I wonder how to interpret this plot. I know, in general, that a qqplot depicts the quantiles of the data against the quantiles of the normal function.
How do I apply this to this plot?
QQ-Plot – How to Interpret QQPlot Correctly
qq-plot
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This QQ plot has the following salient features:
The stairstep pattern, in which only specific, separated heights ("sample quantiles") are attained, shows the data values are discrete. Almost all are whole numbers from $3$ through $21$. A few half-integers appear. Evidently some form of rounding has occurred.
Because the extreme "theoretical quantiles" are at $\pm 3.2$ (roughly), there must be around $1400$ data shown. This is because the extremes for this much Normally distributed data would have Z-scores about $\pm 3.2$. (This estimate of $1400$ is rough, but it's in the right ballpark.)
There is a large number of values at the minimum of $3$, far more than any other value. This is characteristic of left censoring, whereby any value less than a threshold ($3$) is replaced by an indicator that it is less than that threshold--and, for plotting purposes, all such values are plotted at the threshold. (For more on what censoring does to probability plots, see the analysis at https://stats.stackexchange.com/a/30749.)
Apart from this "spike" at $3$, the rest of the points come fairly close to following the diagonal reference line. This suggests the remaining data are not too far from Normally distributed.
A closer look, though, shows the remaining points are initially slightly lower than the reference line (for values between $5$ and $10$) and then slightly greater (for values between $13$ and $20$) before returning to the line at the end (value $21$). This "curvature" indicates a certain form of non-normality.
This particular kind of curvature is consistent with data that are starting to follow an extreme-value distribution. Specifically, consider the following data-generation mechanism:
Collect $k\ge 1$ independent, identically distributed Normal variates and retain just the largest of them.
Do that $n = 1400$ times.
Left-censor the data at a threshold of $3$.
Record their values to two or three decimal places.
Round the values to the nearest integer--but don't round any value that is exactly a half-integer (that is, ends in $.500$).
If we set $k=50$ or thereabouts and adjust the mean and standard deviation of those underlying Normal variates to be $\mu = -10$ and $\sigma = 7.5$, we can produce random versions of this QQ plot and most of them are practically indistinguishable from it. (This is an extremely rough estimate; $k$ could be anywhere between $8$ and $200$ or so, and different values of $k$ would have to be matched with different values of $\mu$ and $\sigma$.) Here are the first six such versions I produced:
What you do with this interpretation depends on your understanding of the data and what you want to learn from them. I make no claim that the data actually were created in such a way, but only that their distribution is remarkably like this one.
This is R code to reproduce the figure (and generate many more like it if you wish).
k <- 50
mu <- -10
sigma <- 7.5
threshold <- 3
n <- 1400
#
# Round most values to the nearest integer, occasionally
# to a half-integer.
#
rnd <- function(x, prec=300) {
y <- round(x * prec) / prec
ifelse(2*y == floor(2*y), y, round(y))
}
q <- c(0.25, 0.95) # Used to draw a reference line
par(mfcol=c(2,3))
set.seed(17)
invisible(replicate(6, {
# Generate data
z <- apply(matrix(rnorm(n*k), k), 2, max) # Max-normal distribution
y <- mu + sigma * z # Scale and recenter it
x <- rnd(pmax(y, threshold)) # Censor and round the values
# Plot them
qqnorm(x, cex=0.8)
m <- median(x)
s <- diff(quantile(x, q)) / diff(qnorm(q))
abline(c(m, s))
#hist(x) # Histogram of the data
#qqnorm(y) # QQ plot of the uncensored, unrounded data
}))
This seems to be a qqplot of the data compared with a standard normal distribution, so I would have thought the $x$ values should the typical values of the population quantiles of a standard normal distribution
So with $105$ observations I would have thought the extreme left $x$ value should be not far away from $\Phi^{-1}\left(\dfrac{0.5}{105}\right) \approx -2.59$ and the one next to it near $\Phi^{-1}\left(\dfrac{1.5}{105}\right) \approx -2.19$, with the extreme right values being the corresponding $\Phi^{-1}\left(\dfrac{104.5}{105}\right) \approx +2.59$ and $\Phi^{-1}\left(\dfrac{103.5}{105}\right) \approx +2.19$. Visually, this seems to be close to what you have in the charts
Best Answer
I explain how to read qq-plots in general here: QQ plot does not match histogram, and walk through constructing one here: PP-plots vs. QQ-plots. Those posts may help you.
Because your data are on the vertical axis, when we see the top right points above the line, we can conclude that they are too far out relative to a true normal, whereas the lower left points aren't far enough out. That is, your data are positively skewed relatively to a true normal. In addition, we can see that the points at the bottom turn perfectly horizontal. That means there is a floor that your data don't drop below. These could be data that cannot be negative, for example.