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