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Normal Plots

Introduction

• We know that a bell-shaped histogram is said to have a normal distribution.
• However, in practice, it is often difficult to look at a histogram and determine how close its distribution is to normal.
• An easier way to tell if a dataset is normally distributed is to look at the normal plot. If the normal plot is close to a straight line, the distribution of the dataset is close to normal.

Normal Scores

• To define normal scores, find z-scores that divide the standard normal curve into n+1 equal areas of 1/(n+1) each. For example, the normal scores when n = 5 are defined by the bins (-∞, -0.97], (-0.97, -0.43], (-0.43, 0.00], (0.00, 0.43], [0.43, 0.97], (0.97, ∞). Each of these bins has area 1/(5+1) = 0.1667. This means that the normal scores for a dataset with n = 5 are -0.97, -0.43, 0.00, 0.43, and 0.97.
• We can also see that the bins (-∞, -0.97], (-∞, -0.43], (-∞, 0.00], (-∞, 0.43], (-∞, 0.97] have areas 1/6 = 0.1667, 1/3 = 0.3333, 1/2 = 0.5000, 2/3 = 0.6667, 5/6 = 0.8333, respectively.

Normal Plots

• A normal plot or Q-Q plot is formed by plotting the normal scores defined in the previous section are plotted on the y-axis vs. the actual sorted data values on the x-axis.
• If the normal plot is close to a straight line, we conclude that the dataset is close to normal.
• Here is a normal plot of the dataset
       3   60   98   145   201

  The dataset values on the y-axis are plotted against the normal scores
       -0.97  -0.43   0.00   0.43   0.97.

• Here is the resulting normal plot:
  
• The normal plot is approximately a straight line, so we conclude that the original dataset is close to normally distributed.

Normal Plots with R

• Example:   Obtain the normal plot (Q-Q Plot) of the NIST-10 data.
  

  # Suppose that the nbs-10.txt file is in c:/workspace.
  setwd("c:/workspace")
  df <- read.csv("nbs-10.txt")
  w <- df$Thickness
  qqnorm(w)
  

Nonnormality

• What if the dataset is not normal?
• Here are some examples of simulated datasets that deviate from normality in various ways. In each case the histogram and the normal plot are shown.
• In these normal plots, the sample quantiles are plotted on the vertical scale (y-axis) and the expected normal scores are plotted on the horizontal axis (x-axis). To interpret the normal plots, tilt your head 90 degrees to your left left side so that your left ear is pointing to the ground and your right ear is pointing to the ceiling. Then the vertical axis of the normal plot will look like it is to the right when going up and will look like it is to the left when going down.

1. Case 1. Data are approximately normal. The histogram has the tails like a normal distribution; the normal plot forms a straight line.
   
2. Data are skewed to the right. The data are closer to the center on the left (down) and further away on the right (up) than they would be if they were normal. Further away on the right means "skewed to the right."
   
3. Data are skewed to the left. The data are further from the center to the left (down) and closer to the center on the right (up) than if they were normal.  Further away on the left means "skewed to the left."
   
4. The distribution of the data has thin tails. The data are closer in to the center of the histogram on both the left (down) and the right (up) than they would be if the data were normal.
   
5. The distribution of the data has thick tails. The data are further away on both the left (down) and the right (up) than they would be if the data were normal.