Understanding Pearson Product-Moment Correlation Coefficient

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14 August 2024
 
LO: Calculate the Pearson product-moment correlation
coefficient using the formulae and using the GDC
 
 
 
Correlation.
 
Up to this point we have used the scatter diagram to see
if there is a relationship between two variables.
 
We have stated that direction is important to determine
the correlation.
 
We have seen that the strength is important when it
comes to the correlation.
 
 
 
Negative Correlation:-
 
Positive Correlation:-
 
When the independent variable increases,
the dependent variable decreases
 
An increase in the independent variable
means that the dependant variable
generally increases.
 
No correlation:-
 
No apparent link between the variables.
Direction
 
For a generally downward trend, we say
that the correlation is 
negative.
 
For a generally upward trend, we say that
the correlation is 
positive.
 
For randomly scattered points, with no
upward or downward trend, we say there
is 
no correlation.
 
 
 
The strength of correlation is usually described as either
strong, moderate or weak
 
Strong
 negative
Strength
 
Strong
 positive
 
Moderate
 positive
 
Weak
 positive
 
Moderate
 negative
 
Weak
 negative
 
 
Correlation
 
Now we are going to seek to classify the strength of the
correlation numerically
 
There are several scales in use; we will study a correlation
coefficient developed by Karl Pearson
We have found the equation of the regression line
of y on x and used the line for prediction purposes
The 
Pearson product-moment correlation coefficient
denoted by 
r
, 
is a measure of the correlation between
two variables 
x
 and 
y
, giving a value between -1 and +1
inclusive.
 
It is widely used in the sciences as a measure of the
strength of linear dependence between two variables
 
 
 
A perfect correlation will have a regression coefficient of
r
 = ± 1
 
Perfect
 negative
linear correlation
Strength
 
Perfect 
positive
linear correlation
 
No
 correlation
 
r
 = 1
 
r
 = -1
 
r
 = 0
The 
PMCC or regression coefficient is used to determine
how nearly the points fall on a straight line, or how nearly
linear they are.
 
If there is no correlation we will have a regression
coefficient of r = 0
 
 
 
A quick way to interpret the r-value is
Strength
 
Strong
 
Weak
 
Very Weak
 
Moderate
 
0 
 |r| 
 
0.5
 
0.5 
 |r| 
 
0.7
 
0.7 
 |r| 
 
0.87
 
0.87 
 |r| 
 
0.95
 
Very weak
 
Weak
 
Moderate
 
Strong
 
 
 
0.95 
 |r| 
 
1
 
Very Strong
 
          
|r| 
= 1
 
Perfect
 
We use the following formulas:-
 
Notes:-
DON’T PANIC – the formulas
are not that bad to use!
There are other versions of
these formulas that you may
see in books, but the ones
here are the easiest to use.
x
 is the independent variable
y
 is the dependent variable
n
 is how many pairs of data
you have.
 
Calculating the Pearson’s correlation coefficient (
r
)
 
Let us see how to use them in
an example
 
 
 
Example.
 
The cost, c, (in pence) of a
number of paperback books with total number
of pages, p, is recorded, as follows :-
No. of pages (x):
 
150   200   300   350   450
Cost (y):
  
500   580   660   820   900
Calculate the PMCC, r, for this data and
interpret the relationship.
 
 
x
150
200
300
350
450
 
y
500
580
660
820
900
 
xy
75000
116000
198000
287000
405000
 
x
2
22500
40000
90000
122500
202500
 
y
2
250000
336400
435600
672400
810000
 
1450
 
3460
 
1081000
 
477500
 
2504400
 
The value 0.9796 indicates positive Very Strong correlation
 
 
Correlation coefficient
The following table is a guide for describing the strength
of linear association using the correlation coefficient.
 
You should use your GDC to find the Correlation coefficient (
r
)
 
 
Example.
 
The cost, c, (in pence) of a
number of paperback books with total number of
pages, p, is recorded, as follows :-
No. of pages (x):
 
150   200   300   350   450
Cost (y):
  
500   580   660   820   900
Use the GDC to Calculate the PMCC, r, for this
data and interpret the relationship.
 
 
Turn on the GDC
 
Press 2
 
Statistics
 
 
Example.
 
The cost, c, (in pence) of a
number of paperback books with total number of
pages, p, is recorded, as follows :-
No. of pages (x):
 
150   200   300   350   450
Cost (y):
  
500   580   660   820   900
Use the GDC to Calculate the PMCC, r, for this
data and interpret the relationship.
 
 
Type the values of No. of pages in the list 1
 
Type the values of Cost in the list 2
Turn on the GDC
Press 2
Statistics
 
 
Example.
 
The cost, c, (in pence) of a
number of paperback books with total number of
pages, p, is recorded, as follows :-
No. of pages (x):
 
150   200   300   350   450
Cost (y):
  
500   580   660   820   900
Use the GDC to Calculate the PMCC, r, for this
data and interpret the relationship.
 
 
F2
 
CALC
Type the values of No. of pages in the list 1
Type the values of Cost in the list 2
Turn on the GDC
Press 2
Statistics
 
 
Example.
 
The cost, c, (in pence) of a
number of paperback books with total number of
pages, p, is recorded, as follows :-
No. of pages (x):
 
150   200   300   350   450
Cost (y):
  
500   580   660   820   900
Use the GDC to Calculate the PMCC, r, for this
data and interpret the relationship.
 
 
F3
 
REG
F2
CALC
Type the values of No. of pages in the list 1
Type the values of Cost in the list 2
Turn on the GDC
Press 2
Statistics
 
 
Example.
 
The cost, c, (in pence) of a
number of paperback books with total number of
pages, p, is recorded, as follows :-
No. of pages (x):
 
150   200   300   350   450
Cost (y):
  
500   580   660   820   900
Use the GDC to Calculate the PMCC, r, for this
data and interpret the relationship.
 
 
F1
 
X
F3
REG
F2
CALC
Type the values of No. of pages in the list 1
Type the values of Cost in the list 2
Turn on the GDC
Press 2
Statistics
 
 
Example.
 
The cost, c, (in pence) of a
number of paperback books with total number of
pages, p, is recorded, as follows :-
No. of pages (x):
 
150   200   300   350   450
Cost (y):
  
500   580   660   820   900
Use the GDC to Calculate the PMCC, r, for this
data and interpret the relationship.
 
 
F1
 
ax + b
F1
X
F3
REG
F2
CALC
Type the values of No. of pages in the list 1
Type the values of Cost in the list 2
Turn on the GDC
Press 2
Statistics
 
 
Example.
 
The cost, c, (in pence) of a
number of paperback books with total number of
pages, p, is recorded, as follows :-
No. of pages (x):
 
150   200   300   350   450
Cost (y):
  
500   580   660   820   900
Use the GDC to Calculate the PMCC, r, for this
data and interpret the relationship.
 
 
r = 0.97964806
 
The value 0.9796 indicates positive Very Strong correlation
F1
ax + b
F1
X
F3
REG
F2
CALC
Type the values of No. of pages in the list 1
Type the values of Cost in the list 2
Turn on the GDC
Press 2
Statistics
 
 
 
Thank you for using resources from
 
https://www.mathssupport.org
 
If you have a special request, drop us an email
 
info@mathssupport.org
 
 
 
For more resources visit our website
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Pearson product-moment correlation coefficient, denoted by "r," is a numerical measure of the strength and direction of a linear relationship between two variables. It ranges from -1 to +1, where 1 signifies a perfect positive linear correlation, -1 a perfect negative linear correlation, and 0 no correlation. This coefficient helps in analyzing the degree of association between variables and is widely used in scientific research to quantify relationships accurately.


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  1. 14 August 2024 Pearson product-moment correlation coefficient (r) LO: Calculate the Pearson product-moment correlation coefficient using the formulae and using the GDC www.mathssupport.org

  2. Correlation. Up to this point we have used the scatter diagram to see if there is a relationship between two variables. We have stated that direction is important to determine the correlation. We have seen that the strength is important when it comes to the correlation. www.mathssupport.org

  3. Direction Negative Correlation:- For a generally downward trend, we say that the correlation is negative. When the independent variable increases, the dependent variable decreases Positive Correlation:- For a generally upward trend, we say that the correlation is positive. An increase in the independent variable means that the dependant variable generally increases. No correlation:- For randomly scattered points, with no upward or downward trend, we say there is no correlation. No apparent link between the variables. www.mathssupport.org

  4. Strength The strength of correlation is usually described as either strong, moderate or weak Strong positive Moderate positive Weak positive Moderate negative Weak negative Strong negative www.mathssupport.org

  5. Correlation We have found the equation of the regression line of y on x and used the line for prediction purposes Now we are going to seek to classify the strength of the correlation numerically There are several scales in use; we will study a correlation coefficient developed by Karl Pearson The Pearson product-moment correlation coefficient denoted by r, is a measure of the correlation between two variables x and y, giving a value between -1 and +1 inclusive. It is widely used in the sciences as a measure of the strength of linear dependence between two variables www.mathssupport.org

  6. Strength The PMCC or regression coefficient is used to determine how nearly the points fall on a straight line, or how nearly linear they are. A perfect correlation will have a regression coefficient of r = 1 If there is no correlation we will have a regression coefficient of r = 0 Perfect negative linear correlation r = -1 Perfect positive linear correlation r = 1 No correlation r = 0 www.mathssupport.org

  7. Strength A quick way to interpret the r-value is r-value Correlation Very weak 0 |r| 0.5 0.5 |r| 0.7 0.7 |r| 0.87 Very Weak Weak Weak Moderate Strong 0.87 |r| 0.95 Very Strong Perfect 0.95 |r| 1 |r| = 1 Strong Moderate www.mathssupport.org

  8. Calculating the Pearsons correlation coefficient (r) We use the following formulas:- ??? ???? Notes:- DON T PANIC the formulas are not that bad to use! There are other versions of these formulas that you may see in books, but the ones here are the easiest to use. x is the independent variable y is the dependent variable n is how many pairs of data you have. ? = ? ? ? ???= ?? ?2 ? ?2 ??= ?2 ? Let us see how to use them in an example ?2 ??= www.mathssupport.org

  9. Example. number of paperback books with total number of pages, p, is recorded, as follows :- No. of pages (x): 150 200 300 350 450 Cost (y): 500 580 660 820 900 Calculate the PMCC, r, for this data and interpret the relationship. x 150 200 300 350 450 900 1450 3460 1081000 477500 2504400 The cost, c, (in pence) of a ? ? ? 1450 3460 5 ???= ?? ???= 1081000 ???= 77600 ?2 ? ?2 ??= xy x2 y2 y 14502 5 75000 116000 198000 287000 405000 22500 40000 90000 122500 202500 250000 336400 435600 672400 810000 500 580 660 820 ??= 477500 ??= 238.75 ?2 ? ?2 ??= 34602 5 ??= 2504400 ??? ???? 77600 = 0.9796 = ? = 238.75 331.78 ??= 331.78 The value 0.9796 indicates positive Very Strong correlation www.mathssupport.org

  10. Correlation coefficient The following table is a guide for describing the strength of linear association using the correlation coefficient. Value r= 0 0 < r < 0.50 0.50 r< 0.70 0.70 r< 0.87 0.87 r< 0.95 0.95 r< 1 r= 1 Strength of association No correlation Very weak correlation Weak correlation Moderate correlation Strong correlation Very strong correlation Perfect correlation You should use your GDC to find the Correlation coefficient (r) www.mathssupport.org

  11. Example. number of paperback books with total number of pages, p, is recorded, as follows :- No. of pages (x): 150 200 300 350 450 Cost (y): 500 580 660 820 900 Use the GDC to Calculate the PMCC, r, for this data and interpret the relationship. The cost, c, (in pence) of a Turn on the GDC Press 2 Statistics www.mathssupport.org

  12. Example. number of paperback books with total number of pages, p, is recorded, as follows :- No. of pages (x): 150 200 300 350 450 Cost (y): 500 580 660 820 900 Use the GDC to Calculate the PMCC, r, for this data and interpret the relationship. The cost, c, (in pence) of a Turn on the GDC Press 2 Statistics Type the values of No. of pages in the list 1 Type the values of Cost in the list 2 www.mathssupport.org

  13. Example. number of paperback books with total number of pages, p, is recorded, as follows :- No. of pages (x): 150 200 300 350 450 Cost (y): 500 580 660 820 900 Use the GDC to Calculate the PMCC, r, for this data and interpret the relationship. The cost, c, (in pence) of a Turn on the GDC Press 2 Statistics Type the values of No. of pages in the list 1 Type the values of Cost in the list 2 F2 CALC www.mathssupport.org

  14. Example. number of paperback books with total number of pages, p, is recorded, as follows :- No. of pages (x): 150 200 300 350 450 Cost (y): 500 580 660 820 900 Use the GDC to Calculate the PMCC, r, for this data and interpret the relationship. The cost, c, (in pence) of a Turn on the GDC Press 2 Statistics Type the values of No. of pages in the list 1 Type the values of Cost in the list 2 F2 CALC F3 REG www.mathssupport.org

  15. Example. number of paperback books with total number of pages, p, is recorded, as follows :- No. of pages (x): 150 200 300 350 450 Cost (y): 500 580 660 820 900 Use the GDC to Calculate the PMCC, r, for this data and interpret the relationship. The cost, c, (in pence) of a Turn on the GDC Press 2 Statistics Type the values of No. of pages in the list 1 Type the values of Cost in the list 2 F2 CALC F3 REG F1 X www.mathssupport.org

  16. Example. number of paperback books with total number of pages, p, is recorded, as follows :- No. of pages (x): 150 200 300 350 450 Cost (y): 500 580 660 820 900 Use the GDC to Calculate the PMCC, r, for this data and interpret the relationship. The cost, c, (in pence) of a Turn on the GDC Press 2 Statistics Type the values of No. of pages in the list 1 Type the values of Cost in the list 2 F2 CALC F3 REG F1 X ax + b F1 www.mathssupport.org

  17. Example. number of paperback books with total number of pages, p, is recorded, as follows :- No. of pages (x): 150 200 300 350 450 Cost (y): 500 580 660 820 900 Use the GDC to Calculate the PMCC, r, for this data and interpret the relationship. The cost, c, (in pence) of a Turn on the GDC Press 2 Statistics Type the values of No. of pages in the list 1 Type the values of Cost in the list 2 F2 CALC F3 REG F1 X ax + b F1 r = 0.97964806 The value 0.9796 indicates positive Very Strong correlation www.mathssupport.org

  18. Thank you for using resources from A close up of a cage Description automatically generated For more resources visit our website https://www.mathssupport.org If you have a special request, drop us an email info@mathssupport.org www.mathssupport.org

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