# Test of Means or Medians for Non Normal Populations?

##### Involved In Discussions
HI All, I have a number of paired data sets of various characteristics that per my Anderson Darling test one or both of the sets are not normally distributed at a 95% confidence level. I would like to see at that same confidence level if the means are different. I don't think I can use a two sided t test in this case. Is there a test for the means on non normally distributed data? Should I use the Mann-Whitney U test or Mood's Median test?

#### Statistical Steven

##### Statistician
Staff member
Super Moderator
HI All, I have a number of paired data sets of various characteristics that per my Anderson Darling test one or both of the sets are not normally distributed at a 95% confidence level. I would like to see at that same confidence level if the means are different. I don't think I can use a two sided t test in this case. Is there a test for the means on non normally distributed data? Should I use the Mann-Whitney U test or Mood's Median test?
If you have PAIRED data, then the differences of the pairs should be approxminately normal if the two set of data came from the same distribtion. If so, you can use the paired t-test.

##### Involved In Discussions
I believe the data is paired in so far that one instrument measured the exact same point as the other. Unfortunately I only have one set of measured data from both instruments. One company measured each characteristic once as did the other company. Each point corresponds to the exact same area so ideally they should be identical. I will try to post an excel file of the two data sets.
Thanks,

##### Involved In Discussions

The best thing to do is post your data.

NC
Hopefully my attachment will get posted
Thanks!

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N

#### NumberCruncher

I've looked at your data. Most of the differences are approximately normally distributed, but there are a few serious outliers in both data sets.

A simple t test suggests that data set 1 is not significantly different to zero, but data set 2 is significantly different to zero. My concern is that, when I take out just 2 outliers, from data set 1, the t test tells me that this is also significantly different to zero.

I am not happy with the t test because of these outliers.

I would suggest using the Mann-Whitney test on the data and see if this confirms the above conclusions.

NC

#### Bev D

##### Heretical Statistician
Staff member
Super Moderator
Brad: your data is still matched pairs. The fact that you have only one reading from each supplier means that the suppliers individual repeatability is confounded with the between supplier reproducibility. If the total measurement error is acceptable in relation to the specifications then all is well. If it isn't then you need to determine if one or both suppliers has substantial repeatability problems. (what is the specification limit?)

I have attached a spreadsheet with the Youden analysis, Matched pair analysis (and I also threw in a gratuitous distribution analysis).

You have a small bias between the two measuremetns in data set 2 but it is very small compared to the total error.

There is no bias between the two measurements in data set 1.

As an aside, the lack of fit to a Normal distribution is very minor. t tests and ANOVAs are relatively unnaffected by this lack of fit. And in any case a simple plot of the data tells you what you need to know. No need for fretting about mathematical precision.

I think that we have collectively forgotten how to plot and think about our data now that sophisticated statistics can be done in milliseconds with JMP or Minitab...

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