# Which one of these sets of numbers looks least normally distributed?

a). 88 91 99 100 100 101 105 105 106 107
b). 0.7 1.3 1.4 1.5 1.6 2.1 2.1 2.2 2.3 3.1
c). 7 8 9 16 20 21 23 29 34 101
d). 11 13 17 19 21 23 24 25 26 28
e). 1.2 1.3 1.5 1.9 2.2 3.0 3.3 3.7 4.0 5.0

You could try a D'Agostino-Pearson test. A Shapiro-Wilk test might not work well because (a) and (b) have ties.

There are sophisticated statistical tests that can be used, but I’m guessing that you’re supposed to be making this judgement by eye.

Make a sketch of each distribution, something like this one for (a):

                    100 105 107
88 91      99 101 105
-|--|-------|||---|||---
|    |


A more or less normal distribution will be roughly symmetric and will cluster in the middle, with a few points trailing off at each end. This one is a bit unbalanced, as it has a longer tail to the left than to the right, and it has a bit of a ‘hole’ in the middle of the concentrated part, but it’s not too far from normal.

Compare it with this one for (c):

          8            21
7 9     16  20 23    29   34                            101
-|||------|---||-|-----|----|-----------------------...---|-


That $101$ is way out of line, and overall this one seems to be concentrated at the left and to trail off to the right.

What about (b)? (This time I’ve left off the numbers, but that shouldn’t cause any real difficulty.)

        -|-----||||----|||-------|---
|


This one really does look fairly normal: it’s concentrated in the centre, it’s roughly symmetric, and it trails off on both ends.

Here’s (e):

        -||-|---|--|-------|--|---|--|---------|-


It’s a bit like (c), but not so extreme. It’s also a bit like (a), but turned around and smoother.

Finally, here’s (d):

        -|-|---|-|-|-||||-|--


It’s a lot like (e), but turned around and more concentrated at its ‘heavy’ end; it certainly looks more like a normal distribution than most of the others.

Just from the sketches it appears that (a) and (c) are the best candidates: they look much less like a normal distribution than any of the other three. And (c) wouldn’t look bad if it the $107$ were $115$, say, while (c) would take a lot of adjustment to look much like a normal distribution, so (c) is the best ‘eyeball’ choice.

For a better way to check "by eye" use a q q plot. non-normality shows up in point departing from a straight line.

• It does look to me that the OP was intended to make a quantile-quantile plot of his data... Aug 12 '12 at 12:19