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I need to make a statement regarding whether the mean of this data is significantly different from zero. The data can not be described by a normal distribution.

The population's size is 27, mean is -0.265, and standard deviation is 0.193.

My questions would be:

  • How would you quantify the compatibility of the data with zero? Can I say with X % confidence that the mean is -0.265 and not 0?
  • What is the typical way to approach this small-sized non-normal population diagnose problems?

Please provide detailed examples and references if possible.

Sample

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    $\begingroup$ TLDR; Statistical test would check if your sample mean is different from the population mean. You have data for whole population so the answer, without conducting the test, is obviously no. $\endgroup$
    – Tim
    Commented Jul 8, 2016 at 14:05
  • $\begingroup$ Can bootstrap help? $\endgroup$
    – German Demidov
    Commented Jul 8, 2016 at 14:11
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    $\begingroup$ @GermanDemidov what for? Bootstrap sampling should imitate the sampling that was done -- in this case no sampling was done so correct bootstrap procedure would be "take the sample as is" multiple times... $\endgroup$
    – Tim
    Commented Jul 8, 2016 at 14:13
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    $\begingroup$ "Can we say with X% that the mean is not 0”. You have the population, you know the true mean, end of story. When using the mean you don't really impose a certain structure on the data, you are not really making any assumptions here $\endgroup$
    – Repmat
    Commented Jul 8, 2016 at 14:40
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    $\begingroup$ Almost always, when people say they have the "entire population," they don't really mean it. If these data truly are the population, then nothing you conclude about them will apply to any other populations or circumstances. If you wish to make any kind of generalizations, then you must view your data as some kind of sample of a larger process or "potential" population. $\endgroup$
    – whuber ♦
    Commented Jul 8, 2016 at 15:16

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I am rather confused by your text saying that you have the population's scores, if you do, and you are not looking at a sample of a population, then you have your answer since you have the mean and the standard deviation. If you have a sample of a population for which you want to make decisions, then I would advise to conduct further analysis.

Since the data appears to be negatively skewed, I would advise you to conduct a non-parametric test on that data. However, you can also reach the same conclusions with a confidence interval. You can see if the mean +/- confidence interval includes the value 0, if it does not, you can assume that the mean of the population is different than 0.

When you have this sort of data you should always see the skewness and kurtosis values, and if they are too off, I would conduct a non-parametric test. Unless you have a big enough sample, which you do not.

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    $\begingroup$ There is no indication at all that a Student t-test wouldn't work for these data. Any concern that might exist attaches to the sampling distribution of the $t$ statistic, not the distribution of the sample itself. In fact, this sample is scarcely distinguishable from one drawn from a Normal distribution, for which the t-test works perfectly. Using either a bootstrap or a t-test, you should obtain a p-value smaller than $10^{-6}$. $\endgroup$
    – whuber ♦
    Commented Jul 8, 2016 at 18:11

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