correlated groups or samples

an index of the linear or straight line relationship between two variables which can be ordered. Correlations can be either positive or negative. A positive correlation indicates that high scores on one variable go with high scores on the other variable. A negative correlation means that high scores on one variable go with low scores on the other variable. The size of a correlation can vary from a minimum of —1.00 through 0 to a maximum of 1.00. The bigger the size of the correlation, regardless of whether it is positive or negative, the stronger the linear relationship between the two variables. A correlation of 0 or close to 0 means that there is either no or no linear relationship between the two variables. There may, however, be a curvilinear relationship between the two variables. Consequently, when the correlation is 0 or close to 0, it is useful to draw a scatter diagram which is a graph of the relationship between the two variables.

Correlations which differ substantially from 0 are statistically significant. The bigger the correlation, regardless of its sign, the more likely it is to be statistically significant. The bigger the sample, the more likely it is that a correlation will be significant. Very small correlations may be statistically significant provided that the sample is big enough.

There are various kinds of correlations. The most widely used is Pearson's product moment correlation coefficient. This name is usually shortened to Pearson's correlation and is symbolized as r. It can be calculated by multiplying the standardized scores of the two variables to obtain their ‘product’. These products are then summed and divided by the number of cases minus one to give the mean population estimate of the products. A product moment is the expected or mean value of a product of two variables. A Pearson's correlation is the same as a standardized regression coefficient. It is used to determine the linear relationship between two variables which are normally distributed. Pearson's correlation can be strongly affected by extreme scores or outliers. Consequently, if the scores are not normally distributed, the scores can be ranked and a Pearson's correlation carried out on these ranked scores. This[Page 39] type of correlation is known as Spearman's rank order correlation coefficient. This name is usually shortened to Spearman's correlation or rho and is symbolized with the Greek letter p. A Pearson's correlation between a dichotomous variable (such as sex) and a normally distributed variable may be called the point-biserial correlation. A Pearson's correlation between two dichotomous variables is called phi.

Table C.10 Correlations, percentage of shared variance and minimum size of sample for the two-tailed 0.05 statistical significance

Squaring a Pearson's correlation gives the coefficient of determination. This provides a clearer indication of the meaning of the size of a correlation as it gives the proportion of the variance that is shared between two variables. For example, a correlation of 0.50 means that the proportion of the variance shared between the two variables is 0.25 (0.502 = 0.25). These proportions are usually expressed as a percentage, which in this case, is 25% (0.25 × 100% = 25%). The percentage of shared variance for 11 correlations which each differ by 0.10 is shown in Table C.10.

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