −X ̄ 1 )^2 +∑(Xi 2 −X ̄ 2 )^2 +...+∑(Xik−X ̄k)^2

(n 1 − 1 )+(n 2 − 1 )+...+(nk− 1 )

Simplified, this formula states:

SSW=

k

∑

k= 1

nk

∑

i= 1

Xik^2 −

k

∑

k= 1

Tk^2

nk

where

Tk=sum of the observations in groupk

Essentially, this formula sums the squares of each observation and then subtracts the total of the observations squared
divided by the number of observations. Finally, we divide this value by the total number of degrees of freedom in
the scenario(N−K).

MSw=

SSw

N−K

3. Calculate the test statistic. The test statistic is as follows:

F=

MSB

MSW