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 TransformhCompute followed by function selections.
TransformhCompute
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 TransformhCompute 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 hGraphshInteractivehLine
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.
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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 10 + 3(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 0.25. 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 TransformhCompute. 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.
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Now use the menu options hGraphshInteractivehLine. 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.
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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 hGraphshControl 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).
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19.5
20.3
20.7
18.9
19.5
20.7
21.4
21.9
22.7
23.8
20.5
Enter the data in the first column, and name the column oz.
Select hGraphshControl. Choose “Individuals, Moving Range” as shown below.
Click on Define. Enter oz into the Process Measurement box. Click on “Individuals” under Charts.
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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 + 3σ = 20.7 + 3(0.7) = 22.8 as the upper limit and µ − 3σ = 20.7 − 3(0.7) = 18.6 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”.
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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 and σ = 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
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Futures Quotes for the Price of Coffee Beans: Tscc04.sav
Incidence of Melanoma Tumors: Tscc05.sav
Percent Change in Consumer Price Index: Tscc06.sav
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