# Spearman's Rank Correlation Coefficient (rS)

# Spearman's Rank Correlation Coefficient (r_{S)}

The following exam style questions aim to give you an idea of what you can expect for this topic in your exams. You can print the page by choosing '**print this page**' from the **menu on the rgiht** OR you can use the onscreen version and check your answers as you go.

Judge X is suspected of being biased. To test this hypothesis, it is believed the rankings attributed by an unbiased judge will have a montonic relationship with the average ranking awarded by all other judges. The below table shows the rankings awarded by Judge X compared to the average of all other judge's rankings.

(a) From the table, which countries does Judge X appear to be biased toward?

[1]

(b) Calculate Spearman's rank correlation coefficient for this data.

[2]

(c) At the 5% sig level the critical value is 0.707, test if the Spearman's rank correlation coefficient for this data is significant or not and interpret your result in the context of the question.

[3]

(a) Countries whose rank is higher than the 'average ranking' = France, Canada, Germany (A1 all three correct)

(b) Average of tied ranks has to be found for both columns (M1) and placed into the GDC (A1) to give: 0.197 (to 3 s.f.)

(c) We would not anticipate this Spearman rank coefficient to be significant given it is close to 0.

(Test statistic, from our sample) 0.197 < 0.707 (critical value, from the 5% significance level). (**A1**)

H_{0}: ρ = 0

H_{A}: ρ ≠ 0

This sample provides no evidence to reject H_{0}, therefore there appears to be **no **significant correlation between Judge X's rankings and those of the other judges. This suggests that Judge X's rankings are biased. (**A1 **- hypothesis and reject H_{0}), (**A1 **- interpret in context)

Below are the prices and rankings, by professional wine tasters and resellers, of ten different bottles of wine. Normally, a higher price (Canadian $) should indicate superior quality.

(a) Give one reason why Spearman's rank correlation coefficient may be more appropriate for this data than Pearson's product moment correlation coefficient?

[1]

(b) Calculate the Spearman's rank correlation coefficient (rs) for this data and comment on whether or not it provides evidence in support of the questions claims about the price of wine and its quality.

[4

(a) The '*wine ranking*' column is ranked data rather than continuous. (scatter graph suggests no correlation so linear or Spearman's could be reasonable in terms of measuring the correlation/absence of correlation).

(b) **(M1)(A1)** for ranking the wine data and correctly averaging the tied rank (2.5)

**A1** correct r_{S} = -0.170 **A****1 **there is no evidence of a relationship between the price of a bottle of wine (above $36) and its quality.

It's believed that US citizens have similar tastes in the genre of television programs they most enjoy watching when compared with India. The table below shows the top 7 genres of television program in each country and the percentage of total viewers.

(a) Complete the scatter graph on the grid below for this data. Describe the correlation, from looking at your scatter graph, between US and Indian television program preferences?

[2]

(b) Write down the rankings of each "genre" for the US and India in the empty table below.

[1]

(c) Estimate what values you would expect to get for Spearman's rank correlation coefficient if US and Indian citizens do show similar tastes in genre of TV program?

[1]

(d) Calculate the Spearman's rank correlation coefficient for this data and determine whether the data supports the hypothesis given in the question.

[3]