Lesson 13 - 1
Comparing Three or More Means
ANOVA
(One-Way Analysis of Variance)
Objectives
• Verify the requirements to perform a
one-way ANOVA
• Test a claim regarding three or more
means using one way ANOVA
Vocabulary
• ANOVA – Analysis of Variance: inferential method that is used
to test the equality of three or more population means
• Robust – small departures from the requirement of normality
will not significantly affect the results
• Mean squares – is an average of the squared values (for
example variance is a mean square)
• MST – mean square due to the treatment
• MSE – mean square due to error
• F-statistic – ration of two mean squares
One-way ANOVA Test Requirements
• There are k simple random samples from k
populations
• The k samples are independent of each
other; that is, the subjects in one group
cannot be related in any way to subjects in a
second group
• The populations are normally distributed
• The populations have the same variance;
that is, each treatment group has a
population variance σ2
ANOVA Requirements Verification
● ANOVA is robust, the accuracy of ANOVA is not
affected if the populations are somewhat nonnormal or do not quite have the same variances
● Particularly if the sample sizes are roughly equal
● Use normality plots
● Verifying equal population variances requirement:
● Largest sample standard deviation is no more than two
times larger than the smallest
ANOVA – Analysis of Variance
Computing the F-test Statistic
1. Compute the sample mean of the combined data set, x
2. Find the sample mean of each treatment (sample), xi
3. Find the sample variance of each treatment (sample), si2
4. Compute the mean square due to treatment, MST
5. Compute the mean square due to error, MSE
6. Compute the F-test statistic:
mean square due to treatment
MST
F = ------------------------------------- = ---------mean square due to error
MSE
k
MST =
Σ
n=1
ni(xi –
-------------k–l
x)2
k
MSE =
Σ
n=1
(ni – 1)si2
------------n–k
MSE and MST
● MSE - mean square due to error, measures how
different the observations, within each sample, are
from each other
 It compares only observations within the same sample
 Larger values correspond to more spread sample means
 This mean square is approximately the same as the
population variance
● MST - mean square due to treatment, measures how
different the samples are from each other
 It compares the different sample means
 Larger values correspond to more spread sample means
 Under the null hypothesis, this mean square is
approximately the same as the population variance
ANOVA – Analysis of Variance Table
Source of
Variation
Sum of
Squares
Degrees
of
Mean
Freedom Squares
Treatment Σ ni(xi – x)2
k-1
MST
Error
Σ (ni – 1)si2
n-k
MSE
Total
SST + SSE
n-1
F-test
Statistic
F Critical
Value
MST/MSE F α, k-1, n-k
Excel ANOVA Output
• Classical Approach:
– Test statistic > Critical value … reject the null hypothesis
• P-value Approach:
– P-value < α (0.05) … reject the null hypothesis
TI Instructions
• Enter each population’s or
treatments raw data into a list
• Press STAT, highlight TESTS
and select F: ANOVA(
• Enter list names for each sample or
treatment after “ANOVA(“ separate by
commas
• Close parenthesis and hit ENTER
• Example: ANOVA(L1,L2,L3)
Summary and Homework
• Summary
– ANOVA is a method that tests whether three, or more,
means are equal
– One-Way ANOVA is applicable when there is only one
factor that differentiates the groups
– Not rejecting H0 means that there is not sufficient
evidence to say that the group means are unequal
– Rejecting H0 means that there is sufficient evidence
to say that group means are unequal
• Homework
– pg 685-691; 1-4, 6, 7, 11, 13, 14, 19
Problem 19 TI-83 Calculator Output
• One-way ANOVA
– F=5.81095
– p=.013532
– Factor
• df=2
• SS=1.1675
• MS=0.58375
– Error
•
•
•
•
df=15
SS=1.50686
MS=.100457
Sxp=0.31695