Would this Normal Quantile Plot be considered normal?

Question

How would I describe this QQ plot? Would it indicate fair normality? Or would it be skewed? I have attached the histogram also.

I have been asked to the answer the following: "By referring to the normal quantile plot below briefly discuss if the measurements are approximately normally distributed."

I have said it looks normally distributed as the majority of the dots are close to the reference line and appears linear based on the minimal sample size.

Would that fly?

Thank you.

With fewer than 20 observations, it is difficult to establish or refute normality. Depending on binning, the histogram of of the sample might be 'bimodal', but that may not be an indication that the population is a mixture of two distributions. — BruceET, Jul 28 '19 at 08:38
You can't really "establish" normality. A sample might appear reasonably consistent with normality but it will also be consistent with many non-normal distributions. — Glen_b, Jul 28 '19 at 14:37
How would I describe it if it is not normally distributed? Would it be considered right skewed? Thank you. — Jaime, Aug 01 '19 at 04:16

BruceET · Answer 1 · 2019-07-28T09:44:07.940

1

Here are histograms and Q-Q plots of three samples of size $n=20$ from a 50:50 mixture of distributions $\mathsf{Norm}(\mu = 35, \sigma=0.6)$ and $\mathsf{Norm}(\mu = 38, \sigma=0.6).$

Because the means are more than two standard deviations apart, the density of the mixture distribution is bimodal. [See this page.] The 'bimodal effect' is somewhat more pronounced in my example than in your question.

The density function is shown below:

lbl="Density of Mixture Dist'n of two Normal Densities"
curve(.5*dnorm(x,35,.6)+.5*dnorm(x,38,.6), 32, 41, ylab="Density", lwd=2, main=lbl)
  abline(h=0, col="green2")

The following R code was run three times to make the first figure:

n = 20;  p = rbinom(n, 1, .5)
x1 = rnorm(n, 35, .6);  x2 = rnorm(n, 38, .6)
x = p*x1 + (1-p)*x2
hist(x, col="skyblue2");   qqnorm(x, pch=19)

edited Jul 28 '19 at 09:44

answered Jul 28 '19 at 09:12

BruceET

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Thank you for the detailed answer, although I think it's above my IQ level at the moment. I have attached a histogram of the dataset for reference. – Jaime Aug 01 '19 at 04:26
I think maybe the histogram has to few bins to reveal the true shape of the sample. – BruceET Aug 01 '19 at 04:54
bins = binomials right? so could i say, "that while it is plausible that the data come from a normal distribution, there are too few binomials to conclude the true distribution of the sample"? Thank you. – Jaime Aug 01 '19 at 05:13
_bins_ means _intervals._ – BruceET Aug 01 '19 at 05:20
sweet thanx. learnt something new. – Jaime Aug 01 '19 at 05:59
Learning something new is usually worthwhile. – BruceET Aug 01 '19 at 06:03

score 0 · Answer 2 · answered Jul 28 '19 at 08:39

You've too few points to say anything conclusive but for posterity, as a rough visual guide, you might expect (1) as many dots above the line as there are below it for every quartile, and (2) a similar amount of dots to the left and right of your 50% quartile with a self-evident visual symmetry.

At the moment, to me your graph looks to depart from (1) substantially, and whilst there is a similar number of points on either side, they don't look visually symmetrical around the 50% quartile.

Thank you for your answer. How would I describe it if I was to state that it is not normally distributed? — Jaime, Aug 01 '19 at 04:27

Would this Normal Quantile Plot be considered normal?

2 Answers2