2. ANOVA
is a statistical method used to test and compare
the means of more than two independent
groups and determine if at least one mean is
different from the others. It is first developed by
Sir Ronald A. Fisher in 1918 and introduced to
educational researchers in 1934 (Tweney, 2005).
3. ANOVA
ANOVA is also called as F-test. The most
commonly used ANOVA test practice are the
one-way ANOVA and two-way ANOVA. One-way
ANOVA is a parametric test also called as single
factor ANOVA. It is used to test the effect of one
independent variable (with more than two
levels) on one dependent variable.
4. Assumptions of ANOVA
1. Independence – the data (continuous not discrete) collected should be independent from one
group to another. No hidden relationships among observations.
2. Equal Variance – in the population, variances within the group should be equal.
3. Normality – the data collected should follow the normal distribution. The assumption of
normality can be tested using chi-square tests for normality or draw histograms to visually check
the distribution of data
5. Hypotheses of ANOVA
1. Null hypothesis (H0) – states that there is no significant
difference among the group and equality of means. All
treatment means are equal.
2. Alternative Hypothesis (H1) – states that there is a
significant difference among the means and group. There is
a treatment effect and at least one population mean is
different from the others.
6. Limitations of ANOVA
1. Although it is a robust test, it is less powerful and may not yield
exact p-values when observations are drawn from a non-normal
distribution.
2. Since it is designed to test all alternatives to the null hypothesis
therefore testing it to a specific hypothesis may give us under par
results.
3. The major disadvantage of the ANOVA test is that it tells us
whenever more than two groups have different sample means but it
does not help us identify which groups are different. Thus, running a
post hoc test is a must.
7. One-way ANOVA Calculation using Excel
Step 1: State the hypotheses.
Step 2: Identify Level of Significance (Alpha Level)
Step 3: Run one-way ANOVA using Excel.
Step 4: Copy the generated ANOVA table for analysis.
Step 5: Decision.
Step 6: State the conclusion.
8. Reject or Accept
Reject the Null Hypothesis if the F –
value (F) > Fcrit
Reject the Null Hypothesis if the P-value
< alpha level
9. A group of investment company wants to test and compare five different brands of flashlight
batteries to identify what particular brand last longer. They randomly selected 30 flashlights and
divided it into five groups having six flashlights each. Prior to this, the company conducted initial
data analyses and found out that the selected samples passed the normality test. Presented
below are the lifetimes of the batteries measured in hours of usage. At 0.05 significance level,
perform one-way ANOVA test to determine if there is a significant difference among the five
brands of batteries being tested.
Brand A Brand B Brand C Brand D Brand E
37 35 35 35 35
30 36 35 28 33
39 34 39 32 32
28 32 38 30 34
30 35 36 27 31
29 36 38 29 33
11. Tukey-Kramer Test
After performing the one-way ANOVA test and finding out
that there is a significant difference among the group of
observations, it is a must that we use another test to exactly
find which at least one group are differs from the other
group.
There are wide variety of multiple comparison test that can
be used, such as: Scheffe’s Test and Dunnett’s Test but the
most common test is the Tukey-Kramer Test also called as
Tukey’s Honestly Significant Difference (Tukey’s HSD) test.
12. Steps on How to Calculate Tukey HSD
Test
Step 1: Find the absolute mean difference between each group.
Step 2: Find the Q critical value or range.
Step 3: Decision.
13. Things to remember
Factor level (k) – a factor (also called an independent variable) is an explanatory variable manipulated by
the experimenter. Each factor has two or more levels. Also known as the number of treatments.
Comparison (C) – refers to the number of groups to be paired. To calculate C, use the following formula.
C = k( k- 1)/2
Number of observations (N) -refers to the total size of the samples or observations
Q critical Value or range
Pooled Variance – average mean of the variance
Q critical value – use the formula
Q critical value = Q (√pooled variance/n.)
