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Spearman’s Rank Correlation Coefficient © Christine Crisp “Teach A Level Maths” Statistics 1

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Spearman’s Rank Correlation Coefficient This presentation develops another method of measuring the relationship between 2 variables. Instead of dealing with the values of the variables as in the product moment correlation coefficient, we assign a number ( rank ) to each variable. We then calculate a correlation coefficient based on the ranks. The calculated value is called the Spearman’s Rank Correlation Coefficient, r s, and is an approximation to the p.m.c.c. The formula is where d is the difference in ranks and n is the number of pairs. The values of r s have the same meanings as those for the p.m.c.c.

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Spearman’s Rank Correlation Coefficient e.g.1 Two judges rank 8 books which have been nominated for a prize. The rankings are given below. Find the Spearman’s rank correlation coefficient and comment on it’s value. ABCDEFGH Judge 124185736 Judge 215486723 Solution:In this example, the data are already ranked. We find d by subtracting the ranks for each book. (I insert signs for d but since we are going to square you can ignore them if you wish. )

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Spearman’s Rank Correlation Coefficient 10 11 30 33 11 1 32768451Judge 2 3 G 7 F 5 E 68142Judge 1 d HDCBA There is a strong positive correlation. In general, the judges are in agreement about the books. This comment is a statistical interpretation. This comment is in the context of the question.

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Spearman’s Rank Correlation Coefficient e.g.2 Find the Spearman’s rank correlation coefficient for the following data which gives the yield per acre for oats and barley over 5 years. Solution: Does it matter whether we choose 1 to be the least or 1 to be the greatest? 709 475 2003 92537578585 Oats 211581682681 Barley 2002200120001999 Source: Alberta: Agriculture, Food and Rural Development ANS: No. ( I’ve chosen 1 for the least. ) We first need to assign ranks.

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Spearman’s Rank Correlation Coefficient e.g.2 Find the Spearman’s rank correlation coefficient for the following data which gives the yield per acre for oats and barley over 5 years. Solution: 709 475 2003 92537578585 Oats 211581682681 Barley 2002200120001999 Source: Alberta: Agriculture, Food and Rural Development 5 2 2003 1345 Oats 1243 Barley 2002200120001999 We first need to assign ranks. We now find the differences in the ranks.

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Spearman’s Rank Correlation Coefficient 33 0102 51243 Barley 2 2003 1345 Oats d 2002200120001999 There is a weak positive correlation. There is little evidence that high ( or low ) yields of both crops appear in the same years.

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Spearman’s Rank Correlation Coefficient SUMMARY Spearman’s rank correlation coefficient is given by where d is the difference in ranks and n is the number of pairs. The values of r s have the same meanings as those for the p.m.c.c.

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Spearman’s Rank Correlation Coefficient Exercise For each of the following, find Spearman’s rank correlation coefficient and interpret your answer in the context of the question. 1. Two students ranked 8 countries in order of preference for a holiday. Their rankings are given below: 6 5 Ho 2 4 Tu 4 8 Sc 376211 st student 587312 nd student ItGeFrGrSp 2. The number of known species of plants (thousands) and birds (hundreds) for 10 Asian countries are as follows: 4 3 Ko 8 16 Ma 7 8 La 5 6 Ja 16 30 In 7193254 Plants 9111263 Birds NeIndChBaAr Source: Earthtrends

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Spearman’s Rank Correlation Coefficient SpGrFrGeScTuHoIt 1 st student12678453 2 nd student13784265 d0 42 -2 1. The students largely agree about the holiday destinations. There is a strong positive correlation. Solution:

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Spearman’s Rank Correlation Coefficient ArBaChIndInJaLaKoMaNe Plants 45321930683167 Birds 3612111657489 2. ArBaChIndInJaLaKoMaNe Plants 23108946175 Birds 14981035267 d 110 11 1-2 There is a very strong evidence that the higher the number of plant species the higher the number of bird species. Solution: Assigning ranks, with 1 the lowest, we get

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Spearman’s Rank Correlation Coefficient The following slides contain repeats of information on earlier slides, shown without colour, so that they can be printed and photocopied. For most purposes the slides can be printed as “Handouts” with up to 6 slides per sheet.

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Spearman’s Rank Correlation Coefficient SUMMARY Spearman’s rank correlation coefficient is given by where d is the difference in ranks and n is the number of pairs. The values of r s have the same meanings as those for the p.m.c.c.

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Spearman’s Rank Correlation Coefficient e.g.1 Two judges rank 8 books which have been nominated for a prize. The rankings are given below. Find the Spearman’s rank correlation coefficient and comment on it’s value. 2 3 G 7 7 F 6 5 E 68142Judge 1 38451Judge 2 HDCBA Solution:In this example, the data are already ranked. We find d by subtracting the ranks for each book. (I insert signs for d but since we are going to square you can ignore them if you wish. )

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Spearman’s Rank Correlation Coefficient 10 11 30 33 11 1 32768451Judge 2 3 G 7 F 5 E 68142Judge 1 d HDCBA There is a strong positive correlation. In general, the judges are in agreement about the books. This comment is a statistical interpretation. This comment is in the context of the question.

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Spearman’s Rank Correlation Coefficient e.g.2 Find the Spearman’s rank correlation coefficient for the following data which gives the yield per acre for oats and barley over 5 years. Solution: 709 475 2003 92537578585 Oats 211581682681 Barley 2002200120001999 Source: Alberta: Agriculture, Food and Rural Development 5 2 2003 1345 Oats 1243 Barley 2002200120001999 We first need to assign ranks. We now find the differences in the ranks.

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Spearman’s Rank Correlation Coefficient 33 0102 51243 Barley 2 2003 1345 Oats d 2002200120001999 There is a weak positive correlation. There is little evidence that high ( or low ) yields of both crops appear in the same years.

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