qqplot.Rmd

# Chart: QQ-Plot {#qqplot}

![](images/banners/banner_qqplot.png)
*This chapter originated as a community contribution created by [hao871563506](https://github.com/hao871563506){target="_blank"}*

*This page is a work in progress. We appreciate any input you may have. If you would like to help improve this page, consider [contributing to our repo](contribute.html).*

```{r, include=FALSE}
knitr::opts_chunk$set(echo = TRUE,message = FALSE,
                      warning = FALSE)
```


## Introduction
In statistics, a Q-Q (quantile-quantile) plot is a probability plot, which is a graphical method for comparing two probability distributions by plotting their quantiles against each other. A point (x, y) on the plot corresponds to one of the quantiles of the second distribution (y-coordinate) plotted against the same quantile of the first distribution (x-coordinate). Thus the line is a parametric curve with the parameter which is the number of the interval for the quantile.


## Interpreting qqplots

## Normal or not (examples using qqnorm)

### Normal qqplot
```{r}
x <- rnorm(1000, 50, 10)
qqnorm(x)
qqline(x, col = "red")
```

The points seem to fall along a straight line. Notice the x-axis plots the theoretical quantiles. Those are the quantiles from the standard Normal distribution with mean 0 and standard deviation 1.

### Non-normal qqplot
```{r}
x <- rexp(1000, 5)
qqnorm(x)
qqline(x, col = "red")
```

Notice the points form a curve instead of a straight line. Normal Q-Q plots that look like this usually mean your sample data are skewed.

## Different kinds of qqplots
The following graph is a conclusion of all the kinds of qqplot:
![](images/qqplot_types.png)
*via [Stack Exchange](https://stats.stackexchange.com/questions/101274/how-to-interpret-a-qq-plot/){target="_blank"}*

- **Normal qqplot**: The normal distribution is symmetric, so it has no skew (the mean is equal to the median).

- **Right skewed qqplot**: Right-skew is also known as positive skew.

- **Left skewed qqplot**: Left-skew is also known as negative skew.

- **Light tailed qqplot**: meaning that compared to the normal distribution there is little more data located at the extremes of the distribution and less data in the center of the distribution.

- **Heavy tailed qqplot**: meaning that compared to the normal distribution there is much more data located at the extremes of the distribution and less data in the center of the distribution.

- **Biomodel qqplot**: illustrate a bimodal distribution.


## qqplot using ggplot
In order to use `ggplot2` to plot a qqplot, we must use a dataframe, so here we convert it to one. We can see that using ggplot to plot a qqplot has a similar outcome as using qqnorm
```{r}
library(ggplot2)
x <- rnorm(1000, 50, 10)
x <- data.frame(x)
ggplot(x, aes(sample = x)) +
  stat_qq() +
  stat_qq_line()
```

However, when we need to plot different groups, ggplot will be very helpful with its coloring by factor.
```{r}
library(ggplot2)
ggplot(mtcars, aes(sample = mpg, colour = factor(cyl))) +
  stat_qq() +
  stat_qq_line()
```

## References
- [Understanding Q-Q Plots](https://data.library.virginia.edu/understanding-q-q-plots/){target="_blank"}: A discussion from the University of Virginia Library on qqplots.
- [How to interpret a QQ plot](https://stats.stackexchange.com/questions/101274/how-to-interpret-a-qq-plot){target="_blank"}: Another resource for interpreting qqplots.
- [A QQ Plot Dissection Kit](http://seankross.com/2016/02/29/A-Q-Q-Plot-Dissection-Kit.html){target="_blank"}: An excellent walkthrough on qqplots by Sean Kross.
- [Probability plotting methods for the analysis of data](https://www.jstor.org/stable/2334448?seq=1#metadata_info_tab_contents){target="_blank"}: Paper on plotting techniques, which discusses qqplots. (Wilk, M.B.; Gnanadesikan, R. (1968))
- [QQ-Plot Wiki](https://en.wikipedia.org/wiki/Q%E2%80%93Q_plot#cite_note-1){target="_blank"}: Wikipedia entry on qqplots