OPMT 1197
Business Statistics
Lecture 21 Analysis of Variance (ANOVA)
We did hypothesis tests of population means and proportions for one sample. In the chi-squared
test of independence we tested for the differences between two or more proportions.
We will now look at multiple samples (instead of just one sample) and test to see if there is a
significant difference between the sample means. The kinds of hypothesis you might test are:
There is a significant difference in the students’ grades between the different business
technologies at BCIT: Marketing, Financial Management, and Business Management.
There is a significant difference in home heating bill costs between the various forms of heating:
electrical heat, gas furnace, or heat pump.
Ex 1: A large accounting firm employees over 100 accountants at its three offices in Vancouver,
Toronto and Montreal. The partners want to see if the level of an accountant’s tax knowledge
differs between the three offices. To assess their tax knowledge a random sample of six accountants
were selected from each office and given an exam on income tax rules and regulations.
Exam Scores for the 18 Accountants:
Vancouver Toronto Montreal
85 71 59
75 75 64
82 73 62
76 74 69
71 69 75
85 82 67
Although we don’t know the actual values of the population means we will use the sample
results to test the hypotheses. We want to see if there is a difference in the means between the
three offices. (This is what we are trying to prove so this will always be our alternative
hypothesis). Our null hypothesis is always there is no difference between the means.
If the means for the three populations are equal, we would expect the three sample means to be
close together. In fact, the closer the three sample means are to one another, the more it supports the
conclusion that the population means are equal. The further apart the sample means are the more
evidence we have to support the conclusion that the population means are different.
Analysis of variance is a statistical procedure used to determine whether the
observed differences in the three sample means are enough to reject H0.
The two variables are office location and exam score. Office location (Vancouver, Toronto,
Montreal) is a categorical variable while Exam Score is a quantitative or numerical variable.
Because the objective is to determine whether the mean exam scores are different between the
three offices, the exam scores are referred to as the dependent or response variable and office
location is referred to at the independent variable or factor.
Pg 1 of 6
, OPMT 1197
Business Statistics
The hypothesis we test is:
H0: µ1 = µ2= µ3
HA: not all population means are equal (or at least one mean is different)
Analysis of variance splits the total variation into the variation between groups and the
variation within groups. If there is significantly more variation between the groups than within
the groups we conclude that at least one of the means is different.
Sum of Squares Total (total variation)
n
g j
SST = (xij x )2
j 1i 1
where x = ith value in group j nj = number of values in group j g = number of groups
ij
Calculate the overall mean of all the values as if they were part of the same group (= grand mean)
n
g j
xij
j 1i 1
Grand Mean: x n = total number of values (n = n1 + n2 + n3 + …)
n
The total sum of squares measures the amount of variation between each data value and the
grand mean. Calculate how far each data value is from the grand mean and then square this
difference. Then add the squared differences to get SST.
Exam Scores Grand Mean Difference Squared Differences
x x x x (x x )2
85 73 12 144
75 73 2 4
82 73 9 81
76 73 3 9
71 73 -2 4
85 73 12 144
71 73 -2 4
75 73 2 4
73 73 0 0
74 73 1 1
69 73 -4 16
82 73 9 81
59 73 -14 196
64 73 -9 81
62 73 -11 121
69 73 -4 16
75 73 2 4
67 73 -6 36
Adds to 0 Sum = 946
Pg 2 of 6
Business Statistics
Lecture 21 Analysis of Variance (ANOVA)
We did hypothesis tests of population means and proportions for one sample. In the chi-squared
test of independence we tested for the differences between two or more proportions.
We will now look at multiple samples (instead of just one sample) and test to see if there is a
significant difference between the sample means. The kinds of hypothesis you might test are:
There is a significant difference in the students’ grades between the different business
technologies at BCIT: Marketing, Financial Management, and Business Management.
There is a significant difference in home heating bill costs between the various forms of heating:
electrical heat, gas furnace, or heat pump.
Ex 1: A large accounting firm employees over 100 accountants at its three offices in Vancouver,
Toronto and Montreal. The partners want to see if the level of an accountant’s tax knowledge
differs between the three offices. To assess their tax knowledge a random sample of six accountants
were selected from each office and given an exam on income tax rules and regulations.
Exam Scores for the 18 Accountants:
Vancouver Toronto Montreal
85 71 59
75 75 64
82 73 62
76 74 69
71 69 75
85 82 67
Although we don’t know the actual values of the population means we will use the sample
results to test the hypotheses. We want to see if there is a difference in the means between the
three offices. (This is what we are trying to prove so this will always be our alternative
hypothesis). Our null hypothesis is always there is no difference between the means.
If the means for the three populations are equal, we would expect the three sample means to be
close together. In fact, the closer the three sample means are to one another, the more it supports the
conclusion that the population means are equal. The further apart the sample means are the more
evidence we have to support the conclusion that the population means are different.
Analysis of variance is a statistical procedure used to determine whether the
observed differences in the three sample means are enough to reject H0.
The two variables are office location and exam score. Office location (Vancouver, Toronto,
Montreal) is a categorical variable while Exam Score is a quantitative or numerical variable.
Because the objective is to determine whether the mean exam scores are different between the
three offices, the exam scores are referred to as the dependent or response variable and office
location is referred to at the independent variable or factor.
Pg 1 of 6
, OPMT 1197
Business Statistics
The hypothesis we test is:
H0: µ1 = µ2= µ3
HA: not all population means are equal (or at least one mean is different)
Analysis of variance splits the total variation into the variation between groups and the
variation within groups. If there is significantly more variation between the groups than within
the groups we conclude that at least one of the means is different.
Sum of Squares Total (total variation)
n
g j
SST = (xij x )2
j 1i 1
where x = ith value in group j nj = number of values in group j g = number of groups
ij
Calculate the overall mean of all the values as if they were part of the same group (= grand mean)
n
g j
xij
j 1i 1
Grand Mean: x n = total number of values (n = n1 + n2 + n3 + …)
n
The total sum of squares measures the amount of variation between each data value and the
grand mean. Calculate how far each data value is from the grand mean and then square this
difference. Then add the squared differences to get SST.
Exam Scores Grand Mean Difference Squared Differences
x x x x (x x )2
85 73 12 144
75 73 2 4
82 73 9 81
76 73 3 9
71 73 -2 4
85 73 12 144
71 73 -2 4
75 73 2 4
73 73 0 0
74 73 1 1
69 73 -4 16
82 73 9 81
59 73 -14 196
64 73 -9 81
62 73 -11 121
69 73 -4 16
75 73 2 4
67 73 -6 36
Adds to 0 Sum = 946
Pg 2 of 6
