CHAPTER 6 NORMAL DISTIBUTIONS

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1 CHAPTER 6 NORMAL DISTIBUTIONS GRAPHS OF NORMAL DISTRIBUTIONS (SECTION 6.1 OF UNDERSTANDABLE STATISTICS) The normal distribution is a continuous probability distribution determined by the value of µ and σ. Similar to the binomial distribution, the normal distribution can be studied with three main functions of SPSS: The PDF & Noncentral PDF function which gives the probability density function value for a value X; The CDF & Noncentral CDF function which, for a value X, gives the probability less than or equal to X; and the Inverse DF which gives the inverse of the CDF. To apply PDF and CDF to a normal distribution, we need to use the menu selections Transform Compute followed by function selections. Transform Compute Dialog Box Responses Enter name of the Target Variable Enter the function formula into the Numeric Expression box: Under the Function group, select PDF & Noncentral PDF for probability; CDF & Noncentral CDF for cumulative probability for CDF; Inverse DF for inverse cumulative probability. Then under Functions and Special Variables, select the sub-function for the normal distribution. For example, the PDF function for a normal distribution is PDF.Normal(quant, mean, stddev) and the CDF function for a binomial distribution is CDF.Normal(quant, mean, stddev). Here mean is the mean value of the normal distribution, stddev is the standard deviation of the normal distribution. The input quant is the value of X, for which we want to find the PDF value or CDF value. You may enter a value for quant, or you may store the values for quant in a variable (column) and enter the variable name for quant. The Inverse DF function for a normal distribution is Idf(prob, mean, stddev). It returns the value of X such that CDF(X) = prob. We can sketch the graph of a normal distribution by following these steps: 1. Name the first column X. Enter the values of X for which we want to compute PDF(X). 2. Name the second column pdf_x. Use Transform Compute to compute the values of PDF(X) and store them in the second column. 3. Sketch the graph by using the following menu options for graphs. Menu Options for Graphs To graph functions in SPSS, we use the menu Graphs Interactive Line Dialog Box Responses Choose 2D Coordinate Drag the variable for horizontal axis into the variable box for the horizontal axis. Drag the variable for vertical axis into the variable box for the vertical axis. Click on Dots and Lines to get into the Dots and Lines tab. 324

2 Part IV: SPSS Guide 325 Under Interpolation click on Spline Click on OK. The graph will show up. There are other options available. See the Help menu for more information. Example Graph the normal distribution with mean µ = 10 and standard deviation σ = 2. Since most of the normal curve occurs over the values µ 3σ to µ + 3σ, we will start the graph at 10 3(2) = 4 and end it at (2) = 16. We will let SPSS set the scale on the vertical axis. To graph a normal distribution, use X as the variable name for the first column and enter X values in this column. Let the first value of X be 4, last value be 16, and increment be Altogether there are 49 X values. Use pdf_x as the variable name for the second column. Allow more decimal digits for pdf_x, say 6. (This can be set in the variable view of data editor.) Use Transform Compute. Let pdf_x be the target variable. Under Numeric Expression enter PDF.Normal(X, 10, 2). Click on OK. The PDF(X) are computed and stored in the second column.

3 326 Technology Guide Understandable Statistics, 8th Edition Now use the menu options Graphs Interactive Line. In the dialog box, choose 2D Coordinate. Then drag X into the variable box for the horizontal axis, pdf_x into the variable box for the vertical axis. Now click on Dots and Lines to get into the Dots and Lines tab. Under Interpolation click on Spline as shown below.

4 Part IV: SPSS Guide 327 Click on OK. The graph follows. CONTROL CHARTS (SECTION 6.1 OF UNDERSTANDABLE STATISTICS) SPSS supports a variety of control charts. The type discussed in Section 6.1 of Understandable Statistics is called an individual chart. The menu selection is Graphs Control followed by the sequence of dialogue box responses described below. Dialogue Box Responses Choose Individuals, Moving Range and click on Define. This brings a new dialogue box. Enter the variable into the Process Measurement box. Click on Individuals under Charts. Click on Options. In the Options dialog box, enter the Number of Sigmas. In the chart, the lines indicating sample mean, and sample mean plus (and minus) Number of Sigmas times sample standard deviation will be displayed. Click on Continue. Click on Statistics. In this dialog box, specify upper limit and lower limit. The lines indicating these two limits will also be displayed. This will help you to see the data lying out of these limits. You may also check the button Actual % outside specification limits to show the percentage of data outside these limits. For information about the other options, see the Help menu. Example In a packaging process, the weight of popcorn that is to go in a bag has a normal distribution with µ = 20.7 oz and σ = 0.7 oz. During one session of packaging, eleven samples were taken. Use an individual control chart to show these observations. The weights were (in oz).

5 328 Technology Guide Understandable Statistics, 8th Edition Enter the data in the first column, and name the column oz. Select Graphs Control. Choose Individuals, Moving Range as shown below. Click on Define. Enter oz into the Process Measurement box. Click on Individuals under Charts.

6 Part IV: SPSS Guide 329 Click on Options. In the Options dialog box, enter 3 as the Number of Sigmas. In the chart, the lines indicating sample mean, sample mean plus 3(sample standard deviation), and sample mean minus 3(sample standard deviation) will be displayed. After that click on Continue. Click on Statistics. In this dialog box, specify upper limit and lower limit. Let us use the population statistics µ + 3 σ = (0.7) = as the upper limit and µ 3 σ = (0.7) = as the lower limit. The lines indicating these two limits will also be displayed in the chart. Also check the button Actual % outside specification limits.

7 330 Technology Guide Understandable Statistics, 8th Edition Click on Continue. Then click on OK. The graph follows. We see that one observation (or 9.1% of the data) is outside the set limits. LAB ACTIVITIES FOR GRAPHS OF NORMAL DISTRIBUTIONS AND CONTROL CHARTS 1. (a) Sketch a graph of the standard normal distribution with µ = 0 and σ = 1. Enter into the first column the data from 3 to 3 in increments of 0.5. Then generate the graph. (b) Sketch a graph of a normal distribution with µ = 10 andσ = 1. Enter into the first column the data from 7 to 7 in increments of 0.5. Compare this graph to that of part (a). Do the height and spread of the graphs appear to be the same? What is different? Why would you expect this difference? (c) Sketch a graph of a normal distribution with µ = 0 and σ = 2. Enter into the first column the data from 6 to 6 in increments of 0.5. Compare this graph to that of part (a). Do the height and spread of the graphs appear to be the same? What is different? Why would you expect this difference? Note, to really compare the graphs, it is best to graph them using the same scales. Redo the graph of part (a) using X from 6 to 6. Then redo the graph in this part using the same X values as in part (a) and Y values ranging from 0 to the high value of part (a). 2. Use one of the following SPSS data files found on the CD-ROM to draw a control chart. In each of the files the target value for the mean µ is stored in the Column2-Row1 position and the target value for the standard deviation is stored in the Column3-Row1 position. Use the targeted MU and SIGMA values to set upper and lower limits. Yield of Wheat: Tscc01.sav PepsiCo Stock Closing Prices: Tscc02.sav PepsiCo Stock Volume of Sales: Tscc03.sav

8 Part IV: SPSS Guide 331 Futures Quotes for the Price of Coffee Beans: Tscc04.sav Incidence of Melanoma Tumors: Tscc05.sav Percent Change in Consumer Price Index: Tscc06.sav