Descriptive Statistics > Q Q plots
What is a Q Q Plot?
Q Q Plots (Quantile-Quantile plots) are plots of two quantiles against each other. A quantile is a fraction where certain values fall below that quantile. For example, the median is a quantile where 50% of the data fall below that point and 50% lie above it. The purpose of Q Q plots is to find out if two sets of data come from the same distribution. A 45 degree angle is plotted on the Q Q plot; if the two data sets come from a common distribution, the points will fall on that reference line.
The image above shows quantiles from a theoretical normal distribution on the horizontal axis. It’s being compared to a set of data on the y-axis. This particular type of Q Q plot is called a normal quantile-quantile (QQ) plot. The points are not clustered on the 45 degree line, and in fact follow a curve, suggesting that the sample data is not normally distributed.
How to Make a Q Q Plot
Sample question: Do the following values come from a normal distribution?
7.19, 6.31, 5.89, 4.5, 3.77, 4.25, 5.19, 5.79, 6.79.
Step 1: Order the items from smallest to largest.
- 3.77
- 4.25
- 4.50
- 5.19
- 5.79
- 5.89
- 6.31
- 6.79
- 7.19
Step 2: Draw a normal distribution curve. Divide the curve into n+1 segments. We have 9 values, so divide the curve into 10 equally-sized areas. For this example, each segment is 10% of the area (because 100% / 10 = 10%).
Step 3: Find the z-value (cut-off point) for each segment in Step 3. These segments are areas, so refer to a z-table (or use software) to get a z-value for each segment.
The z-values are:
- 10% = -1.28
- 20% = -0.84
- 30% = -0.52
- 40% = -0.25
- 50% = 0
- 60% = 0.25
- 70% = 0.52
- 80% = 0.84
- 90% = 1.28
- 100% = 3.0
Step 4: Plot your data set values (Step 1) against your normal distribution cut-off points (Step 3). I used Open Office for this chart:
Note: This example used the standard normal distribution, but if think your data could have come from a different normal distribution (i.e. one with a different mean and standard deviation) then you could use that instead.
Q Q Plots and the Assumption of Normality
The assumption of normality is an important assumption for many statistical tests; you assume you are sampling from a normally distributed population. The normal Q Q plot is one way to assess normality. However, you don’t have to use the normal distribution as a comparison for your data; you can use any continuous distribution as a comparison (for example a Weibull distribution or a uniform distribution), as long as you can calculate the quantiles. In fact, a common procedure is to test out several different distributions with the Q Q plot to see if one fits your data well.
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References
Agresti A. (1990) Categorical Data Analysis. John Wiley and Sons, New York.
Gonick, L. (1993). The Cartoon Guide to Statistics. HarperPerennial.
Vogt, W.P. (2005). Dictionary of Statistics & Methodology: A Nontechnical Guide for the Social Sciences. SAGE.
Wheelan, C. (2014). Naked Statistics. W. W. Norton & Company