How to do a Variance Table Problem

33

36

38

33

42

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:​

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.

\(X\)

\(X - \mu\)

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

\(\mu\) = 36.4

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

\(X\)

\(X - \mu\)

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

\(\mu\) = 36.4

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

\(X\)

\(X - \mu\)

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

\(\mu\) = 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.

\(\mu\) = 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

\(\mu\) = 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

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

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

Have Fun.