# 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:
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.

$$X$$

$$X - \mu$$

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

 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.