10.2 Correlation

CGMA BA1

10. Inter-relationships between variables

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Correlation

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Two variables are correlated when they tend to change together. In other words, when the value of one variable changes, the other variable tends to change in a predictable way.

Correlation coefficient

This method measures the relationship between two variables, for example when a linear relationship exists between them.

The formula for the correlation coefficient is:

$r = \frac{n \sum x y - \sum x \sum y}{( n \sum x ^{2} - ( \sum x ) ^{2} ) ( n \sum y ^{2} - ( \sum y ) ^{2} )}$

X = the independent variable
Y = is the dependent variable
N = is the number of data points
Do mind that property always lies in the range of -1 to +1 where
r = +1 denotes perfect positive correlation linear correlation.
r = -1 perfect negative correlation.
r = 0 denotes no correlation.
If the correlation is closer to 1 or -1 is said to be stronger and if it’s closer to 0 is said to be weak.

The following is an example to apply the concept

	Units	Cost
1	10	120
2	20	150
3	30	170
4	40	230
5	50	270

Calculate the correlation coefficient

Solution

	X	Y	XY	X²	Y²
1	10	120	1200	100	14400
2	20	150	3000	400	22500
3	30	170	5100	900	28900
4	40	230	9200	1600	52900
5	50	270	13500	2500	72900
	150	940	32000	5500	191600

The formula is:

r = [ n∑xy - (∑x)(∑y) ] / sqrt( [ n∑x² - (∑x)² ] [ n∑y² - (∑y)² ] )

Step 1: Numerator = 5(32000) - (150)(940) = 160000 - 141000 = 19000

Step 2: First bracket under square root = 27500 - 22500 = 5000

Step 3: Second bracket under square root = 1149600 - 883600 = 266000

Step 4: Product inside square root = 5000 × 266000 = 1,330,000,000

Step 5: Square root of 1,330,000,000 = 36469

Step 6: r = 19000 / 36469 = 0.521

Final answer: r = 0.521

This means that there is strong correlation between the units produced and the cost incurred.

Please follow through with your own calculator using the formula that is listed above.

The coefficient of determination

This gives the proportion of changes in y that can be explained / caused by changes in x.
To get the coefficient of determination you must square the correlation coefficient figure.
For example, if the correlation coefficient is 0.90 then the coefficient of determination is:

$0.9 0^{2} = 0.81$

Interpretation

This means that 81% changes in y are caused / explained by the changes in x.
Do remember y is the dependent variable and x is the independent variable.

Prediction of costs

As you have seen on the high - low method these methods can be used to predict costs.
Unless told in the exam when asked to forecast future costs always use the high - low method as it is short and easy to use.

Rank correlation

Rank correlation is similar to the correlation coefficient, but it uses ranks instead of the original values. You use it when the data is naturally ordered (like grades or positions) or when you want to compare two sets of rankings.

For example, an organization conducting interviews may want to know the relationship between face-to-face interview grades and assessment test scores.

The formula for rank correlation is as follows:

R = 1 - ( n ∑d² / n(n² - 1) )

Interpretations

Correlation of +1 indicates a perfect association of ranks
Correlation of zero indicates no association between ranks
Correlation of -1 indicates a perfect negative association of ranks

The closer correlation is to zero, the weaker the association between the ranks.

The following data was collected from 2 interviews which the job candidates took last week.

Names	Grade (face to face)	Score - assessment
Katie	A	60
Ben	B	70
Russo	C	75
Ketlin	D	52
Jerry	E	55
Maison	F	49

Solution

Rank for face to face	Rank on assessment	d	d²
1	3	-2	4
2	2	0	0
3	1	2	4
4	5	-1	1
5	4	1	1
6	6	0	0
			10

1 - (6(10) / A·36 - D)

= 1 - (60 / 210)

= 1 - 0.29

= 0.71

Such a high positive indicates that a high relationship between the face-to-face interview and the assessment test, which is good news. If the answer was negative that could have indicated that there is a vice versa relationship between the two data sets which will be worrying for the HR manager since they will have to scrap the process.

Correlation

Two variables correlated when they change together in a predictable way
Independent variable = $x$ ; dependent variable = $y$

Correlation Coefficient

Measures strength and direction of linear relationship between two variables
Formula: $r = \frac{n \sum x y - \sum x \sum y}{( n \sum x ^{2} - ( \sum x ) ^{2} ) ( n \sum y ^{2} - ( \sum y ) ^{2} )}$
Range: $- 1 \leq r \leq + 1$
- $r = + 1$ : perfect positive; $r = - 1$ : perfect negative; $r = 0$ : no correlation
- Closer to $\pm 1$ = stronger; closer to $0$ = weaker

Coefficient of Determination

Proportion of changes in $y$ explained by changes in $x$
Calculated by squaring $r$ : e.g., $r = 0.90 \Rightarrow r^{2} = 0.81$ means 81% of changes in $y$ explained by $x$

Predicting Costs

Correlation methods can forecast future costs
Default method in exams: high-low method (unless told otherwise)

Rank Correlation

Used when data is ranked/ordered rather than numerical
Formula: $R = 1 - \frac{6 \sum d ^{2}}{n ( n ^{2} - 1 )}$ where $d$ = difference between paired ranks
Interpretation same as $r$ : $+ 1$ perfect positive, $0$ no association, $- 1$ perfect negative

Next | 11.1 Fundamentals of time series model

Correlation

Two variables are correlated when they tend to change together. In other words, when the value of one variable changes, the other variable tends to change in a predictable way.

Correlation coefficient

This method measures the relationship between two variables, for example when a linear relationship exists between them.

The formula for the correlation coefficient is:

$r = \frac{n \sum x y - \sum x \sum y}{( n \sum x ^{2} - ( \sum x ) ^{2} ) ( n \sum y ^{2} - ( \sum y ) ^{2} )}$

X = the independent variable
Y = is the dependent variable
N = is the number of data points
Do mind that property always lies in the range of -1 to +1 where
r = +1 denotes perfect positive correlation linear correlation.
r = -1 perfect negative correlation.
r = 0 denotes no correlation.
If the correlation is closer to 1 or -1 is said to be stronger and if it’s closer to 0 is said to be weak.

The following is an example to apply the concept

	Units	Cost
1	10	120
2	20	150
3	30	170
4	40	230
5	50	270

Calculate the correlation coefficient

Solution

	X	Y	XY	X²	Y²
1	10	120	1200	100	14400
2	20	150	3000	400	22500
3	30	170	5100	900	28900
4	40	230	9200	1600	52900
5	50	270	13500	2500	72900
	150	940	32000	5500	191600

The formula is:

r = [ n∑xy - (∑x)(∑y) ] / sqrt( [ n∑x² - (∑x)² ] [ n∑y² - (∑y)² ] )

Step 1: Numerator = 5(32000) - (150)(940) = 160000 - 141000 = 19000

Step 2: First bracket under square root = 27500 - 22500 = 5000

Step 3: Second bracket under square root = 1149600 - 883600 = 266000

Step 4: Product inside square root = 5000 × 266000 = 1,330,000,000

Step 5: Square root of 1,330,000,000 = 36469

Step 6: r = 19000 / 36469 = 0.521

Final answer: r = 0.521

This means that there is strong correlation between the units produced and the cost incurred.

Please follow through with your own calculator using the formula that is listed above.

The coefficient of determination

This gives the proportion of changes in y that can be explained / caused by changes in x.
To get the coefficient of determination you must square the correlation coefficient figure.
For example, if the correlation coefficient is 0.90 then the coefficient of determination is:

$0.9 0^{2} = 0.81$

Interpretation

This means that 81% changes in y are caused / explained by the changes in x.
Do remember y is the dependent variable and x is the independent variable.

Prediction of costs

As you have seen on the high - low method these methods can be used to predict costs.
Unless told in the exam when asked to forecast future costs always use the high - low method as it is short and easy to use.

Rank correlation

For example, an organization conducting interviews may want to know the relationship between face-to-face interview grades and assessment test scores.

The formula for rank correlation is as follows:

R = 1 - ( n ∑d² / n(n² - 1) )

Interpretations

Correlation of +1 indicates a perfect association of ranks
Correlation of zero indicates no association between ranks
Correlation of -1 indicates a perfect negative association of ranks

The closer correlation is to zero, the weaker the association between the ranks.

The following data was collected from 2 interviews which the job candidates took last week.

Names	Grade (face to face)	Score - assessment
Katie	A	60
Ben	B	70
Russo	C	75
Ketlin	D	52
Jerry	E	55
Maison	F	49

Solution

Rank for face to face	Rank on assessment	d	d²
1	3	-2	4
2	2	0	0
3	1	2	4
4	5	-1	1
5	4	1	1
6	6	0	0
			10

1 - (6(10) / A·36 - D)

= 1 - (60 / 210)

= 1 - 0.29

= 0.71

CGMA BA1

10. Inter-relationships between variables

Our CGMA course is currently in development and is a work-in-progress.

Correlation

4 min read

Font

Discuss

Feedback

Two variables are correlated when they tend to change together. In other words, when the value of one variable changes, the other variable tends to change in a predictable way.

Correlation coefficient

This method measures the relationship between two variables, for example when a linear relationship exists between them.

The formula for the correlation coefficient is:

$r = \frac{n \sum x y - \sum x \sum y}{( n \sum x ^{2} - ( \sum x ) ^{2} ) ( n \sum y ^{2} - ( \sum y ) ^{2} )}$

X = the independent variable
Y = is the dependent variable
N = is the number of data points
Do mind that property always lies in the range of -1 to +1 where
r = +1 denotes perfect positive correlation linear correlation.
r = -1 perfect negative correlation.
r = 0 denotes no correlation.
If the correlation is closer to 1 or -1 is said to be stronger and if it’s closer to 0 is said to be weak.

The following is an example to apply the concept

	Units	Cost
1	10	120
2	20	150
3	30	170
4	40	230
5	50	270

Calculate the correlation coefficient

Solution

	X	Y	XY	X²	Y²
1	10	120	1200	100	14400
2	20	150	3000	400	22500
3	30	170	5100	900	28900
4	40	230	9200	1600	52900
5	50	270	13500	2500	72900
	150	940	32000	5500	191600

The formula is:

r = [ n∑xy - (∑x)(∑y) ] / sqrt( [ n∑x² - (∑x)² ] [ n∑y² - (∑y)² ] )

Step 1: Numerator = 5(32000) - (150)(940) = 160000 - 141000 = 19000

Step 2: First bracket under square root = 27500 - 22500 = 5000

Step 3: Second bracket under square root = 1149600 - 883600 = 266000

Step 4: Product inside square root = 5000 × 266000 = 1,330,000,000

Step 5: Square root of 1,330,000,000 = 36469

Step 6: r = 19000 / 36469 = 0.521

Final answer: r = 0.521

This means that there is strong correlation between the units produced and the cost incurred.

Please follow through with your own calculator using the formula that is listed above.

The coefficient of determination

This gives the proportion of changes in y that can be explained / caused by changes in x.
To get the coefficient of determination you must square the correlation coefficient figure.
For example, if the correlation coefficient is 0.90 then the coefficient of determination is:

$0.9 0^{2} = 0.81$

Interpretation

This means that 81% changes in y are caused / explained by the changes in x.
Do remember y is the dependent variable and x is the independent variable.

Prediction of costs

As you have seen on the high - low method these methods can be used to predict costs.
Unless told in the exam when asked to forecast future costs always use the high - low method as it is short and easy to use.

Rank correlation

For example, an organization conducting interviews may want to know the relationship between face-to-face interview grades and assessment test scores.

The formula for rank correlation is as follows:

R = 1 - ( n ∑d² / n(n² - 1) )

Interpretations

Correlation of +1 indicates a perfect association of ranks
Correlation of zero indicates no association between ranks
Correlation of -1 indicates a perfect negative association of ranks

The closer correlation is to zero, the weaker the association between the ranks.

The following data was collected from 2 interviews which the job candidates took last week.

Names	Grade (face to face)	Score - assessment
Katie	A	60
Ben	B	70
Russo	C	75
Ketlin	D	52
Jerry	E	55
Maison	F	49

Solution

Rank for face to face	Rank on assessment	d	d²
1	3	-2	4
2	2	0	0
3	1	2	4
4	5	-1	1
5	4	1	1
6	6	0	0
			10

1 - (6(10) / A·36 - D)

= 1 - (60 / 210)

= 1 - 0.29

= 0.71

Correlation

Two variables correlated when they change together in a predictable way
Independent variable = $x$ ; dependent variable = $y$

Correlation Coefficient

Measures strength and direction of linear relationship between two variables
Formula: $r = \frac{n \sum x y - \sum x \sum y}{( n \sum x ^{2} - ( \sum x ) ^{2} ) ( n \sum y ^{2} - ( \sum y ) ^{2} )}$
Range: $- 1 \leq r \leq + 1$
- $r = + 1$ : perfect positive; $r = - 1$ : perfect negative; $r = 0$ : no correlation
- Closer to $\pm 1$ = stronger; closer to $0$ = weaker

Coefficient of Determination

Proportion of changes in $y$ explained by changes in $x$
Calculated by squaring $r$ : e.g., $r = 0.90 \Rightarrow r^{2} = 0.81$ means 81% of changes in $y$ explained by $x$

Predicting Costs

Correlation methods can forecast future costs
Default method in exams: high-low method (unless told otherwise)

Rank Correlation

Used when data is ranked/ordered rather than numerical
Formula: $R = 1 - \frac{6 \sum d ^{2}}{n ( n ^{2} - 1 )}$ where $d$ = difference between paired ranks
Interpretation same as $r$ : $+ 1$ perfect positive, $0$ no association, $- 1$ perfect negative

Next | 11.1 Fundamentals of time series model

Correlation

Two variables are correlated when they tend to change together. In other words, when the value of one variable changes, the other variable tends to change in a predictable way.

Correlation coefficient

This method measures the relationship between two variables, for example when a linear relationship exists between them.

The formula for the correlation coefficient is:

$r = \frac{n \sum x y - \sum x \sum y}{( n \sum x ^{2} - ( \sum x ) ^{2} ) ( n \sum y ^{2} - ( \sum y ) ^{2} )}$

X = the independent variable
Y = is the dependent variable
N = is the number of data points
Do mind that property always lies in the range of -1 to +1 where
r = +1 denotes perfect positive correlation linear correlation.
r = -1 perfect negative correlation.
r = 0 denotes no correlation.
If the correlation is closer to 1 or -1 is said to be stronger and if it’s closer to 0 is said to be weak.

The following is an example to apply the concept

	Units	Cost
1	10	120
2	20	150
3	30	170
4	40	230
5	50	270

Calculate the correlation coefficient

Solution

	X	Y	XY	X²	Y²
1	10	120	1200	100	14400
2	20	150	3000	400	22500
3	30	170	5100	900	28900
4	40	230	9200	1600	52900
5	50	270	13500	2500	72900
	150	940	32000	5500	191600

The formula is:

r = [ n∑xy - (∑x)(∑y) ] / sqrt( [ n∑x² - (∑x)² ] [ n∑y² - (∑y)² ] )

Step 1: Numerator = 5(32000) - (150)(940) = 160000 - 141000 = 19000

Step 2: First bracket under square root = 27500 - 22500 = 5000

Step 3: Second bracket under square root = 1149600 - 883600 = 266000

Step 4: Product inside square root = 5000 × 266000 = 1,330,000,000

Step 5: Square root of 1,330,000,000 = 36469

Step 6: r = 19000 / 36469 = 0.521

Final answer: r = 0.521

This means that there is strong correlation between the units produced and the cost incurred.

Please follow through with your own calculator using the formula that is listed above.

The coefficient of determination

This gives the proportion of changes in y that can be explained / caused by changes in x.
To get the coefficient of determination you must square the correlation coefficient figure.
For example, if the correlation coefficient is 0.90 then the coefficient of determination is:

$0.9 0^{2} = 0.81$

Interpretation

This means that 81% changes in y are caused / explained by the changes in x.
Do remember y is the dependent variable and x is the independent variable.

Prediction of costs

As you have seen on the high - low method these methods can be used to predict costs.
Unless told in the exam when asked to forecast future costs always use the high - low method as it is short and easy to use.

Rank correlation

For example, an organization conducting interviews may want to know the relationship between face-to-face interview grades and assessment test scores.

The formula for rank correlation is as follows:

R = 1 - ( n ∑d² / n(n² - 1) )

Interpretations

Correlation of +1 indicates a perfect association of ranks
Correlation of zero indicates no association between ranks
Correlation of -1 indicates a perfect negative association of ranks

The closer correlation is to zero, the weaker the association between the ranks.

The following data was collected from 2 interviews which the job candidates took last week.

Names	Grade (face to face)	Score - assessment
Katie	A	60
Ben	B	70
Russo	C	75
Ketlin	D	52
Jerry	E	55
Maison	F	49

Solution

Rank for face to face	Rank on assessment	d	d²
1	3	-2	4
2	2	0	0
3	1	2	4
4	5	-1	1
5	4	1	1
6	6	0	0
			10

1 - (6(10) / A·36 - D)

= 1 - (60 / 210)

= 1 - 0.29

= 0.71