The following data set reflects one sample of eight individuals measured on one variable. The data are presented in a format suitable for entry into statistical software.
| Outcome | |
|---|---|
| 1 | .00 |
| 2 | .00 |
| 3 | 3.00 |
| 4 | 5.00 |
| 5 | 4.00 |
| 6 | 7.00 |
| 7 | 4.00 |
| 8 | 9.00 |
The following tables represent typical output from statistical software. Options, labels, and layout vary from program to program.
The frequency distribution can be used to determine the descriptive statistics.
| Frequency | Percent | Valid Percent | Cumulative Percent | ||
|---|---|---|---|---|---|
| Valid | 0.00 | 2 | 25.0 | 25.0 | 25.0 |
| 3.00 | 1 | 12.5 | 12.5 | 37.5 | |
| 4.00 | 2 | 25.0 | 25.0 | 62.5 | |
| 5.00 | 1 | 12.5 | 12.5 | 75.0 | |
| 7.00 | 1 | 12.5 | 12.5 | 87.5 | |
| 9.00 | 1 | 12.5 | 12.5 | 100.0 | |
| Total | 8 | 100.0 | 100.0 |
The table of descriptive statistics shows the key elements to be calculated.
| N | Mean | Std. Deviation | |
|---|---|---|---|
| Outcome | 8 | 4.000 | 3.117 |
Mean: The mean (or arithmetic average) is calculated as an unbiased estimate of the population mean. Here, the mean is determined as the average of the scores weighted by their frequencies:
\[M = \frac{\sum(fY)}{N} = \frac{(2 \times 0) + (1 \times 3) + (2 \times 4) + (1 \times 5) + (1 \times 7) + (1 \times 8)}{8} = 4.000\]
Sum of Squares: The Sum of Squares is the basic measure of the variability of the scores. Formally, it is the sum of the weighted deviations of the scores about the mean.
\[SS = \sum f (Y - M) = 2 (0 - 4)^2 + 1(3 - 4)^2 + 2(4 - 4)^2 + 1(5 - 4)^2 + 1(7 - 4)^2 + 1(8 - 4)^2 = 68.000\]
Mean Squares: Mean Squares (also known as Variance) is a function of the Sum of Squares. It is calculated as an unbiased estimate of the population variance.
\[MS = \frac{SS}{(N - 1)} = \frac{68.000}{7} = 9.714\]
Standard Deviation: Standard Deviation is a function of Mean Squares. It is also calculated as an unbiased estimate of the population standard deviation.
\[SD = \sqrt{MS} = \sqrt{9.714} = 3.117\]