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Below is a normal probability plot of residuals from my lecture enter image description here

The NSCORE(z score) is quite confusing. For example, the first nscore is -1.54664, which should be 0.061 or 61% percentile, it doesn't match 0.1 or 0.074468. I don't think it's round error

The second question is, what we need to prove is the errors at each value of the predictor are Normally distributed. But there is just some unique predictor values such as 2,4 in the picture, and we can't collect many different response values at each predictor value, does it mean that we can consider all residuals to be residuals at one predictor value since errors are all indenpendent?

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    $\begingroup$ You need to describe what's going on more accurately, as a number of things are unclear. Also, what you describe as a "normal probability plot" doesn't contain a plot -- that's a table of numbers. Your nscores don't seem to be right; your percentile ranks are symmetric about $\frac12$, which suggests your normal scores should be symmetric about $0$. How are your "nscore" values calculated, exactly? $\endgroup$ – Glen_b Jan 30 '16 at 13:04
  • $\begingroup$ @Glen_b This is the context onlinecourses.science.psu.edu/stat501/node/281 $\endgroup$ – whoisit Jan 30 '16 at 13:23
  • $\begingroup$ They don't explain there as far as I can see; is there any other information they give you on normal scores? $\endgroup$ – Glen_b Jan 30 '16 at 13:40
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That table embodies at least two errors.

The first is that the NSCORE values were computed for ten data values rather than nine. We may speculate that the example originally involved $n=10$ values and later was changed to nine, but the NSCORE column was not updated.

The second is that the formula for MTB_PCT is based on a misunderstanding of what the software actually does. The formula the software really uses is

$$pp(i) = \frac{i - 3/8}{n + 1 - 2(3/8)} = \frac{i - 0.375}{n + 0.25}$$

where "$pp(i)$" is the "plotting point" for the $i^\text{th}$ smallest residual.

These conclusions are forcibly demonstrated by carrying out the calculation as I have described it. In R, for instance, these ten NSCORE values could be computed with the command

qnorm((1:10 - 0.375)/10.25)

Here is its output (rounded to five decimal places for comparison to the table):

-1.54664 -1.00049 -0.65542 -0.37546 -0.12258  0.12258  0.37546  0.65542  1.00049  1.54664

They are exactly as shown in the question, without any rounding differences at all.

For $n=9$ residuals, Minitab should produce these plotting points:

qnorm((1:9 - 0.375)/9.25)
-1.49415 -0.93197 -0.57164 -0.27439  0.00000  0.27439  0.57164  0.93197  1.49415

I explain this mysterious-looking formula for $pp$ at http://www.quantdec.com/envstats/notes/class_02/characterizing_distributions.htm in the "Percentiles and EDF plots" section. Reading probability plots is discussed in the next page, "Probability Plots," at http://www.quantdec.com/envstats/notes/class_03/probability.htm.

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