Pearson Correlation Coefficient (r)

The Pearson Correlation Coefficient (or the Pearson Product Moment Correlation) is the measure of the strength of the linear association between two quantitative variables. The formula is:

formula1, where formula2and formula3

are the standard scores for x and y that show how many standard deviations x and y are from the mean and n is the sample size.

Note: make sure you use sample statistics instead of the population parameters.

Correlation_coefficientThe coefficient (r) ranges from -1 to 1, where -1 represents a perfect negative, +1 a perfect positive relationship and 0 represents absence of a relationship. It is important to notice that the correlation coefficient will never be exactly +/-1 or 0, but rather a strong correlation will approach +/- 1 and a weak correlation will approach 0 (see figure).

Properties of the correlation coefficient:

  • Unitless – the coefficient is an abstract index, not a measurement
  • Not affected by a change of scale
  • Symmetric in x and y – the coefficient between x and y is the same as it is between y and x
  • Has a pitfall – it is useful to first look at the scatterplot to make sure that outliers do not reduce the strength of the correlation

To find the correlation coefficient table in SPSS, go to Analyze – Correlate – Bivariate. In order to understand the correlation coefficient in the resulting table, you need to look at three things:

  • Sign – if the coefficient is positive, there is a positive correlation; if it is negative, the correlation is negative.
  • Location in the range from 0 to 1 – is the correlation weak, moderately weak, moderate, moderately strong, or strong?
  • Number of stars (maximum 2) – this shows the significance of the correlation, in other words the evidence of a relationship. The correlation may be significant whether or not it is strong or weak.

Use this and this website to better understand the concept and look at examples. Click here to see the formula and the properties of the correlation coefficient. Click here, if you are interested in the biography of Karl Pearson who developed the coefficient.

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