Spearman’s Rank C.C.Spearman’s Rank C.C.

Measuring CorrelationMeasuring Correlation

Beyond The ScattergraphBeyond The Scattergraph

For GCSE Mathematics you simply draw a line For GCSE Mathematics you simply draw a line of best fit through the data and talk about of best fit through the data and talk about “strong” or “weak” correlation in vague terms.“strong” or “weak” correlation in vague terms.

For GCSE Statistics you learn how to For GCSE Statistics you learn how to CALCULATE the amount of correlation so that CALCULATE the amount of correlation so that you can compare one scattergraph with another.you can compare one scattergraph with another.

You don’t even need to draw the scattergraph!You don’t even need to draw the scattergraph!

Spearman’s Rank C.C. FormulaSpearman’s Rank C.C. Formula

This formula is on the formula sheet so you don’t need to learn it!

This formula is on the formula sheet so you don’t need to learn it!

Perfect Negative

Correlation

No Correlation

Perfect Positive

Correlation

Interpretation of S.R.C.C.Interpretation of S.R.C.C.

Values close to 0 suggest no correlationValues close to 0 suggest no correlation 0.4 to 0.6 (0.5) is “weak correlation”0.4 to 0.6 (0.5) is “weak correlation” 0.7 and higher suggests “strong 0.7 and higher suggests “strong

correlation”correlation” This applies to negative values tooThis applies to negative values too There are no hard and fast rules about There are no hard and fast rules about

when “weak” becomes “strong” when “weak” becomes “strong” If If rr > 1 then you went wrong !!!!!! > 1 then you went wrong !!!!!!

Outline of ProcedureOutline of Procedure

Let’s say you are exploring Let’s say you are exploring nn heights and heights and weights in an investigationweights in an investigation

Rank the heights i.e. put them in order Rank the heights i.e. put them in order (1 = biggest, (1 = biggest, nn = smallest) = smallest)

Rank the weights Rank the weights (1 = biggest, (1 = biggest, nn = smallest) = smallest)

dd = difference between the two ranks for = difference between the two ranks for each personeach person

Square and add these differencesSquare and add these differences

Problems With Equal RanksProblems With Equal Ranks

What if two things are 3rd= ?What if two things are 3rd= ? One has to be third, the other fourth.One has to be third, the other fourth. To be fair, each takes the average:To be fair, each takes the average:

(3 + 4) ÷ 2 = 3.5(3 + 4) ÷ 2 = 3.5 What if three things are 5th= ?What if three things are 5th= ? Call them 5th, 6th and 7thCall them 5th, 6th and 7th Give each one the average rank:Give each one the average rank:

(5 + 6 + 7) ÷ 3 = 6(5 + 6 + 7) ÷ 3 = 6

Fertiliser v. Plant GrowthFertiliser v. Plant Growth

CropCrop AA BB CC DD EE

FertiliserFertiliser 12.812.8 17.117.1 8.38.3 6.76.7 10.210.2

YieldYield 103103 108108 8989 7575 105105

First, Rank The Data:First, Rank The Data:

CropCrop AA BB CC DD EE

FertiliserFertiliser 12.812.8 17.117.1 8.38.3 6.76.7 10.210.2

FertiliserFertiliserRANKRANK 22 11 44 55 33

YieldYield 103103 108108 8989 7575 105105

YieldYieldRANKRANK 33 11 44 55 22

Second, Find The Rank Differences:Second, Find The Rank Differences:

CropCrop AA BB CC DD EE

FertiliserFertiliser 12.812.8 17.117.1 8.38.3 6.76.7 10.210.2

FertiliserFertiliserRANKRANK 22 11 44 55 33

YieldYield 103103 108108 8989 7575 105105

YieldYieldRANKRANK 33 11 44 55 22

RankRankDifferenceDifference -1-1 00 00 00 11

Third, Square The Rank Differences:Third, Square The Rank Differences:

CropCrop AA BB CC DD EE

FertiliserFertiliser 12.812.8 17.117.1 8.38.3 6.76.7 10.210.2FertiliserFertiliserRANKRANK 22 11 44 55 33

YieldYield 103103 108108 8989 7575 105105

YieldYieldRANKRANK 33 11 44 55 22

RankRankDifferenceDifference -1-1 00 00 00 11

d^2d^2 11 00 00 00 11

Now Find “Sigma D Squared”Now Find “Sigma D Squared”

CropCrop AA BB CC DD EE

FertiliserFertiliser 12.812.8 17.117.1 8.38.3 6.76.7 10.210.2FertiliserFertiliserRANKRANK 22 11 44 55 33

YieldYield 103103 108108 8989 7575 105105YieldYield

RANKRANK 33 11 44 55 22

RankRankDifferenceDifference -1-1 00 00 00 11

d^2d^2 11 00 00 00 11

Finally Use The Formula:Finally Use The Formula:

n = 5 (there were 5 crops)

-> There is very strong correlation between the amount of fertiliser and the crop yield.

Here’s One For You To Try…Here’s One For You To Try…

AA 5050 175175 270270 375375 425425 580580 710710 790790 890890 980980

BB 1.801.80 1.201.20 2.002.00 1.001.00 1.001.00 1.201.20 0.800.80 0.600.60 1.001.00 0.850.85

These data show the distances in metres (A) of shops from the Museum in Barcelona and the prices in Euros (B) for a 50cl bottle of water.

Hypothesis: Shops closer to the museum charge more for their water than those further away.

Do the data support this hypothesis?

ProcedureProcedure

AA 5050 175175 270270 375375 425425 580580 710710 790790 890890 980980

RARA 1010 99 88 77 66 55 44 33 22 11

BB 1.801.80 1.201.20 2.002.00 1.001.00 1.001.00 1.201.20 0.800.80 0.600.60 1.001.00 0.850.85

RBRB 22 3.53.5 11 66 66 3.53.5 99 1010 66 88

Differences and SquaresDifferences and Squares

AA 5050 175175 270270 375375 425425 580580 710710 790790 890890 980980

RARA 1010 99 88 77 66 55 44 33 22 11

BB 1.801.80 1.201.20 2.002.00 1.001.00 1.001.00 1.201.20 0.800.80 0.600.60 1.001.00 0.850.85

RBRB 22 3.53.5 11 66 66 3.53.5 99 1010 66 88

d^2d^2 6464 30.2530.25 4949 11 00 2.252.25 2525 4949 1616 4949

ConclusionConclusion

Do the data support our hypothesis?