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Situation: Comparing mean scores of level of satisfaction across multiple institutions and overall.

Let's say there are 50 institutions, all with different sample sizes.

Level of satisfaction is rated 1 to 5.

I was asked to perform a t-Test to determine outliers and significance of the score differences, with a control range for 1 SD and 2 SD.

Eg. Institute = Mean score

Institute A = 3.8 Institute B = 3.9 Institute C = 4.1 . . . Overall = 4.0

This is so that we can rank the institutions based on their mean score.

So here's my question:

  1. How do I make a fair comparison of the mean scores? It is possible?
  2. How does a t-Test determine statistical significance in this scenario?
  3. Do I look at P-value or overlap in distribution to assess usefulness of the test?
  4. I conducted multiple t-Tests for 2 groups (eg. Institute A & Overall, Institute B & Overall, Institute C & Overall, ...) but I'm aware that it is not appropriate to conduct multiple t-Tests comparing 2 groups at a time. What should I do?

Please kindly help. I am quite lost and confused.

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    $\begingroup$ Read about the analysis of variance. $\endgroup$ – Michael R. Chernick Jan 6 '17 at 12:53
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While typing I saw Michael Chernick comment, which I completely agree with.

To perform a statistical comparisons of the means of two or more groups you require an ANOVA or analysis of variance.

Three notes:

  • As stated, if you want to test the institutes for significant differences amongst eachother use an ANOVA. This will tell you if any one of the institutes has a significantly different mean than the others. However, without follow-up tests (which require multiple testing) the ANOVA will not tell you which!
  • If the ranking of mean per institute is all you require, you might opt out of significance testing all together and just rank the means (i.e. the best approximation of the 'true and unknown' mean per institute is the mean value per institute you draw from your data, irrespective of your uncertainty).
  • Also, take heed when comparing to the overall mean, as the group of interest is also part of the overall mean.
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