I'm using a learning tool program and came across a study question
that asked me to compute the variance of sample means. After getting
the answer wrong, the program gave me the correct answer of 20.083 as
the variance of the sample means which are located in the table below.
However, the program did not clearly show me how to compute this. I
need to know how to calculate the variance of the sample means to
learn this problem correctly. The total number of students is 300,
which were divided into 4 versions of 75 each. The exam used by the
professor in the problem is worth 200 points. The problem:

ANOVA: Mean squares and the common population variance
In an effort to counteract student cheating, the professor of a large
class created four versions of a midterm exam, distributing the four
versions among the  students in the class, so that each version was
given to  students. After the exam, the professor computed the
following information about the scores (the exam was worth  points):

Population        Sample Mean           Sample Variance
75                    152.6                    376.1
75                    146.3                    394.5
75                    155.6                    289.7
75                    156.0                    435.1

The professor is willing to assume the populations of the scores from
which the above samples were drawn have the same mean and the same
variance. Using the column of sample means, how do I compute the
variance of the sample means giving me the answer of 20.083?

To calculate the mean squared varience, take the individual mean for
each group and subtract it from the grand mean, square this value, sum
the 4 values, and divide over the degrees of freedom.

Between Groups Varience = [(Xbar1-XbarGrand)^2 +(Xbar2-XbarGrand)^2 +
(Xbar3-XbarGrand)^2 + (Xbar4-XbarGrand)^2] / k-1

In the above n =population / group

The Grand mean = Sum of all scores / n or for simplicity in this case,
the sum of the group means / 4

K-1 = degrees of freedom where k is the number of groups or means.

So.  Here we go.

First calculate your grand mean:

XbarGrand (Grand Mean) = (152.6+146.3+155.6+156.0)/4 = 610.5/4 = 152.625


Now, plug you grand mean and group means into the formula above: 

Between Groups Varience = [(152.5-152.625)^2 +(146.3-152.625)^2 +
(155.6-152.625)^2 + (156.0-152.625)^2] / k-1

[.015625+40.0056+8.850+11.39]/4-1 = 60.261/3 = 20.08


Hope this helps.

Patrick

Thankyou for the help. I applied what was said and it worked perfectly.

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