Abstract

Perform a wide variety of statistical significance tests including standard deviation, mean, sum and sample size estimation. It’s ideal for marketing research use.

Market Research Statistical Calculator

The Quirks.com free statistics calculator lets you perform a wide variety of statistical significance tests including standard deviation, mean, sum and sample size estimation. It’s ideal for marketing research use.

IMPORTANT: Do not use commas in any of the fields (e.g. use 5000 not 5,000).

Compare three or more means

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Use this selection to test for differences between three or more means. An analysis of variance (ANOVA) will be performed.

In an ANOVA, the F-ratio is used to compare the variance between the groups to the variance within the groups. For example, suppose we have two groups of data. In the best of all possible worlds, all the people in group one would have very similar scores. That is, the group is cohesive, and there would be very little variability (diversity) in scores within the group. All the people in group two would also have similar scores (although different than group one). Again, there is very little variability within the group. Both groups have very little variability within their group, however, there might be substantial variability between the groups. The ratio of the between groups variability (numerator) to the within groups variability (denominator) is the F-ratio. The larger the F-ratio, the more certain we are that there is a difference between the groups.

If the probability of the F-ratio is less than or equal to your critical alpha level, it means that there is a significant difference between at least two of groups. The F-ratio does not tell which group(s) are different from the others...just that there is a difference.

If the F-ratio is not significant, we stop there, because we know that no group(s) are significantly different from the others. However, if the F-ratio is significant, then we do "post-hoc" (after the fact) tests to determine which group(s) are different from the others. There are a wide variety of post-hoc tests, but one of the most common is to do a series of special t-tests between all the combinations of groups. This software uses the lsd (least significant difference) t-test for all post-hoc comparisons. For the post-hoc tests, the same critical alpha level is used that was used to test for the significance of the F-ratio.

Example

A company has offices in four cities with sales representatives in each office. At each location, the average number of sales per salesperson was calculated. The company wants to know if there are significant differences between the four offices with respect to the average number of sales per sales representative.

Number of groups: 4
Critical alpha level for post-hoc testing: .05

	Mean	SD	N
Group 1	3.29	1.38	7
Group 2	4.90	1.45	10
Group 3	7.50	1.38	6
Group 4	6.00	1.60	8

Analysis of variance
Source	df	SS	MS	F	p
Factor	3	62.8	20.9	9.78	<.001
Error	27	57.8	2.13
Total	30	120.6

Post-hoc lsd t-tests
Groups	lsd t-test
1 & 2	t(15)=2.23, p=.041
1 & 3	t(11)=5.17, p<.001
1 & 4	t(13)=3.58, p=.003
2 & 3	t(12)=3.44, p=.004

ANOVA's are complex and it is advisable to present the results in tables to make it easier for the reader to understand. However, in the text, you could describe the analysis like this:

A one-way analysis of variance was performed to compare the average sales per salesperson at four company offices. The F-ratio was significant at the .05 critical alpha level, F(3,27)=9.78.p<.001. Thus, it was concluded that there was a significant difference between at least two of the offices.

Since the F-ratio was significant, post-hoc lsd t-tests were performed to determine the nature of the differences. Office 1 had the lowest sales per salesperson (M=3.29, SD=1.38), and it was signifcantly less than the other three offices. Office 3 had the highest sales per salesperson (M=7.50, SD=1.38), and it was signifcantly more than office 2.