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ALL VARIANCES ARE SOME VARIATION OF THE FOLLOWING:
MS  SD
2
 (X  M )

2

N 1
Or:
SS
df
SS
df
Let’s say we have three different ways of teaching columnar addition. One way emphasizes rote
memorization of sequence of steps, one way uses manipulatives such as popsicle sticks, and the
last way uses a calculator. We randomly assign 15 to one of three groups. We teach 3 groups of
2nd graders, 5 to a group for a whole semester, then give a posttest of 20 addition problems.
Here are the scores of each group:
ROTE
7
5
3
4
1
MANIP
5
9
12
12
7
Sum of X = __
Mean = __
Sum of X = __
Mean = __
CALCULAT
21
15
17
18
14
Sum of X = __
Mean = __
GRAND MEAN = __________
Calculate a oneway ANOVA on this data
VARIANCE BETWEEN (MS BETWEEN)
Caused by:
1. individual differences
2. measurement error
3. treatment
Calculate the MEAN SQUARE BETWEEN in the blank space using the
following formula:
MSb  SD
2
2
b

 n( M  M )
SS
 b
k 1
df b
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Now, fill in the following blanks:
SUM OF SQUARES BETWEEN =
__________
DEGREES OF FREEDOM BETWEEN = __________
MEAN SQUARE BETWEEN =
__________
Now, calculate the MEAN SQUARE WITHIN in the blank space using the
following formula:
VARIANCE WITHIN (MS WITHIN)
Caused by:
1. individual differences
2. measurement error
( X  M )
SDw  (n  1)
2
2
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Now, fill in the following blanks:
SUM OF SQUARES WITHIN =
__________
DEGREES OF FREEDOM WITHIN = __________
MEAN SQUARE WITHIN =
__________
The above are all that is needed to calculate the ANOVA, but the SS Total and df Total are
calculated to check our arithmetic. We don’t finish the equation for total variance by dividing
the SS Total by the df Total, however. We stop when we have the SS Total and the df Total,
because the SS between plus the SS Within should equal the SS Total. And, the df between
plus the df within should equal the df Total.
TOTAL VARIANCE (MS TOTAL)
Not actually calculated. Use the blank space below but stop with numerator
(SS Total) and denominator
(df Total)
SD
2
t

2
(
X

M
)

N 1
Now, fill in the following blanks:
SUM OF SQUARES TOTAL =
__________
DEGREES OF FREEDOM TOTAL = __________
MEAN SQUARE TOTAL = NOT CALCULATED
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Now, you have everything you need to calculate the F statistic:
MS b
F
MS w
Calculate the F for this ANOVA in the following space:
F = __________
Now, since you are doing this by hand, you must enter an F table with the
degrees of freedom for the numerator of the equation for F and the degrees of
freedom for the denominator of the equation for F.
The df for the numerator (MS between) is ______.
The df for the denominator (MS within) is ______.
Here is a small section of an F table. The upper number is the .05 level, the
lower number the .01 level:
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The critical F from an F table for these degrees of freedom, .05 level, is _____.
The critical F from an F table for these degrees of freedom, .01 level, is _____.
Does your calculated F exceed either of these critical Fs and if so, which ones?
__________
The effect size for a oneway ANOVA is eta squared. Here is the formula – use
it in the space to calculate eta squared for this data:
2 
The rule of thumb for eta squared is:
.01 – a small effect size
.06 – a medium effect size
SS b
SS t
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.14 – a large effect size
Now, you can fill out a standard ANOVA SOURCE TABLE:
SOURCE
SS
df
MS
F
p
η2
Between
Within
Total
Now, write a brief summary. Don’t worry about a multiple comparison test.
You will run that on SPSS. If your ANOVA was significant, all you can say is
that at least one pair of means differ and a multiple comparison test needs to
be calculated to identify which of the three pairs differ. If not significant, then
we have no evidence that any of the three pairs differ.