77 Normal Probability Plot

77.1 Definition

The Normal Probability Plot is a special case of the QQ Plot (see Chapter 76): one axis shows the sample quantiles and the other axis shows the theoretical quantiles of a Normal distribution.

If the data are approximately Normal, the plotted points should follow a straight line. Systematic curvature indicates departures from Normality — for example, an S-shape indicates skewness, and curvature in both tails (points bowing above the line at the right end and below at the left) indicates heavier tails than Normal (leptokurtosis).

77.2 R Module

77.2.1 Public website

The Normal Probability Plot is available on the public website as part of several modules:

https://compute.wessa.net/rwasp_percentiles.wasp (Percentiles module)
https://compute.wessa.net/rwasp_fitdistrnorm.wasp (ML Fitting of Normal Density)

77.2.2 RFC

The Normal Probability Plot can be found in RFC under the menu “Distributions / ML Fitting”.

To compute the Normal Probability Plot on your local machine, the following script can be used in the R console:

# Two contrasting datasets
set.seed(42)
x_normal <- rnorm(100, mean = 50, sd = 10)    # Normal
x_skewed <- rexp(100, rate = 0.1)             # Right-skewed

par(mfrow = c(1, 2))

qqnorm(x_normal,
       main = "Normal data",
       xlab = "Theoretical quantiles",
       ylab = "Sample quantiles")
qqline(x_normal, col = "red")

qqnorm(x_skewed,
       main = "Right-skewed data",
       xlab = "Theoretical quantiles",
       ylab = "Sample quantiles")
qqline(x_skewed, col = "red")

par(mfrow = c(1, 1))

The qqnorm function plots the sample quantiles against the theoretical Normal quantiles. The qqline function adds a reference line through the first and third quartiles. Points near the line indicate approximate Normality; systematic deviations indicate departures.

77.3 Purpose

The Normal Probability Plot is a quick diagnostic for checking whether a Normal model is plausible before using methods that rely on Normality assumptions.

For formal tests of fit (beyond visual inspection), see Hypothesis Testing.

77.4 Pros & Cons

77.4.1 Pros

The Normal Probability Plot has the following advantages:

It provides a sensitive visual check of Normality that is easy to interpret: points near the line indicate approximate Normality.
It highlights the nature of the departure — an S-shape indicates skewness; curved tails indicate excess kurtosis.
It does not depend on any tuning parameter (unlike the histogram, which depends on bin width).
It is available in virtually all statistical software.

77.4.2 Cons

The Normal Probability Plot has the following disadvantages:

It is specific to the Normal distribution. To compare against other distributions, the more general QQ Plot must be used (see Chapter 76).
For small samples (\(n < 30\)), departures from Normality are difficult to distinguish from sampling variability.
It does not provide a summary statistic or \(p\)-value; formal normality tests (see Hypothesis Testing) are needed to quantify the evidence.

77.5 Example

The example below uses the monthly birth counts from Belgium (\(n = 73\)), the same series shown in Chapter 76. Because the series exhibits a long-term downward trend, the distribution is bimodal — clearly non-Normal.

births <- c(126.304, 125.511, 125.495, 130.133, 126.257, 110.323,
             98.417, 105.749, 120.665, 124.075, 127.245, 146.731,
            144.979, 148.210, 144.670, 142.970, 142.524, 146.142,
            146.522, 148.128, 148.798, 150.181, 152.388, 155.694,
            160.662, 155.520, 158.262, 154.338, 158.196, 160.371,
            154.856, 150.636, 145.899, 141.242, 140.834, 141.119,
            139.104, 134.437, 129.425, 123.155, 119.273, 120.472,
            121.523, 121.983, 123.658, 124.794, 124.827, 120.382,
            117.395, 115.790, 114.283, 117.271, 117.448, 118.764,
            120.550, 123.554, 125.412, 124.182, 119.828, 115.361,
            114.226, 115.214, 115.864, 114.276, 113.469, 114.883,
            114.172, 111.225, 112.149, 115.618, 118.002, 121.382,
            120.663)

qqnorm(births,
       main = "Normal QQ Plot — Belgium monthly births",
       xlab = "Theoretical quantiles",
       ylab = "Sample quantiles (thousands)")
qqline(births, col = "red")

Many points fall outside the reference line, and the curved pattern at both ends reveals non-Normality. The S-shape is a visual signature of a bimodal (two-peaked) distribution, consistent with the histogram shown in Chapter 76.

77.6 Task

Apply qqnorm and qqline to the student survey submission times from Chapter 62. Do the submission times appear Normally distributed? Describe the shape of any departure from the reference line and explain what it implies about the distribution.