The distribution used for a One-Way ANOVA is a
new one. It is called the F distribution,
named after Sir
Ronald Fisher, an English
statistician. The F statistic is a fraction
with two sets of degrees for freedom; one for the
numerator and one for the denominator. The
notation used therefore has two degrees of freedom
and looks like F ~ F4,20 where 4
is the degrees for freedom for the numerator and
20 is the degrees of freedom for the denominator.
The F distribution is derived from the Student's
t-distribution, where the F distribution are are
squares of the corresponding values of the
t-distribution. One-Way ANOVA expands the
t-test for comparing more than two groups.
Since the derivation of this is beyond the scope
of this class we will not look at it here but
rather just use the concept.
Two estimates are made to calculate the F ratio.
1. Variation between samples
An estimate of σ2
that is the variance of the sample means
multiplied by n (when the sample sizes are the
same). If the sample sizes are different,
the variance between samples is weighted to
account for the different sample sizes.
Also called the variation due to treatment or
FACTOR (i.e. the different populations) or
explained variation.
2. Variation within samples
An estimate of σ2
that is the average of the sample variances (aka
pooled variance). When the sample sizes
are different, the variance within samples is
weighted.
Also called the variation due to ERROR or
unexplained variation.
To calculate the F ratio (aka
the F-Statistic)
The F-statistic is a 'roll-up' calculation where we
take the mean square between groups (MSbetween)
and divide it by the mean square within groups (MSwithin).
The MSbetween is the quotient of the sum
of squares between groups (SSbetween) and
the degrees of freedom between groups (dfbetween).
In turn the MSwithin is the quotient of
the sum of squares within groups (SSwithin)
and the degrees of freedom within groups (dfwithin).
We use the letter k to represent the number of
different groups under consideration and n for the
total number of samples taken from ALL groups.
This means that the degrees of freedom for the
numerator is k - 1 and the degrees of freedom for
the denominator is n - k.
To summarize the formulas
F = (MSbetween
) / (MSwithin )
MSbetween = ( SSbetween
)/ ( dfbetween)
MSwithin
= ( SSwithin )/(dfwithin )
To see the rest of the formulas needed to find the F
ratio, refer to Chapter 13 of Introductory
Statistics
Fortunately for us the
TI-83/84 will do the work for us
Recall that the hypotheses for a One-Way ANOVA are
Ho: µ1 = µ2 =
µ3 = ... = µk
Ha: At least two of the group means
are not equal.
Using a TI83/84 calculator, you will enter each
of the group samples into a separate list (L1, L2,
L3, ...) and then accessing STAT, TESTS, find
ANOVA(, press Enter. You then need to type
L1, L2, L3,..) where the ... will be all the lists
where the samples are. Press Enter.
The calculator will give you the following
information.
The F-statistic
The p-value
Factor (the between stuff)
df
SS
MS
Error (the within stuff)
df
SS
MS
Based on the p-value when compared to the given
level of significance, you make your decision
about the hypothesis test.
If using some other computer software, above
information is usually displayed in a table
similar to
Source of
Variation
Sum of
Squares (SS)
Degrees of
Freedom (df)
Mean Square (MS)
F
Factor
(between)
SS (Factor)
k - 1
MS(Factor)
MS(Factor)/MS(Error)
Error
(within)
SS (Error)
n - k
MS(Error)
Total
SS (Total)
n - 1
Please continue to the next section
of this lesson.
