ACCA F5 Performance Management - 2010 - Study text

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Chapter 8: Quantitative analysis in budgeting

Cost estimation: linear regression analysis

 The purpose of linear regression analysis

 The linear regression formulae

 Applying the linear regression formulae

2 Cost estimation: linear regression analysis

2.1 The purpose of linear regression analysis

Linear regression analysis is a statistical technique for calculating a line of best fit

from a set of data:

y = a + bx

The data is in ‘pairs’, which means that there are a number of different values for x,

and for each value of x there is an associated value of y in the data.

Linear regression analysis can be used to estimate fixed costs and the variable cost

per unit from historical data for total costs. It is an alternative to the high-low

method.

Linear regression analysis can also be used to predict future sales by projecting the

historical sales trend into the future (on the assumption that sales growth is rising at

a constant rate, in a ‘straight line’).

Regression analysis and high-low analysis compared

There are important differences between linear regression analysis and the high-low

method.

 High-low analysis uses just two sets of data for x and y, the highest value for x

and the lowest value for x. Regression analysis uses as many sets of data for x

and y as are available.

 Because regression analysis calculates a line of best fit for all the available data, it

is likely to provide a more reliable estimate than high-low analysis for the values

of a and b.

 In addition, regression analysis can be used to assess the extent to which values

of y depend on values of x. For example, if a line of best fit is calculated that

estimates total costs for any volume of production, we can also calculate the

extent to which total costs do seem to be linked (or ‘correlated’) to the volume of

production. This is done by calculating a correlation co-efficient, which is

explained later.

 Regression analysis uses more complex arithmetic than high-low analysis, and a

calculator or small spreadsheet model is normally needed

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In summary, linear regression analysis is a better technique than high-low analysis

because:

 it is more reliable and

 its reliability can be measured.

2.2 The regression analysis formulae

The formulae for estimating costs using regression analysis are provided in a

formulae sheet in your examination. You might be required to apply the formulae to

data provided in an examination question.

Linear regression analysis is used to calculate values for a and b in the linear cost

equation: y = a + bx.

The linear regression formulae for calculating a and b are shown below. The

number of pairs of data that are used in the calculation is n.

The regression analysis formulae are as follows. They will be given to you in your

examination, so you do not need to learn them. However, you may be required to

use them:

a =

∑

−

∑

b =

∑

xy –

∑

n ∑ x

–(∑x)

where:

n = the number of pairs of data used for x and y

x, y represent the values of x (volume of activity or output) and y (total cost)

The value of ‘b’ is an item in the formula for calculating ‘a’. It is therefore necessary

to calculate ‘b’ first before calculating ‘a’.

Understanding the formulae

These formulae might seem complicated, but they involve the application of fairly

simple arithmetic.

 You should think of the pairs of data as a list of values set out in two columns,

with one column for the values of x and the second column for the associated

values of y. Typically x represents the volume of activity and y is the total cost.

 You should then prepare two more columns.

 There should be an additional column for x². Calculate the square of each value

of x in the x column and write the answer in the x² column.

 The fourth column is for xy. For each pair of data for x and y, multiply the value

of x by the value of y, and enter the answer in the xy column.

 You must then add up the totals for each of the four columns. ∑x is the total of

all the values in the x column. ∑y is the total of all the values in the y

column.∑x² is the total of all the values in the x² column and ∑xy is the total of

all the values in the xy column.

Chapter 8: Quantitative analysis in budgeting

 You now have all the values you need to calculate a value for b in the formula y

= a + bx.

 The only other item you might not be sure about is (∑x)². This is not the same as

∑x². It is the square of the value for ∑x.

 Having calculated a value for b, you can then use the formula for calculating the

value of a: this includes the value for b, which is why you need to calculate b

first.

2.3 Applying the regression analysis formulae

The formulae might seem complicated, but they are actually fairly straightforward

provided that you understand the meaning of the ∑ items in the formulae, and

remember that n is the number of pairs of data used to make the estimate. Study the

following example carefully. It uses the same data as one of the previous examples

for the high-low method.

Example

A company has recorded the following costs in the past six months:

Month Activity

Total cost

Direct labour hours,

in 000

$000

January 5.8

40.3

February 7.7

47.1

March 8.2

48.7

April 6.1

40.6

May 6.5

44.5

June 7.5

47.1

Required

Using linear regression analysis, prepare an estimate of total costs in July if output is

expected to be 7,000 direct labour hours.

Answer

The starting point is to draw a table showing the values of x (output) and y (total

cost). For each value of x and y, you should calculate the value of:

 x

and

 xy.

For reasons to be explained later, the value of y

will also be calculated here.

There are six pairs of data, so there should be six different values for x, y, x

, xy and y

Add the figures in each column to obtain totals for:

 ∑x (= the sum of the values of x)

 ∑y (= the sum of the values of y)

 ∑x

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 ∑xy

 ∑y

. (The reason for calculating ∑y

will be explained later.)

Month Output Total cost

000 units

$000

xy y

January 5.8

40.3

33.64

233.74 1,624.09

February 7.7

47.1

59.29

362.67 2,218.41

March 8.2

48.7

67.24

399.34 2,371.69

April 6.1

40.6

37.21

247.66 1,648.36

May 6.5

44.5

42.25

289.25 1,980.25

June 7.5

47.1

56.25

353.25 2,218.41

41.8

268.3

295.88

1,885.91 12,061.21

∑x

= ∑y

= ∑x

= ∑xy

= ∑y

There are six pairs of data, so n = 6.

We now have all the figures we need to calculate values for ‘a’ and ‘b, starting with ‘b’.

b =

n ∑ xy – ∑ x∑ y

n ∑ x

– ∑x

()

3.585 b

28.04

100.52

1,747.24–1,775.28

11,214.94–11,315.46

)41.8(–)295.88( 6

)268.3)(41.8(–)1,885.91( 6

===

This is the estimate of the variable cost per unit. The value of ‘b’ is used to calculate

a value for ‘a’, as follows.

19.74 = $000)(in a

24.98 –44.72

)41.8( 3.585

–

268.3

–

∑∑

Answer (a)

Estimated fixed costs each month = a = $19,740

Estimated variable cost per unit = b = $3.585

Therefore, linear cost function, y, is estimated to be:

y = 19,740 + 3.585x

For simplicity, this could be rounded to:

y = 19,700 + 3.6x.

Chapter 8: Quantitative analysis in budgeting

Answer (b)

Using y = 19,700 + 3.6x, when monthly output is expected to be 7,000 hours, the

expected total costs will be:

Fixed 19,700

Variable (7,000 × $3.6) 25,200

44,900

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Correlation and the correlation coefficient

 Correlation

 Degrees of correlation

 Correlation coefficient, r

 Coefficient of determination, r

3 Correlation and the correlation coefficient

3.1 Correlation

Linear regression analysis can be applied to any sets of data where the data is in

pairs (x and y). It could be used, for example, to calculate a line of best fit for total

weekly costs of an activity by taking pairs of data for total weekly costs (= y) and the

associated values for the activity. However, the activity level might be any of the

following:

 units of output each week, if it is assumed that weekly costs vary with the

volume of output produced

 direct labour hours each week, if it is assumed that weekly costs vary with the

number of direct labour hours worked

 machine hours operated each week, if it is assumed that weekly costs vary with

the number of machine hours.

Which activity is the best one to choose as ‘x’?

Since a line of best fit can be calculated between any two variables using regression

analysis, important questions are:

 How can we assess the reliability of the line of best fit?’

 How do we know whether our choice of activity as ‘x’ is a good one?

The answer to both questions is that the reliability of the regression formula can be

assessed using a statistic called the coefficient of correlation.

Correlation is a measure of the strength of the relationship between two variables.

Variables are said to be correlated if a change in one variable results in a change in

the other variable.

If you plot a graph of the data relating to two variables, you should be able to see if

any visible relationship exists between them (such a graph is known as a

scattergraph or scatterchart). If a visible relationship is seen to exist, the data on the

graph can be plotted to show the ‘line of best fit’. The ‘line of best fit’ is of the form

y = a + bx (linear cost function) and indicates that a ‘possible’ linear relationship

exists between two variables.

Chapter 8: Quantitative analysis in budgeting

3.2 Degrees of correlation

The following scattergraphs show the different degrees of correlation that may be

seen to exist between two variables.

Perfect positive correlation

Perfect correlation is seen to exist when all the data points plotted lie in an exact

straight line and a linear relationship exists between the two variables.

Perfect positive correlation means that high values of are variable are associated

with high values of another variable. Alternatively, low values of one variable may

be associated with low values of another variable.

Perfect negative correlation

Perfect negative correlation is seen to exist when all the data points plotted lie in an

exact straight line and that high values of one variable are associated with low

values of another variable. Alternatively, low values of one variable may be

associated with high value of another variable.

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No correlation (uncorrelated)

‘Uncorrelated’ means that no correlation is seen to exist between the variables.

Positive correlation (but not perfect correlation)

Positive correlation means that there appears to be some correlation between the

values of y and x, and in general the value of y increases as the value of x increases.

However, the correlation is not perfect because all the data does not lie exactly on a

straight line on the scattergraph.

Negative correlation

Chapter 8: Quantitative analysis in budgeting

Negative correlation means that a relationship exists between two variables, and the

value of y declines as the value of x increases. However, the correlation is not

perfect.

As an alternative to drawing a scattergraph to see whether a visible relationship

exists between the two variables x and y, the correlation between them can be

measured by calculating a

correlation coefficient.

3.3 Correlation coefficient r

An advantage of using the regression analysis method is that the reliability of the

estimates can be assessed statistically, by calculating a correlation coefficient.

The formula for the correlation coefficient (r) will be given to you in the

examination.

r =

∑

xy –

∑

n∑ x

– ∑ x

()

n ∑ y

– ∑ y

()

This formula might seem difficult, but it is fairly similar to the formula for

calculating ‘b’ in the linear cost equation. The only additional value that we need to

calculate the correlation coefficient is a value for [n∑y

– (∑y)

 If you look back at the previous example, you will see that the value for ∑y

has

already been calculated. In the example, there is a separate column for y

Calculate the square of each value of y in the y column and enter it in the same

row in the y

column

Then add up all the values of y

in the y

column to get

∑y

 (∑y)

is a different value. It is the square of the value for ∑y.

The value of the correlation coefficient, r, in the example in paragraph 2.3 is

therefore calculated as follows. Remember that most of the values for the formula

have been calculated in paragraph 2.3 as parts of the formula for calculating the

value of b.

0.97+ =

103.55

100.52

(382.37) (28.04)

100.52

](268.3)-21)[6(12,061. (28.04)

100.52

The correlation coefficient r is + 0.97.

Significance of the correlation coefficient

The value of the correlation coefficient must always be in the range – 1 to + 1.

 A value of – 1 indicates that there is perfect negative correlation between the

values for y and the values for x that have been used in the regression analysis

estimates. Perfect negative correlation means that all the values for x and y,

plotted on a graph, would lie on a straight downward-sloping line.

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 A value of + 1 indicates that there is perfect positive correlation between the

values for y and the values for x that have been used in the regression analysis

estimates. Perfect positive correlation means that all the values for x and y,

plotted on a graph, would lie on a straight upward-sloping line.

 A value of r = 0 indicates no correlation at all between the values of x and y.

For cost estimation, a value for r close to + 1 would indicate that the cost estimates

are likely to be very reliable.

As a general guide, a value for r between + 0.90 and + 1 indicates good correlation

between the values of x and y, suggesting that the formula for costs can be used

with reasonable confidence for cost estimation.

If you calculate a value for r that is more than +1 or is a greater negative value than

– 1, your calculation will be wrong.

3.4 Coefficient of determination r

The square of the correlation coefficient, r

, is called the coefficient of determination.

The value of r

shows how much the variations in the value of y, in the data used to

calculate the regression analysis formula, can be explained by variations in the value

of x.

Significance of coefficient of determination

The value of the coefficient of determination must always be in the range 0 to +1.

 If the value of r is + 0.70, this means that on the basis of the data used in the

regression analysis formula, 0.49 or 49% (= 0.70

) of variations in the value of y

can be explained by variations in the value of x.

 Similarly if the value of r is – 0.80, this means that on the basis of the data used

in the regression analysis formula, 0.64 or 64% (= 0.80

) of variations in the value

of y can be explained by variations in the value of x. Since r is negative, this

means that y falls in value as the value of x increases.

In the example above, where r = + 0.97, we can say that from the data used to

produce a formula for total costs, 94.09% (0.97 × 0.97) of the variations in total cost

can be explained by variations in the volume of output. This would suggest that the

formula obtained for total costs is likely to be fairly reliable for estimating future

costs for any given (budgeted) volume of production.

The coefficient of correlation and the coefficient of determination can therefore be

used to give a statistical measurement to the reliability of estimates of y from a

given value for x, using a line of best fit that has been calculated by linear regression

analysis. As you might imagine, this can be a useful item of management

information for the purpose of forecasting or planning.

ACCA F5 Performance Management - 2010 - Study text - Emile Woolf Publishing

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