Lecture, Chapter 12

Differences Among Several Means

Concept: use the F-statistic, which is (Variation among the samples) / (Variation within the samples)

Definition: (n times variance of means) / (mean of sample variances)

This can be done by ...

Analysis of Variance (ANOVA)

Collect, for each sample, statistics for n, x, and x². Summarize as follows:

= N ΣΣx ΣΣx² Σ [(Σx)²/n]

Summary statistics: Based on the totals, calculate:

Calculations:

Where

k = number of treatments (or data sets)

n = sample size of each data set

N = Σn = total 1

SST = ΣΣx² - (ΣΣx)² / N, = total 3 - (total 2)² / total 1

SS(Tr) = Σ[(Σx)²/n] - (ΣΣx)² / N = total 4 - (total 2)² / total 1

SSE = SST -SS(Tr)

MS(Tr) = SS(Tr) / (k-1)

MSE = SSE / (N-k)

F = MS(Tr) / MSE

When calculating the critical value of F,

DF (numerator) = DF (Treatments) = k - 1, and

DF (denominator) = DF (Error) = N - k.

Minitab example for ANOVA (One-way analysis of variance:

Tests whether the difference among several means is significant

All data goes into column 1; column 2 tells which data set the entry comes from.

In this example, the first 4 items are from set 1, the next 4 from set 2, and the last 4 from set 3.

Here is the same calculation, with Excel. The data from each sample is in a different column.

To run this test, call up the Analysis Tool-Pack, and ask to do a Single-Factor ANOVA:

65	80	72	Anova: Single Factor
69	84	76
71	86	77	SUMMARY
75	90	79	Groups	Count	Sum	Average	Variance
			Column 1	4	280	70	17.33333333
			Column 2	4	340	85	17.33333333
			Column 3	4	304	76	8.666666667
			ANOVA
			Source of Variation	SS	df	MS	F	P-value	F crit
			Between Groups	456	2	228	15.78461538	0.001141	4.256495
			Within Groups	130	9	14.44444444
			Total	586	11