Normal quantile plots

The normal quantile plot

• see that normal distributions of data (or being normal enough) important
• only tools we have to assess this are histograms and maybe boxplots
• a better tool is normal quantile plot:
  • plot data against what you expect if data actually normal
  • look for points to follow a straight line, at least approx
• ggplot code: aes sample; geoms stat_qq and stat_qq_line

Packages

The usual:

library(tidyverse)

── Attaching core tidyverse packages ──────────────────────── tidyverse 2.0.0 ──
✔ dplyr     1.1.2     ✔ readr     2.1.4
✔ forcats   1.0.0     ✔ stringr   1.5.0
✔ ggplot2   3.4.2     ✔ tibble    3.2.1
✔ lubridate 1.9.2     ✔ tidyr     1.3.0
✔ purrr     1.0.1     
── Conflicts ────────────────────────────────────────── tidyverse_conflicts() ──
✖ dplyr::filter() masks stats::filter()
✖ dplyr::lag()    masks stats::lag()
ℹ Use the conflicted package (<http://conflicted.r-lib.org/>) to force all conflicts to become errors

Kids learning to read

Rows: 44
Columns: 2
$ group <chr> "t", "t", "t", "t", "t", "t", "t", "t", "t", "t", "t", "t", "t",…
$ score <dbl> 24, 61, 59, 46, 43, 44, 52, 43, 58, 67, 62, 57, 71, 49, 54, 43, …

ggplot(kids, aes(x = group, y = score)) + geom_boxplot()

Each group looks more or less normal, or at least close to symmetric.

Get the groups separately

kids %>% filter(group == "t") -> treatment
kids %>% filter(group == "c") -> control

to check

treatment %>% count(group)

# A tibble: 1 × 2
  group     n
  <chr> <int>
1 t        21

control %>% count(group)

# A tibble: 1 × 2
  group     n
  <chr> <int>
1 c        23

The treatment group

ggplot(treatment, aes(sample = score)) + 
  stat_qq() + stat_qq_line()

only problem here is lowest value a little too low (mild outlier).

Control group

ggplot(control, aes(sample = score)) + 
  stat_qq() + stat_qq_line()

This time, highest value a little too high, but again, no real problem with normality.

Facetting more than one sample

Use the whole data set and facet by groups

ggplot(kids, aes(sample = score)) + 
  stat_qq() + stat_qq_line() + facet_wrap(~group)

Blue Jays attendances, skewed to right

ggplot(jays, aes(x = attendance)) + geom_histogram(bins = 6)

On a normal quantile plot

ggplot(jays, aes(sample = attendance)) + 
  stat_qq() + stat_qq_line()

• Attendances at low end too bunched up: skewed to right.
• Right-skewness can also show up as highest values being too high, or as a curved pattern in the points.

More normal quantile plots

• How straight does a normal quantile plot have to be?
• There is randomness in real data, so even a normal quantile plot from normal data won’t look perfectly straight.
• With a small sample, can look not very straight even from normal data.
• Looking for systematic departure from a straight line; random wiggles ought not to concern us.
• Look at some examples where we know the answer, so that we can see what to expect.

Normal data, large sample

d <- tibble(x=rnorm(200))
ggplot(d, aes(x=x)) + geom_histogram(bins=10)

The normal quantile plot

ggplot(d,aes(sample=x))+stat_qq()+stat_qq_line()

Normal data, small sample

• Not so convincingly normal, but not obviously skewed:

d <- tibble(x=rnorm(20))
ggplot(d, aes(x=x)) + geom_histogram(bins=5)

The normal quantile plot

Good, apart from the highest and lowest points being slightly off. I’d call this good:

ggplot(d, aes(sample=x)) + stat_qq() + stat_qq_line()

Chi-squared data, df = 10

Somewhat skewed to right:

d <- tibble(x=rchisq(100, 10))
ggplot(d,aes(x=x)) + geom_histogram(bins=10)

The normal quantile plot

Somewhat opening-up curve:

ggplot(d,aes(sample=x))+stat_qq()+stat_qq_line()

Chi-squared data, df = 3

Definitely skewed to right:

d <- tibble(x=rchisq(100, 3))
ggplot(d, aes(x=x)) + geom_histogram(bins=10)

The normal quantile plot

Clear upward-opening curve:

ggplot(d,aes(sample=x))+stat_qq()+stat_qq_line()

t-distributed data, df = 3

Long tails (or a very sharp peak):

d <- tibble(x=rt(300, 3))
ggplot(d, aes(x=x)) + geom_histogram(bins=15)

The normal quantile plot

Low values too low and high values too high for normal.

ggplot(d,aes(sample=x))+stat_qq()+stat_qq_line()

Summary

On a normal quantile plot:

• points following line (with some small wiggles): normal.
• kind of deviation from a straight line indicates kind of nonnormality:
  • a few highest point(s) too high and/or lowest too low: outliers
  • else see how points at each end off the line:

• short-tailed distribution OK for \(t\) (mean still good), but others problematic (depending on sample size).