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 can be used to assist in calculating the standardized scores.
| N | Mean | Std. Deviation | |
|---|---|---|---|
| Outcome | 8 | 4.000 | 3.117 |
Standardized (z) Score: A standardized score is a deviation score divided by the standard deviation. For the first score in the distribution:
\[z = \frac{(Y - M)}{SD} = \frac{(0 - 4.000)}{3.117} = -1.283\]This is repeated for each score in the distribution. In software programs, these would be calculated and presented back in the data set:
| Outcome | zOutcome | |
|---|---|---|
| 1 | .00 | -1.283 |
| 2 | .00 | -1.283 |
| 3 | 3.00 | -0.321 |
| 4 | 5.00 | 0.321 |
| 5 | 4.00 | 0 |
| 6 | 7.00 | 0.963 |
| 7 | 4.00 | 0 |
| 8 | 9.00 | 1.604 |
Percentile Rank: The 50th percentile (the Median) and the 25th and 75th percentiles (collectively known as the Interquartile Range) are the most commonly calculated. Assuming a perfectly normal distribution:
From the Standard Normal Distribution table: for PR = .250, z = -.680
From the Standard Normal Distribution table: for PR = .500, z = 0.000
From the Standard Normal Distribution table: for PR = .750, z = +.680
Percentile Point: Percentiles provide the scores associated with particular percentile ranks. These can be estimated using the z score formula and the descriptive information from the original variable:
For PR = .250: -.680 = (Y - 4.000) / 3.117; Y = 1.880
For PR = .500: 0.000 = (Y - 4.000) / 3.117; Y = 4.000
For PR = .750: 0.000 = (Y - 4.000) / 3.117; Y = 6.120
Mean (of the z Scores): 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:
\[M = \frac{\sum(fY)}{N} = \frac{(2 \times -1.283) + (1 \times -.321) + (2 \times 0) + (1 \times .321) + (1 \times .963) + (1 \times 1.604)}{8} = 0.000\]
Sum of Squares (of the z Scores): 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.
\[SS = \sum f (Y - M) = 2 (-1.283 - 0)^2 + 1(-.321 - 0)^2 + 2(0.000 - 0)^2 + 1(.321 - 0)^2 + 1(.963 - 0)^2 + 1(1.604 - 0)^2 = 7.000\]
Mean Squares (of the z scores): 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.
\[MS = \frac{SS}{(N - 1)} = \frac{7.000}{7} = 1.000\]
Standard Deviation (of the z Scores): Standard Deviation is a function of Mean Squares. It is also calculated as an unbiased estimate of the population standard deviation.
\[SD = \sqrt{MS} = \sqrt{1.000} = 1.000\]