# Interpretation of Shapiro test

I'm pretty new to statistics and I need your help. I have a small sample looking as follows:

H4U 0.269 0.357 0.2 0.221 0.275 0.277 0.253 0.127 0.246

I run the Shapiro test using R: shapiro.test(precisionH4U$H4U) and I got the following result: W = 0.9502, p-value = 0.6921 Now, if I assume the significance level at 0.05 than the p-value is larger then alpha (0.6921 > 0.05) and I cannot rejest the H0 hypothesis about the normal distribution, but does it allow me to say that the sample have normal distribution? Thanks! - add comment ## 4 Answers No - you cannot say "the sample has a normal distribution" or "the sample comes from a population which has a normal distribution", but only "you cannot reject the hypothesis that the sample comes from a population which has a normal distribution". In fact the sample does not have a normal distribution (see the qqplot below), but you would not expect it to as it is only a sample. The question as to the distribution of the underlying population remains open. qqnorm( c(0.269, 0.357, 0.2, 0.221, 0.275, 0.277, 0.253, 0.127, 0.246) )  - the qqplot looks pretty like normal i think... you can try qqnorm(rnorm(9)) several times... – Tomas Sep 21 '11 at 7:31 @Tomas: Perhaps better to say "the qqplot looks as if it could have come from a normal population". It might instead have come from a distribution with heavier tails. – Henry Sep 21 '11 at 13:48 Yes, qqnorm(runif(9)) can produce similar result. So we cannot actually say anything... – Tomas Sep 21 '11 at 13:55 add comment Considering that you are pretty new to statistics, I suspect that you are thinking about this because these are residuals of an estimate of a mean and you want to know whether the assumption of normality is valid for confidence estimates using a$t$-distribution.$t$-tests are quite robust to violations of this assumption, the data look vaguely normal in Henry's q-q plot, and the Shapiro test doesn't indicate that the data come from a population with a non-normal distribution, so I would say that a$t\$-test is appropriate.

I further speculate that you are looking at proportions, in which case you could use a binomial distribution if you were concerned about violations of assumptions.

If it was some other concern that got you to Shapiro tests, you can ignore everything I just said.

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You got it right, I wanted to know if I can use t test for my sample. Thanks! –  Jakub Sep 18 '11 at 14:44

As Henry already said you can't say it's normal. Just try to run the following command in R several times:

shapiro.test(runif(9))


This will test the sample of 9 numbers from uniform distribution. Many times the p-value will be much larger than 0.05 - which means that you cannot conclude that the distribution is normal.

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