ANOVA: A Test of Analysis
of Variance
By Harry Lee and Manik
Kuchroo
What is the ANOVA Test?
• Remember the 2-Mean T-Test?
• For example: A salesman in car sales wants
to find the difference between two types of
cars in terms of mileage:
• Mid-Size Vehicles
• Sports Utility Vehicles
Car Salesman’s Sample
The salesman took an independent SRS
from each population of vehicles:
Level
n
Mean
StDev
Mid-size 28 27.101 mpg 2.629 mpg
SUV
26 20.423 mpg 2.914 mpg
If a 2-Mean TTest were done on this data:
T = 8.15
P-value = ~0
What if the salesman wanted to compare
another type of car, Pickup Trucks in addition
to the SUV’s and Mid-size vehicles?
Level
Midsize
SUV
Pickup
n
28
26
8
Mean
27.101 mpg
20.423 mpg
23.125 mpg
StDev
2.629 mpg
2.914 mpg
2.588 mpg
This is an example of when we would
use the ANOVA Test.
In a 2-Mean TTest, we see if the
difference between the 2 sample means
is significant.
The ANOVA is used to compare multiple
means, and see if the
difference between multiple sample
means is significant.
Let’s Compare the Means…
Yes, we see that no two of
these confidence intervals
overlap, therefore the means
are significantly different.
This is the question that the
ANOVA test answers
Do these
sample means look
mathematically.
significantly different from each other?
More Confidence Intervals
Not
Significant
Significant
What if the confidence intervals were
different? Would these confidence
intervals be significantly different?
ANOVA Test Hypotheses
H0: µ1 = µ2 = µ3 (All of the means are equal)
HA: Not all of the means are equal
For Our Example:
H0: µMid-size = µSUV = µPickup
The mean mileages of Mid-size vehicles, Sports
Utility Vehicles, and Pickup trucks are all equal.
HA: Not all of the mean mileages of Mid-size vehicles,
Sports Utility Vehicles, and Pickup trucks are equal.
F Statistic
• Like any other test, the ANOVA test has
its own test statistic
• The statistic for ANOVA is called the F
statistic, which we get from the F Test
• The F statistic takes into consideration:
– number of samples taken (I)
– sample size of each sample (n1, n2, …, nI)
– means of the samples ( x1, x2, …, xI)
– standard deviations of each sample (s1, s2,
…, sI)
Explaining the F-Statistic
• The F statistic determines if the variation
between sample means is significant
VariationAmongSample Means
VariationAmongIndividuals In Each Sample
This is what we are doing when we look at the
95% confidence intervals.
Another Look at the CI’s
From this picture, we can see that the
variation between sample means is greater
than the variation in each sample;
therefore, F is large.
F Statistic Equation
Rewritten as a formula, the F
Statistic looks like this:
Means (Squared)
Weighing
n1 ( x1  x ) 2  n2 ( x2  x ) 2  ...  nI ( xI  x ) 2
I 1
F
(n1  1) s12  (n2  1) s22  ...  (nI  1) sI2
N I
Weighing
Standard Deviations (Squared)
The F Statistic
Degrees of Freedom
• The ANOVA test has 2 degrees of freedom:
– N-I (Total number sampled – Number of Groups)
– I-1 (Number of Groups – 1)
• Some sample
distributions with
different degrees
of freedom:
How About Our Example:
Data:
Level
n
Mean
StDev
Midsize 28 27.101 mpg 2.629 mpg
SUV
26 20.423 mpg 2.914 mpg
Pickup 8 23.125 mpg 2.588 mpg
F value = 40.05
P-value = ~0 (Found from a table or using
the Fcdf calculator command).
Conditions
As useful as the ANOVA test is, we can only
use it if a number of conditions are met:
• We must take an independent SRS from
each population that we sample
• All populations have the same standard
deviation. (No population’s standard
deviation is double another’s)
• All of the populations must be normally
distributed
Testing the Conditions
• The salesman had originally taken independent
SRS’s.
• The second condition is fulfilled since no
sample has more than twice the standard
deviation of any other.
• To test the third condition, whether the
populations being sampled are normally
shaped, we must look at the histograms of
each sample:
Sample Histograms
All of the histograms
appear to be relatively
normally shaped.
Try a Problem
• Researchers are trying to see if the
English AP scores from four different
Massachusetts private schools are
different. From each school, a random
sample of students in the past year was
taken and compared. Here are the results
from the samples:
Results
School
BB&N
Roxbury Latin
Winsor
Belmont Hill
n
23
25
26
29
Mean
4.3
3.9
4.2
3.1
StDev
0.4
0.6
0.3
0.3
Is there any significant difference between these
schools’ AP English scores? (Assume that the
populations are normally distributed)
Hypotheses
• H0: = µBB&N µRL = µWinsor = µBelHill
The mean AP English Test scores in
BB&N, Roxbury Latin, Winsor, and
Belmont Hill are all the same.
• HA: The mean AP English Test scores in
BB&N, Roxbury Latin, Winsor, and
Belmont Hill are not all the same.
Conditions
• Random samples taken
• All of the standard deviations are the same
– No standard deviation is more than twice any
other.
• All of the populations are normally
distributed
Doing out the F Statistic
F Curve
• Plug the F statistic into the F distribution (df = 3,
99). The shaded area has a p-value of nearly 0.
Interpretation
Since all the conditions were met, we have
conclusive evidence (df = 3,99, p = 0) to
reject the null hypothesis that the mean AP
English Test scores in BB&N, Roxbury
Latin, Winsor, and Belmont Hill are all the
same.
Thanks For Watching
• A special thanks to Mr. Coons for all the
help and advice.