Pearson Product Moment Correlation Coefficient
The Pearson Product Moment Correlation Coefficient is a statistical test which is used to show the strength and direction of a
correlation between two sets of data in which the individual values are paired against each other. In this sense, it is similar to
the Spearman’s Rank Correlation Coefficient, though the former uses real data values rather than the rankings of the data in
sequence.
The Pearson Product Moment Correlation Coefficient can be used by researchers to test a variety of geographical
variables against each other to see if there is a meaningful correlation. Here are some examples of possible
geographical investigations that could use this test:
• Settlements - how house prices change as one moves away from the CBD.
• Rivers - the variance in bedload size with distance from the source.
• Development - how life expectancy changes with average calorie intake.
• Coasts - how favourability scores vary with the length of time sea defences have been in place.
• Weather - how percentage cloud cover and precipitation volume may be correlated.
• Industry - how the air miles of a garment vary with the recommended retail price.
• Tourism - how average tourist spend changes with distance to the home area or country.
• Demography - how the birth rate of a region or country changes with the average age of first time mothers.
Distance from CBD Nitrogen dioxide level
How to carry out a Pearson Product Moment Correlation (km) (µgm-3)
Coefficient calculation: 0 61
For this example, a geographical researcher is examining how 1 64
air pollution levels vary with distance from the CBD of a
major city. Nitrogen dioxide levels were recorded and 2 82
mapped across the city at regular intervals along a transect. 3 79
The researcher believed that as the centre of the city was
4 60
quite old and contained few roads wide enough for large
volumes of traffic, it would not have as high a level of 5 49
nitrogen dioxide as the city outskirts. The researcher
6 38
therefore formulated the following hypothesis:
7 25
“There is a correlation between the level of nitrogen
8 34
dioxide in the air and the distance from the CBD.”
9 28
The researcher started by tabulating the observed data and
10 20
calculating the mean values.
11 13
12 15
mean ( x ) = 6.0 mean ( y ) = 43.7
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The deviation ( dx and dy ) was then calculated for each data set by subtracting the x or y value from its corresponding mean.
These figures were squared to remove any negative values. Finally, the sum of the squared deviations was noted, as was dx
multiplied by dy for each value. This produced the following results table:
x dx dx2 y dy dy2 dxdy
0 6 36 61 -17.3 299.6 -103.8
1 5 25 64 -20.3 412.4 -101.5
2 4 16 82 -38.3 1467.5 -153.2
3 3 9 79 -35.3 1246.6 -105.9
4 2 4 60 -16.3 265.9 -32.6
5 1 1 49 -5.3 28.2 -5.3
6 0 0 38 5.7 32.4 0
7 -1 1 25 18.7 349.4 -18.7
8 -2 4 34 9.7 93.9 -19.4
9 -3 9 28 15.7 246.2 -47.1
10 -4 16 20 23.7 561.3 -94.8
11 -5 25 13 30.7 942.0 -153.5
12 -6 36 15 28.7 823.2 -172.2
x=6 Ʃ = 182 y = 43.7 Ʃ = 6768.8 Ʃ = -1008.0
The Pearson Product Moment Correlation Coefficient value ( r ) was then calculated using the formula:
r= Ʃ ( dxdy ) The negative calculated value for r indicates a strong negative correlation.
√ Ʃ (dx2) x Ʃ (dy2) However, before one can fully accept the hypothesis, a significance test should
also be carried out. This tells the researcher the extent to which one can be sure
r= -1008.0 that the results are meaningful and the level to which one can be sure that the
results did not occur by chance.
√ 182 x 6768.8
To calculate this, a significance table is required. The researcher compares the r
r= -1008.0 value with the critical value for the appropriate number of sets of paired data. If
the calculated value (regardless of the correlation direction) is greater than the
√ 1231921.6
critical value, the hypothesis can be accepted. A full significance table for the
Pearson Product Moment Correlation Coefficient can be downloaded from the
r= -1008.0
island geographer site.
1109.9
The degrees of freedom ( df ) = n-1 where n is the number of paired observations.
Therefore, in this case, the degrees of freedom is 12.
r = -1008.0
To a 95% significance level, the critical value in this example is 0.458. Therefore,
This is known as the calculated the calculated value is greater than the critical value and the researcher can fully
value. accept their hypothesis.
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