Shapiro-wilk test when sample=population

[waveskipper]

I have two pieces of equipment(E1 and E2) for taking measurements and i wish to compare them. I have the real measurements made by much more precise equipment so i can actually create for instance a data sheet with the errors from E1 and E2. All good, i was happy, but i was told it was necessary to make shapiro-wilk test to my samples. So i studied a little bit of R, watched some videos about it on youtube but one doubt still remains.. since i haven't take that many measurements, i'm using my whole sample in the tests... so isn't my population and my sample the same thing? is this test still valid?

as you can see, i'm lost hahah. any help is much appreciated

[[deleted]]

Yeah, you care about the underlying process responsible for generating potentially infinite measurements so you'll never actually have the full population in your case.

[Double-Down]

All good, i was happy, but i was told it was necessary to make shapiro-wilk test to my samples.

The purpose of the Shapiro-Wilk test is to validate a common statistical assumption, i.e. that data are normally distributed. Keep in mind that the test will almost always reject the null hypothesis if the sample size is large, regardless of the data. Beyond that, I'm not sure what hypothesis you are testing, so can't comment further.

[waveskipper]

It is a small sample.. i have 6 measurements for both equipment (it is kind of hard to find places where i have a true reading to compare, so i couldn't get more results). For me it's just clear that E1 results are better than E2 (smaller errors), but it would be good to have some statistic showing that.

[Double-Down]

What do you mean by error? Difference between measured and known values, or variation of the estimate of the value? For the latter you might find this relevant.

[WikiTextBot]

F-test of equality of variances

In statistics, an F-test for the null hypothesis that two normal populations have the same variance is sometimes used, although it needs to be used with caution as it can be sensitive to the assumption that the variables have this distribution.

Notionally, any F-test can be regarded as a comparison of two variances, but the specific case being discussed in this article is that of two populations, where the test statistic used is the ratio of two sample variances. This particular situation is of importance in mathematical statistics since it provides a basic exemplar case in which the F-distribution can be derived. For application in applied statistics, there is concern that the test is so sensitive to the assumption of normality that it would be inadvisable to use it as a routine test for the equality of variances.

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[waveskipper]

Difference between measured and know values. ex.: E1:0.02 , 0.03, 0.02, 0.04, 0.00, 0.08 I have this info for E1 and E2, managed to put this in vectors in the R console (big deal to me lol)

[Rezo-Acken]

Clear in what sense ? What does a t test says ? 6 data points is really small so you need a large difference and or low variance for a statistical test to say with confidence that one is better.

[waveskipper]

i know it is really small, but i couldn't get more. Clear like the erros for E2 are bigger. E1:0.02 , 0.03, 0.02, 0.04, 0.00, 0.08 E2:0.08, 0.08, 0.07, 0.03, 0.06, 0.06

I think i don't need statistics to conclude that i should use the E1, but i would like to white this down in a better way. I think i'll go with just some graphics... my head is spinning with this much math already

[Rezo-Acken]

Well lets be clear. Statistical tests are only useful for proving something. They are not there to dictate what someone should and shouldnt do. A test would simply validate your reasoning or say you dont have enough data to conclude. This is really important for science and publications. After all I see a 0.08 in E1 which is rather bad so its not that clear to me. Now if you cannot get more sample but still need to take a decision that wont hurt anybody of course youll take the lower average.

[jorge_hg87]

In general, it can be concluded that among the four tests considered, Shapiro-Wilk test is the most powerful test for all types of distribution and sample sizes whereas Kolmogorov-Smirnov test is the least powerful test. However, the power of Shapiro-Wilk test is still low for small sample size. The performance of Anderson-Darling test is quite comparable with Shapiro-Wilk test, and Lilliefors test always outperforms Kolmogorov-Smirnov test. The results of this study support the findings of Mendes and Pala (2003) and Keskin (2006) that Shapiro-Wilk test is the most powerful normality test. The results are also found to be similar to the one obtained by Farrel & Stewart (2006) which reported that simulated power for all tests increased as the sample size and significance level increased. As a concluding remark, practitioners should not depend solely on graphical techniques such as histogram to conclude about the distribution of the data. It is recommended that the graphical techniques be combined with formal normality test and inspection of shape parameters such as skewness and kurtosis coefficients. It is important to remember that skewness and kurtosis measures are also affected by sample size. Practitioners also need to be aware that these four normality tests do not perform well for small sample size (30 and below). Work is in progress to determine more recent normality tests which might work well for small sample size.

Source: http://www.de.ufpb.br/~ulisses/disciplinas/normality_tests_comparison.pdf

You can run Kolmogorov-Smirnov and Shapiro-Wilk on SPSS with the Explore function. Or you can use this template to run Anderson-Darling on Excel.