Spearman Rho Correlation

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Spearman Rho Correlation

• Introduction
• Spearman's rank correlation coefficient or
  Spearman's rho is named after Charles Spearman
• Used Greek letter ? (rho) or as rs (non-
  parametric measure of statistical dependence
  between two variables)
• Assesses how well the relationship between two
  variables can be described using a monotonic
  function
• Monotonic is a function (or monotone function) in
  mathematic that preserves the given order.
• If there are no repeated data values, a perfect
  Spearman correlation of 1 or -1 occurs when each
  of the variables is a perfect monotone function
  of the other

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Spearman Rho Correlation

• A correlation coefficient is a numerical measure
  or index of the amount of association between two
  sets of scores. It ranges in size from a maximum
  of 1.00 through 0.00 to -1.00
• The sign indicates a positive correlation
  (the scores on one variable increase as the
  scores on the other variable increase)
• The - sign indicates a negative correlation
  (the scores on one variable increase, the scores
  on the other variable decrease)

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Spearman Rho Correlation

• Calculation
• Often thought of as being the Pearson correlation
  coefficient between the ranked (relationship
  between two item) variables
• The n raw scores Xi, Yi are converted to ranks
  xi, yi, and the differences di  xi - yi between
  the ranks of each observation on the two
  variables are calculated
• If there are no tied ranks, then ? is given by
  this formula

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Spearman Rho Correlation

• Calculation
• Often thought of as being the Pearson correlation
  coefficient between the ranked (relationship
  between two item) variables
• The n raw scores Xi, Yi are converted to ranks
  xi, yi, and the differences di  xi - yi between
  the ranks of each observation on the two
  variables are calculated
• If there are no tied ranks, then ? is given by
  this formula

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Spearman Rho Correlation

• Interpretation
• The sign of the Spearman correlation indicates
  the direction of association between X (the
  independent variable) and Y (the dependent
  variable)
• If Y tends to increase when X increases, the
  Spearman correlation coefficient is positive
• If Y tends to decrease when X increases, the
  Spearman correlation coefficient is negative
• A Spearman correlation of zero indicates that
  there is no tendency for Y to either increase or
  decrease when X increases

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Spearman Rho Correlation

• Interpretation cont/
• Alternative name for the Spearman rank
  correlation is the "grade correlation the "rank"
  of an observation is replaced by the "grade"
• When X and Y are perfectly monotonically related,
  the Spearman correlation coefficient becomes 1
• A perfect monotone increasing relationship
  implies that for any two pairs of data values
  Xi, Yi and Xj, Yj, that Xi - Xj and Yi - Yj
  always have the same sign

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Spearman Rho Correlation

• Example 1
• Calculate the correlation between the IQ of a
  person with the number of hours spent in the
  class per week
• Find the value of the term d²i
• 1. Sort the data by the first column (Xi). Create
  a new column xi and assign it the ranked values
  1,2,3,...n.
• 2. Sort the data by the second column (Yi).
  Create a fourth column yi and similarly assign it
  the ranked values 1,2,3,...n.
• 3. Create a fifth column di to hold the
  differences between the two rank columns (xi and
  yi).

IQ, Xi Hours of class per week, Yi
106 7
86 0
100 27
101 50
99 28
103 29
97 20
113 12
112 6
110 17
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Spearman Rho Correlation

• Example 1 cont/
• 4. Create one final column to hold the value of
  column di squared.

IQ (Xi ) Hours of class per week (Yi) rank xi rank yi di d²i
86 0 1 1 0 0
97 20 2 6 -4 16
99 28 3 8 -5 25
100 27 4 7 -3 9
101 50 5 10 -5 25
103 29 6 9 -3 9
106 7 7 3 4 16
110 17 8 5 3 9
112 6 9 2 7 49
113 12 10 4 6 36
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Spearman Rho Correlation

• Example 1- Result
• With d²i found, we can add them to find ? d²i
  194
• The value of n is 10, so
• ? 1- 6 x 194
  10(10² - 1)
• ?   -0.18
• The low value shows that the correlation between
  IQ and hours spent in the class is very low

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Spearman Rho Correlation

• Example 2
• 5 college students have the following rankings
  in Mathematics and Science subject. Is there an
  association between the rankings in Mathematics
  and Science subject.

Student Ashley David Owen Steven Frank
Mathematic class rank 1 2 3 4 5
Science class rank 5 3 1 4 2
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Spearman Rho Correlation

• Example 2 cont/
• Compute Spearman Rho
• n number of paired ranks
• d difference between the paired ranks
• (when two or more observations of one variable
  are the same, ranks are assigned by averaging
  positions occupied in their rank order)

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Spearman Rho Correlation
Example 2 cont/
Maths Rank Science Rank X-Y (X-Y) 2
X Y D d2
1 5 -4 16
2 3 -1 1
3 1 2 4
4 4 0 0
5 2 3 9
30

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Spearman Rho Correlation
Example 2- Result
Result ? -0.5 There is a moderate negative
correlation between the Math and Science subject
rankings of students Students who rank high as
compared to other students in their Math
subject generally have lower Science subject
ranks and those with low Math rankings have
higher Science subject rankings than those with
high Math rankings. (The formula for Pearson r
and Spearman rho are equivalent when there are no
tied)
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Research Article

• Title
• Motivation and Attitude in Learning English
  among UiTM Students in the Northern Region of
  Malaysia
• Purpose
• To describe the relationship between the
  students motivation and attitude to their
  English Language performance

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Article Cont/

• Method
• Used a correlational research design
• Independent variables
• Motivation, attitude, and personal
  characteristics variables, as measured by a
  self-report questionnaire
• Dependent variable
• English Language performance, measured by the
  UiTM Preparatory English (BEL100) examination
  result

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Article Cont/

• Method
• Sampling Design
• The subjects were 139 students from the
  Perlis Campus, 248 from the Kedah Campus and 233
  from the Pulau Pinang Campus.
• The selection criterion used in attaining the
  samples was to choose those students who had just
  received their BEL100 examination result
  regardless of their status whether as the first
  timer or repeater for that particular paper.

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Article Cont/

• Method
• Questionnaire
• Research instrument used was questionnaire that
  comprised questions on personal characteristics,
  motivation and attitudes.
• The instrument was adopted and adapted from
  Gardner andLambert (1972) so that it is more
  appropriate, intelligible and meaningful for the
  sample concerned.
• The reliability test of the instrument
  produced a Cronbach Alfa of 0.757, which was
  satisfactory and acceptable.

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Article Cont/

• Method
• Data Analysis
• The data collected were computed and analyzed
  using the SPSS 12.
• Each students score on the questionnaire was
  matched to his or her BEL100 examination grade.
• The statistical procedures used in this study
  were the descriptive statistics mean standard
  deviation scores, frequency percentage, t-test,
  Spearman Rho Rank-Order Correlation Coefficient
  ANOVA.

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Article Cont/

• Result- Correlation between motivation in
  learning English English language performance
• Spearman Rho rank-order correlation coefficient
  test
• Very weak relationship between Intrinsic
  Motivation English language performance, which
  is -.020.
• One-way ANOVA test
• Intrinsic Motivation Critical value of F at
  alpha .05 is 2.70. The obtained F value is
  1.63, which is less than the critical value.
• Extrinsic Motivation Computed value for the
  correlation test which is .043 and the obtained
  value of F for the one-way ANOVA, 2.39.
• This justifies there is no significant
  differences between Overall Motivation (Intrinsic
  Motivation Extrinsic Motivation) English
  language performance.

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Article Cont/

• Result- Attitude in learning English English
  language performance
• Spearman Rho rank-order correlation coefficient
  test
• Exists a significant correlation (alpha
  .01) between the attitude in learning English
  English language performance, which is -.152.
• One-way ANOVA test
• F value from the one-way ANOVA test is 6.66,
  which is greater than the critical value.
• Mean scores
• Respondents received A - M 3.06, B- M
  2.99, C - M 2.93, D - M 2.80), it can be
  concluded that the respondents who obtained an A
  (high achievers) have better attitude in learning
  English compared to the low achievers.
• This justifies that there is a significant
  difference between the attitude in learning
  English English language performance.

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Spearman Rho Correlation

• References
• http//en.wikipedia.org/wiki/Spearman's_rank_corre
  lation_coefficient
• http//davidmlane.com/hyperstat/A62436.html
• http//www.wellesley.edu/Psychology/Psych205/Spear
  man.html
• www.ccsenet.org/journal.html
• www.statisticallysignificantconsulting.com
• http//www.wikihow.com/Calculate-
  Spearman's-Rank-Correlation-Coefficient
• http//en.wikipedia.org/wiki/Spearman27s_rank_cor
  relation_coefficient

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• Prepared by
• Rubiyatul Huda
• Norsafrina
• Salina

Thank you