The descriptive statistics can be used to assist in calculating the correlation.
> lapply(CorrelationData, function(x) c(length(x), mean(x), sd(x)))
$Outcome1
[1] 4.00000 2.00000 2.44949
$Outcome2
[1] 4.00000 6.00000 2.44949
> cov(Outcome1,Outcome2)
[1] 3
> cor(Outcome1,Outcome2)
[1] 0.5
The inferential statistics show the key elements to be calculated.
> cor.test(Outcome1,Outcome2)
alternative hypothesis: true correlation is not equal to 0
95 percent confidence interval:
-0.8876337 0.9868586
sample estimates:
cor
0.5
Descriptive Statistics: The descriptive statistics are calculated separately for each variable.
Sum of Cross Products: The Sum of Cross Products (“SCP”) is not easily determined solely from the summary statistics of the output, but rather from the data.
\[SCP = \sum ( X - M_X ) ( Y - M_Y ) = ( 0 - 2.000 ) ( 4 - 6.000 ) + ( 0 - 2.000 )( 7 - 6.000 ) + ( 3 - 2.000 )( 4 - 6.000 ) + (5 - 2.000)(9 - 6.000) = 9.000\]
Covariance: The Covariance (“COV”) is a function of the Sum of Cross Products and the sample size:
\[COV = \frac{SCP}{(N - 1)} = \frac{9.000}{(4 - 1)} = 3.000\]
Pearson Correlation Coefficient: The Pearson Correlation Coefficient (“r”) is a function of the Covariance and the Standard Deviations of both variables:
\[r = \frac{COV}{(SD_X) (SD_Y)} = \frac{3.000}{(2.449) (2.449)} = .500\]
Correlations provide a measure of statistical relationship between two variables.
For the participants (N = 4), the scores on Outcome 1 (M = 2.00, SD = 2.45) and Outcome 2 (M = 6.00, SD = 2.45) were moderately correlated, r(2) = .50.
Note that correlations can also have inferential information associated with them (and that this information should be summarized if it is available and of interest).
For the participants (N = 4), the scores on Outcome 1 (M = 2.00, SD = 2.45) and Outcome 2 (M = 6.00, SD = 2.45) were moderately but not statistically significantly correlated, r(2) = .50, 95% CI [-0.89, 0.99], p = .500.