How to do a Variance Table Problem

You're given a set of either n = 5 or n = 7 delicious pieces of data, like this:

First, you're asked to compute measures of centrality: the mean, median, and mode of these data.

To get the mean, you add up the data and divide by n.
- \(\mu\) = (33+36+38+33+42)/5 = 36.4
The get the median, sort the data in order and find the middle value.
- 33 33 36 38 42
- 36 is the median.
The mode is the most popular value. There can be none at all or more than one, but I make sure there is one, and only one.
- In this data set, 33 is the most commonly occurring value. It be the mode.
- 33 33 36 38 42

Then you have to calculate measures of dispersion. These are range, variance, and standard deviation.

Range is calculated by subtracting the smallest value from the largest value.
- 42 - 33 = 9.
- The range is 9.
Variance and standard deviation are calculated using a variance table. The variance table breaks down the following formula for variance into steps:

Variance = \sigma^2 = \dfrac{\Sigma (X-\mu)^2}{n}

33
36
38
33
42

The data are listed in the first column. The mean of the data, as we've already calculated, is \(\mu\) = 36.4.

Now fill out the second column by subtracting \(\mu\) from each value.

\(X\)

\(X - \mu\)

\((X - \mu)^2\)

33	33 - 36.4 = -3.4
36	36 - 36.4 = -0.4
38	38 - 36.4 = 1.6
33	33 - 36.4 = -3.4
42	42 - 36.4 = 5.6

\(\mu\) = 36.4

Now square the values in this second column and put the results in the third column. These are your squares.

\(X\)

\(X - \mu\)

\( (X - \mu)^2 \)

33	-3.4	(-3.4)² = 11.56
36	-0.4	(-0.4)² = 0.16
38	1.6	(1.6)² = 2.56
33	-3.4	(-3.4)² = 11.56
42	5.6	(5.6)² = 31.36

\(\mu\) = 36.4

Now add up those squared values to get the sum of squares, \(\Sigma (X - \mu)^2\).

\(X\)

\(X - \mu\)

\( (X - \mu)^2 \)

33	-3.4	11.56
36	-0.4	0.16
38	1.6	2.56
33	-3.4	11.56
42	5.6	31.36

\(\mu\) = 36.4

\(X\)

\(X - \mu\)

\( (X - \mu)^2 \)

Sum of squares = \(\Sigma (X - \mu)^2\) = 57.2

The \(\Sigma\) symbol means summation, i.e., add up all the squares.

33	-3.4	11.56
36	-0.4	0.16
38	1.6	2.56
33	-3.4	11.56
42	5.6	31.36

\(\mu\) = 36.4

\(X\)

\(X - \mu\)

\( (X - \mu)^2 \)

Sum of squares = \(\Sigma (X - \mu)^2\) = 57.2

Divide the sum of squares by n to get the variance.

Variance = \(\sigma^2 = \frac{57.2}{5} \) = 11.44

33	-3.4	11.56
36	-0.4	0.16
38	1.6	2.56
33	-3.4	11.56
42	5.6	31.36

\(\mu\) = 36.4

\(X\)

\(X - \mu\)

\( (X - \mu)^2 \)

Sum of squares = \(\Sigma (X - \mu)^2\) = 57.2

Variance = \(\sigma^2 = \frac{57.2}{5} \) = 11.44

Finally, take the square root of the variance to get the standard deviation.

Standard deviation = \(\sigma\) = 3.38231

Mean =	36.4
Median =	36
Mode =	33
Range =	9
Variance =	11.44
Standard Deviation =	3.38231

So now we have all the answers to the variance table problem:

Plus, you have to have a filled-out table.

Have Fun.