The following data set reflects a between-subjects design with one factor (that has two levels). The data are presented in a format suitable for entry into statistical software.
| Factor | Outcome | |
|---|---|---|
| 1 | 1.00 | .00 |
| 2 | 1.00 | .00 |
| 3 | 1.00 | 3.00 |
| 4 | 1.00 | 5.00 |
| 5 | 2.00 | 4.00 |
| 6 | 2.00 | 7.00 |
| 7 | 2.00 | 4.00 |
| 8 | 2.00 | 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.
| Group | N | Mean | Std. Deviation | Std. Error Mean | |
|---|---|---|---|---|---|
| Outcome | Level 1 | 4 | 2.000 | 2.445 | 1.225 |
| Level 2 | 4 | 6.000 | 2.445 | 1.225 |
The table of inferential statistics shows the key elements to be calculated.
| t | df | p | Mean Difference | SE Difference | Lower CI | Upper CI | Cohen’s d | |
|---|---|---|---|---|---|---|---|---|
| Outcome | -2.309 | 6.000 | 0.060 | -4.000 | 1.732 | -8.238 | 0.238 | -1.633 |
Mean Difference (Raw Effect): The mean difference is the difference between the two sample means (raw effect).
\[M_{DIFF} = M_1 - M_2 = 2.000 − 6.000 = −4.000\]
Within Groups Statistics: When multiple groups are used, it is necessary to get an estimate of the pooled (combined) within group variabilities.
\[SS_1 = ( SD_1^2 ) ( df_1) = ( 2.44949^2 ) ( 3 ) = 18.000\] \[SS_2 = ( SD_2^2 ) ( df_2) = ( 2.44949^2 ) ( 3 ) = 18.000\] \[SS_{WITHIN} = SS_1 + SS_ 2 = 18.000 + 18.000 = 36.000\] \[df_{WITHIN} = df_1 + df_ 2 = 3 + 3 = 6\] \[MS_{WITHIN} = \frac{SS_{WITHIN}}{df_{WITHIN}} = \frac{36.000}{6} = 6.000\] \[SD_{WITHIN} = \sqrt{MS_{WITHIN}} = \sqrt{6.000} = 2.449\]
Standard Error of the Difference: The standard error of the difference is a function of the two groups’ individual standard errors.
When the two sample sizes are equal:
\[SE_{DIFF} = \sqrt{ SE_1^2 + SE_2^2 } = \sqrt{ 1.225^2 + 1.225^2 } = 1.732\]
Or an expanded version of the formula can be used when the two sample sizes are either equal or unequal:
\[SE_{DIFF} = \sqrt{ \left( \frac{MS_{WITHIN}}{n_1} \right) + \left( \frac{MS_{WITHIN}}{n_2} \right) } = \sqrt{ \left( \frac{6.000}{4} \right) + \left( \frac{6.000}{4} \right) } = 1.732\]
Statistical Significance: The t statistic is the ratio of the mean difference (raw effect) to the standard error of the difference.
\[t = \frac{M_{DIFF}}{SE_{DIFF}} = \frac{-4.000}{1.732} = -2.309\] \[df = ( n_1 - 1 ) + ( n_2 - 1) = N - 2 = 8 - 2 = 6\]With df = 6, tCRITICAL = 2.447
Because t < tCRITICAL, p > .05
Confidence Interval: For this test, the appropriate confidence interval is around (centered on) the mean difference (raw effect).
\[CI_{DIFF} = M_{DIFF} \pm (t_{CRITICAL}) ( SE_{DIFF}) = -4.000 \pm (2.447) (1.732) = [ −8.238, 0.238 ]\]
Effect Size: Cohen’s d Statistic provides a standardized effect size for the difference between the two means.
\[d = \frac{M_{DIFF}}{SD_{WITHIN}} = \frac{-4.000}{2.449} = -1.633\]