Constructing Normal Quantile Plots. In Exercises 17–20, use the given data values to identify the corresponding z scores that are used for a normal quantile plot, then identify the coordinates of each point in the normal quantile plot. Construct the normal quantile plot, then determine whether the data appear to be from a population with a normal distribution.


Earthquake Depths A sample of depths (km) of earthquakes is obtained from Data Set 24 “Earthquakes” in Appendix B: 17.3, 7.0, 7.0, 7.0, 8.1, 6.8.

Verified step by step guidance

1. Step 1: Sort the given data values in ascending order. The sorted data values are: 6.8, 7.0, 7.0, 7.0, 8.1, 17.3.
2. Step 2: Assign ranks to each data value based on their position in the sorted list. For tied values, assign the average rank. For example, the three 7.0 values will share the average rank of their positions.
3. Step 3: Calculate the cumulative probability for each rank using the formula: P = (rank - 0.5) / n, where n is the total number of data points. This gives the cumulative probability for each data value.
4. Step 4: Find the z-scores corresponding to the cumulative probabilities using the standard normal distribution table or an appropriate statistical software. These z-scores represent the theoretical quantiles for a normal distribution.
5. Step 5: Plot the data values (x-axis) against their corresponding z-scores (y-axis) to construct the normal quantile plot. Analyze the plot to determine if the data points follow a straight line, which would suggest the data are from a population with a normal distribution.

Key Concepts

Z-scores

Z-scores are standardized scores that indicate how many standard deviations an element is from the mean of a dataset. They are calculated by subtracting the mean from the data point and then dividing by the standard deviation. In the context of normal quantile plots, z-scores help in transforming the data into a standard normal distribution, allowing for a comparison of the data's distribution to a theoretical normal distribution.

Normal Quantile Plot

A normal quantile plot is a graphical tool used to assess if a dataset follows a normal distribution. It plots the z-scores of the data against the expected z-scores from a standard normal distribution. If the points on the plot form a roughly straight line, it suggests that the data are normally distributed; deviations from this line indicate departures from normality.

Normal Distribution

Normal distribution is a probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean. It is characterized by its bell-shaped curve and is defined by two parameters: the mean and the standard deviation. Understanding normal distribution is crucial for interpreting the results of normal quantile plots and determining the likelihood of data belonging to a normal population.