Section 7.3 Assessing Normality: Using Normal Probability Plots

Section 7.3: Assessing Normality

Objective: Use Normal Probability Plots to Assess Normality

This section introduces the concept of normality in data distributions and explains how normal probability plots can be used to assess whether a dataset is approximately normally distributed. Understanding normality is essential for many statistical methods that assume data follows a normal distribution.

What is a Normal Score?

• Normal Score: A normal score is a value derived from the standard normal distribution, typically used to compare observed data to what would be expected if the data were normally distributed.

• Normal scores are often used in constructing normal probability plots.

What is a Normal Probability Plot?

• Normal Probability Plot: A graphical technique for assessing whether a dataset is approximately normally distributed.

• In a normal probability plot, the observed data values are plotted against the expected normal scores. If the points form a roughly straight line, the data are likely to be normally distributed.

Steps for Drawing a Normal Probability Plot by Hand

To construct a normal probability plot manually, follow these steps:

1. Order the Data: Arrange the observed data values in ascending order.

2. Assign Ranks: Assign each data value a rank (from 1 to n, where n is the sample size).

3. Calculate Normal Scores: For each rank, calculate the corresponding expected normal score (often using the inverse cumulative distribution function of the standard normal distribution).

4. Plot the Points: Plot each observed data value against its corresponding normal score.

Interpreting Normal Probability Plots

• If the plotted points form a straight line, the data are likely to be normally distributed.

• Deviations from linearity suggest departures from normality, such as skewness or heavy tails.

• Linearity: The closer the points are to a straight line, the stronger the evidence for normality.

Example: Drawing a Normal Probability Plot by Hand

The following table shows finishing times (in seconds) for six randomly selected races of a greyhound named Bustiers Bourbon at the 5/16 mile track at Greyhound Park in Dubuque, Iowa. The goal is to assess whether the variable "finishing time" is normally distributed.

Finishing Time (seconds)
31.84
32.04
31.89
32.52
31.52
31.87

To draw the plot by hand, follow the four steps above. If the points are approximately linear, the data may be considered normally distributed.

Example: Drawing a Normal Probability Plot Using Technology

Statistical software or graphing calculators can automate the process of creating normal probability plots. The same data from the previous example can be entered into software such as StatCrunch to generate the plot and assess normality.

Finishing Time (seconds)
31.84	32.52
32.04	31.52
31.89	31.87

Interpret the plot: If the points lie close to a straight line, there is evidence that the variable is normally distributed.

Example: Assessing Normality with a Larger Dataset

The following table shows the time (in minutes) 100 randomly selected riders spent waiting in line for the Demon Roller Coaster. The objective is to determine whether the variable "wait time" is normally distributed.

Wait Time (minutes)
33	7	3	167	31	6	22	12	18	21	53
15	41	31	21	41	9	8	14	16	24	36
...

To assess normality, construct a normal probability plot using the data. If the plot is approximately linear, the wait times may be considered normally distributed. Otherwise, the data may be skewed or have outliers.

Key Formula: Calculating Normal Scores

• For each data point with rank $i$ in a sample of size $n$, the expected normal score is: $z_i = \Phi^{-1}\left(\frac{i - 0.5}{n}\right)$ where $\Phi^{-1}$ is the inverse cumulative distribution function (quantile function) of the standard normal distribution.

Summary Table: Steps for Assessing Normality

Step	Description
1	Order the data from smallest to largest
2	Assign ranks to each data value
3	Calculate expected normal scores for each rank
4	Plot observed values against normal scores
5	Assess linearity of the plot to determine normality

Additional info:

• Normal probability plots are also known as Q-Q (quantile-quantile) plots.

• Statistical software can provide additional tests for normality, such as the Shapiro-Wilk test.