8.55 Construct a normal quantile-quantile plot of these data, which represent the diameters of 36 rivet heads in 1/100 of an inch: 6.72 6.77 6.82 6.70 6.78 6.70 6.62 6.75 6.66 6.66 6.64 6.76 6.73 6.80 6.72 6.76 6.76 6.68 6.66 6.62 6.72 6.76 6.70 6.78 6.76 6.67 6.70 6.72 6.74 6.81 6.79 6.78 6.66 6.76 6.76 6.72 Test for Normality Using a Quantile Plot How do we know whether the normal distribution is an adequate model for the data you have? We will see in Chapter 8 that there are structural reasons why some processes produce data with at least an approximately normal distribution. Besides that, we know that normal distributions are symmetric, and their shape is like a bell. A histogram showing a highly skewed sample may rule out a normal model, but histograms are not usually a reliable representation of the distribution, unless the sample is very large. For a small to moderate sized sample, a “probability plot” gives us better visual evidence: if the data is well-described by a normal distribution, the plotted points will fall approximately on a straight line. While this still involves some subjectivity, deviations such as curvature (especially at the ends) are easy to spot. Given a data set of n observations, the following instructions will allow you to create a “normal quantile plot” using Excel. You can also create a quantile plot in MATLAB using the command qqplot. Create a spreadsheet as follows: 1. In column A enter the heading “Rank” in cell A1 followed by the numbers 1, 2, …, n in rows 2 through n +1 . This can be accomplished quickly by entering 1 in A2 and then using Fill/Series. In the Series text box, choose “series in columns,” type “linear,” and enter n as the stop value. 2. Label column B “quantiles” and convert the ranks in column A to quantiles using the formula (j – 0.5)/n, replacing n with the actual sample size. That is, enter =(A2 -.5)/n in cell B2 and copy this formula into the cells below. (Note, this is different from the formula given in the text.) 3. The next step produces the normal quantiles. Label column C as you wish to refer to the normal distribution (in C1) and enter the following formula into C2: =NORMINV(B2) Next copy C2 and drag down to highlight the rest of the C column next to the ranks. This action fills the column with normal quartiles corresponding to a data set of size n. 4. Enter your data into column D, using any appropriate label. The data needs to be sorted from smallest to larges so highlight the column of data and use the Sort command. Excel will warn you that there is data next to this column, but “continue with the current selection” to sort only this column. 5. Finally, highlight columns C and D and insert a scatter (X,Y) chart that graphs column C versus column D. If the resulting plot looks roughly linear (ignoring “wiggles”), we may conclude that the data can be assumed normal (or close to normal). It will help to insert a trend line (right click on any point and select Add a Trendline). Assignment: Your imaginary employer wants to analyze the rivet head data in problem 8.55 of the text, and asks you to check to see if the data appears normally distributed. Create a normal quantile plot for the data #55 using Excel or MATLAB, and insert this plot into a short report (a paragraph) that answers this question. Explain to your employer what you did and your conclusion. Use complete sentences, not hand-written.
The correct answer and explanation is:
Report
To assess whether the rivet head diameters follow a normal distribution, I constructed a normal quantile-quantile (Q-Q) plot using the given data. The data consists of the diameters of 36 rivet heads measured in hundredths of an inch.
Methodology:
- The data was sorted in ascending order.
- Theoretical quantiles corresponding to a standard normal distribution were calculated using the formula (j−0.5)/n(j – 0.5)/n, where jj is the rank of each data point and nn is the total number of data points.
- These theoretical quantiles were plotted against the ordered data values.
- A reference line representing an ideal normal distribution was added for comparison.
Results:
The Q-Q plot demonstrates that the data points align closely with the ideal normal line, indicating that the data is approximately normally distributed. Small deviations or “wiggles” are visible but do not significantly stray from the line, which suggests that these variations are likely due to random noise rather than systematic non-normality.
Conclusion:
Based on the Q-Q plot, the rivet head diameter data can be reasonably modeled by a normal distribution. This finding supports the assumption that the manufacturing process or measurement variation produces normally distributed outcomes.