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Quadrant count ratio

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The quadrant count ratio (QCR) is a measure of the association between two quantitative variables. The QCR is not commonly used in the practice of statistics; rather, it is a useful tool in statistics education because it can be used as an intermediate step in the development of Pearson's correlation coefficient.

Definition and properties

To calculate the QCR, the data are divided into quadrants based on the mean of the X {\displaystyle X} and Y {\displaystyle Y} variables. The formula for calculating the QCR is then:

q = n ( Quadrant I ) + n ( Quadrant III ) − n ( Quadrant II ) − n ( Quadrant IV ) N , {\displaystyle q={\frac {n({\text{Quadrant I}})+n({\text{Quadrant III}})-n({\text{Quadrant II}})-n({\text{Quadrant IV}})}{N}},}

where n(Quadrant) {\displaystyle {\text{n(Quadrant)}}} is the number of observations in that quadrant and N {\displaystyle N} is the total number of observations.

The QCR is always between −1 and 1. Values near −1, 0, and 1 indicate strong negative association, no association, and strong positive association (as in Pearson's correlation coefficient). However, unlike Pearson's correlation coefficient the QCR may be −1 or 1 without the data exhibiting a perfect linear relationship.

Example

Data from 35 Category 5 Hurricanes showing the relationship between wind speed (X) and pressure (Y). The blue and green lines represent the means of the X and Y values, respectively. The Quadrants have been labeled. The points have been jittered to reduce overlap of observations.

The scatterplot shows the maximum wind speed (X) and minimum pressure (Y) for 35 Category 5 Hurricanes. The mean wind speed is 170 mph (indicated by the blue line), and the mean pressure is 921.31 hPa (indicated by the green line). There are 6 observations in Quadrant I, 13 observations in Quadrant II, 5 observations in Quadrant III, and 11 observations in Quadrant IV. Thus, the QCR for these data is ( 6 + 5 ) − ( 13 + 11 ) 35 = − 0.37 {\displaystyle {\frac {(6+5)-(13+11)}{35}}=-0.37} , indicating a moderate negative relationship between wind speed and pressure for these hurricanes. The value of Pearson's correlation coefficient for these data is −0.63, also indicating a moderate negative relationship..

See also

References

  1. Kader, Gary, D.; Christine A. Franklin (November 2008). "The Evolution of Pearson's Correlation Coefficient". Mathematics Teacher. 102 (4): 292–299. doi:10.5951/MT.102.4.0292.{{cite journal}}: CS1 maint: multiple names: authors list (link)
  2. Holmes, Peter (Autumn 2001). "Correlation: From Picture to Formula". Teaching Statistics. 23 (3): 67–71. doi:10.1111/1467-9639.00058. S2CID 123667316.
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