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 table of descriptive statistics can be used to determine the inferential statistics.
| N | Mean | Std. Deviation | Std. Error Mean | |
|---|---|---|---|---|
| Outcome | 8 | 4.000 | 3.117 | 1.102 |
The table of inferential statistics shows the key elements to be calculated.
| t | df | p | Mean Difference | Lower CI | Upper CI | Cohen’s d | |
|---|---|---|---|---|---|---|---|
| Outcome | -2.722 | 7.000 | 0.030 | -3.000 | -5.606 | -.394 | -.963 |
Standard Error of the Mean: The standard error of the mean provides an estimate of how spread out the distribution of all possible random sample means would be.
\[SE_M = \frac{SD}{\sqrt{N}} = \frac{3.117}{\sqrt{8}} = 1.102\]
Mean Difference (Raw Effect): The Mean Difference is the difference between the sample mean and a user-specified test value or population mean.
\[M_{DIFF} = M - \mu = 4.000 − 7.000 = −3.000\]
Statistical Significance: The t statistic is the ratio of the mean difference (raw effect) to the standard error of the mean.
\[t = \frac{M_{DIFF}}{SE_M} = \frac{-3.000}{1.102} = -2.722\]With df = 7, tCRITICAL = 2.365
Because t > tCRITICAL, p < .05
Confidence Interval: For this design, the appropriate confidence interval is around (centered on) the mean difference (raw effect).
\[CI_{DIFF} = M_{DIFF} \pm (t_{CRITICAL} ) (SE_M) = -3.000 \pm (2.365) (1.102) = [ -5.606, -0.394 ]\]
Effect Size: Cohen’s d Statistic provides a standardized effect size for the mean difference (raw effect).
\[d = \frac{M_{DIFF}}{SD} = \frac{-3.000}{3.117} = 0.963\]