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STANDARD SCORES AND THE NORMAL DISTRIBUTION Subtopics  HYPERLINK "" \l "introduction" \t "_self" Introduction  HYPERLINK "" \l "z" \t "_self" Standard (z) scores  HYPERLINK "" \l "normal" \t "_self" The Normal Distribution  HYPERLINK "" \l "key_concepts" Key Concepts  HYPERLINK "" \l "exercises" \t "_self" Exercises  HYPERLINK "" \l "further" \t "_self" For Further StudySPSS Tools New with this topic  HYPERLINK "../Tools/histogram.html" \t "_blank" Histogram Review  HYPERLINK "../Tools/getting_started.html" \t "_blank" Getting Started  HYPERLINK "../Tools/compute.html" \t "_blank" Compute  HYPERLINK "../Tools/descriptives.html" \t "_blank" Descriptives  HYPERLINK "../Tools/frequencies.html" \t "_blank" Frequencies  HYPERLINK "../Tools/explore.html" \t "_blank" Explore  HYPERLINK "../Tools/boxplot.html" \t "_blank" Boxplot Introduction There is an old saying that you cant compare apples and oranges. Fortunately, this is not always the case. Any interval or ratio variable can be converted to a standard unit of measurement called a standard score. This is especially convenient whenever variables are normally distributed.  Standard (z) scores Examine the variable descriptions in the codebook for the  HYPERLINK "../Codebooks/countries.html" \t "_blank" countries data. Notice that different variables are measured in radically different units of measurement including, among other things, dollars, people, and percentages. Any of these can easily be transformed into a standard (z) score which, by definition, will have a mean of 0 and a standard deviation of 1 regardless of the original unit of measurement. The formula for converting raw scores on any variable X to z scores is:  INCLUDEPICTURE "C:\\User Files\\CP\\HTML\\jlkorey\\POWERMUTT\\Topics\\normal_distributions_files\\image001.gif" \* MERGEFORMAT \d , where: zi is the standard score for case i, Xi is the raw score for case i,  is the mean of the variable, and  is the standard deviation of the variable. You can use the  HYPERLINK "../Tools/compute.html" \t "_blank" compute tool to convert any interval variable to z-scores once you know the mean and the standard deviation. An easier way is to let the  HYPERLINK "../Tools/descriptives.html" \t "_blank" descriptives procedure do the work for you " just check the box to "Save standardized values as variables."  The Normal Distribution Many variables are normally distributed. A typical curve of the normal distribution is shown in the figure below. (A normal curve is sometimes called a bell-shaped curve.) Normal curves have certain defining characteristics. The most frequent values are found in the middle of the distribution, and taper off the further away one goes from the middle. The distribution is symmetric, meaning that the upper half of the distribution is a mirror image of the lower half. Taken together, the result is that the  HYPERLINK "descriptive_statistics.html" \l "mean" \t "_blank" mean,  HYPERLINK "measurement_basics.html" \l "median" \t "_blank" median, and  HYPERLINK "descriptive_statistics.html" \l "mode" \t "_blank" mode are all the same.  INCLUDEPICTURE "C:\\User Files\\CP\\HTML\\jlkorey\\POWERMUTT\\Topics\\normal_distributions_files\\normal_distribution.gif" \* MERGEFORMAT \d  While many variables are normally distributed, many are not. An easy way to tell if a variable is at least more or less normally distributed is to construct a histogram of the variable, and compare the result to a normal curve. The next figure describes an index of the self-identified ideology (liberal, moderate, conservative) of residents of different states derived from analysis of CBS/New York Times polls by Gerald C. Wright et al. HYPERLINK "" \l "note1" [1] The index ranges from -100 (all residents of a state identifying as conservative) to 100 (all residents identifying as liberal). The distribution is approximately bell shaped. In other words, there are a few very conservative and a few very liberal ones, but more are toward the middle of the road.  HYPERLINK \l "figure1"  INCLUDEPICTURE "C:\\User Files\\CP\\HTML\\jlkorey\\POWERMUTT\\pup.jpg" \* MERGEFORMAT \d   INCLUDEPICTURE "C:\\User Files\\CP\\HTML\\jlkorey\\POWERMUTT\\Topics\\normal_distributions_files\\figure1.gif" \* MERGEFORMAT \d  Consider, on the other hand, the distribution of voting records in the U. S. Senate. The following histogram uses DW-NOMINATE scores produced for members of the U.S. Senate by Jeff Lewis and Keith Poole. This index ranges from about -1 (most liberal) to about 1 (most conservative). HYPERLINK "" \l "note2" [2] The distribution is not even close to forming a normal curve. While the two measures are not directly comparable, senators would seem to be far more polarized than their constituents.  HYPERLINK \l "figure23"  INCLUDEPICTURE "C:\\User Files\\CP\\HTML\\jlkorey\\POWERMUTT\\pup.jpg" \* MERGEFORMAT \d   INCLUDEPICTURE "C:\\User Files\\CP\\HTML\\jlkorey\\POWERMUTT\\Topics\\normal_distributions_files\\figure2.gif" \* MERGEFORMAT \d  There are other ways to examine a variable in order to determine whether it is normally distributed. A  HYPERLINK "descriptive_statistics.html" \l "boxplots" \t "_blank" boxplot provides another tool. If a variable is at least approximately normally distributed, the median (the 50th percentile) will be midway in the inter-quartile range (the range between the 25th and 75th percentiles), the length of the top and bottom whiskers above and below the box will be about the same, and there will be few if any outliers or extreme values beyond the whiskers. There are also a couple of descriptive statistics that help measure departures from the normal distribution. Skewness measures departures from normality due to the impact of very high or very low values. In a perfectly normal distribution, it will have the value 0. If the mean is higher than the median (because the mean is inflated by some very high values), a distribution will have a positive skew. If the reverse is the case (due to some extremely low values), the skew will be negative. Kurtosis measures peakedness, the tendency of values to cluster near the middle of the distribution. In a perfectly normal distribution, it will have the value 0. A positive kurtosis indicates that values are more closely clustered toward the middle than would be the case in a normal distribution, while a negative kurtosis indicates that values are more spread out. Many statistical techniques that require at least  HYPERLINK "levels_of_measurement.html" \l "interval" \t "_blank" interval level measurement also require that variables be normally distributed. It is a good idea, therefore, to begin data analysis with some exploratory research into the distribution of the variables in the dataset. Considerable caution should be exercised in analyzing variables with markedly non-normal distributions. If variables are normally distributed, standard scores become extremely useful. It turns out that, in a normal distribution, 68 percent of cases will be within one standard deviation of the mean (that is, will have a z score within the range of 1), 95 percent will be within two standard deviations of the mean, and 99.7 percent will be within 3 standard deviations of the mean. In fact, if a variable is normally distributed, you can, by converting raw scores to z scores: convert a score to a percentile (a score in, for example, the 90th percentile would be one for which 90 percent of cases had that score or lower). determine the probability that a case will be above or below a certain number, or between two numbers. Most statistics texts include a table of the normal distribution for these purposes. There are also applets (little applications) on the Internet that do the same thing.  Key Concepts  HYPERLINK "" \l "histogram" histogram  HYPERLINK "" \l "normal" normal distribution  HYPERLINK "" \l "kurtosis" kurtosis  HYPERLINK "" \l "skewness" skewness  HYPERLINK "" \l "z" standard (z) score  Exercises Note: In SPSS, histograms can be produced using either the  HYPERLINK "../Tools/frequencies.html" \t "_blank" frequencies or the  HYPERLINK "../Tools/explore.html" \t "_blank" explore procedure. There is also a separate procedure specifically designed to produce  HYPERLINK "../Tools/histogram.html" \t "_blank" histograms. Except for explore, these procedures include the option of superimposing a normal curve on the histogram. Skewness and kurtosis can be produced with  HYPERLINK "../Tools/frequencies.html" \t "_blank" frequencies,  HYPERLINK "../Tools/descriptives.html" descriptives, or  HYPERLINK "../Tools/explore.html" \t "_blank" explore. Z-scores can be produced with  HYPERLINK "../Tools/descriptives.html" \t "_blank" descriptives or  HYPERLINK "../Tools/compute.html" \t "_blank" compute. 1. Start SPSS, and open countries.sav. Look at the  HYPERLINK "../Codebooks/countries.html" \t "_blank" countries codebook. Calculate the means and standard deviations for any two interval or ratio variables. Now  HYPERLINK "../Tools/compute.html" \t "_blank" compute two new variables by converting each of the original variables to z scores. For the new variables, calculate means and standard deviations. Use  HYPERLINK "../Tools/descriptives.html" descriptives to accomplish the same purpose. 2. Pick several variables in the Countries file, and obtain histograms, comparing the results to a normal distribution. Are the variables at least roughly normally distributed? Why or why not? 3. Open senate.sav. Look at the  HYPERLINK "../Codebooks/senate.html" \t "_blank" senate codebook. The file includes several measures of the voting behavior of senate members:  HYPERLINK "../Codebooks/senate.html" \l "acu" \t "_blank" acu. These are ratings of senators voting records compiled by the American Conservative Union (ACU). The ACU, like many interest groups, selected what it regarded as key votes on which to score elected representatives. A score of 100 would indicate a perfect conservative record, while a score of 0 would indicate a perfect liberal record.  HYPERLINK "../Codebooks/senate.html" \l "ada" \t "_blank" ada. Compiled by the liberal Americans for Democratic Action, these scores are roughly a mirror image of the ACU scores. A score of 100 would indicate a perfect liberal record, while a score of 0 would indicate a perfect conservative record.  HYPERLINK "../Codebooks/senate.html" \l "dwnom" \t "_blank" dwnom. This is a measure designed by political scientists Keith Poole and Howard Rosenthal to provide a more comprehensive index of roll call behavior. Scores range from about -1 (most liberal) to about 1 (most conservative).  HYPERLINK "../Codebooks/senate.html" \l "unity" \t "_blank" unity. Congressional Quarterly's Party Unity Score measures the percent of the time each senator voted in agreement with his or her party when majorities of the parties were on opposite sides of a roll call. Vermont's Bernie Sanders and Connecticut's Joe Lieberman (who were elected as independents, but who caucus with the Democratic Party for purposes of organizing the chamber and, in return, receiving their committee assignments) are treated as Democrats for purposes of this variable. Using  HYPERLINK "../Tools/explore.html" explore, examine the distributions of each of these measures. (Ask for histograms rather than stem and leaf plots.) Are these variables normally distributed? Repeat the analysis, this time using  HYPERLINK "../Codebooks/senate.html" \l "party" \t "_blank" party as a factor. What do these distributions look like? Use  HYPERLINK "../Tools/boxplots.html" boxplots to compare the distributions of acu, ada, and dwnom scores, broken down by party. (Note: Joseph Lieberman of Connecticut and Bernie Sanders of Vermont are coded as independents for the party variable. To treat party as a dummy variable either, 1)  HYPERLINK "../Tools/recode.html" recode to treat these two senators as Democrats (since they caucus with the Democratic Party), 2) go to SPSS Variable View and make 3 a missing value for this variable, or 3) use  HYPERLINK "../Tools/select_cases.html" select cases to exclude these senators from your analysis.) Does the distribution for dwnom look anything like those for acu and ada? (Why not?) Now convert all three variables to standard scores and run the boxplots again. You should notice a dramatic difference. Repeat, but using house.sav. Look at the  HYPERLINK "../Codebooks/house.html" \t "_blank" house codebook. 5.Open states.sav. Look at the  HYPERLINK "../Codebooks/states.html" \t "_blank" states codebook. Obtain the means and standard deviations for several interval or ratio level variables. In Data View, find the scores on these variables for your state. Convert these scores to z scores. On which variables is your state least typical? 6. Open house.sav. Look at the  HYPERLINK "../Codebooks/house.html" \t "_blank" house codebook. Repeat exercise 5 for your congressional district. (If you are not sure which district you are in, go to  HYPERLINK "http://www.vote-smart.org/" \t "_blank" http://www.vote-smart.org/.) 7. Go to  HYPERLINK "http://psych.colorado.edu/~mcclella/java/normal/normz.html" \t "_blank" http://psych.colorado.edu/%7Emcclella/java/normal/normz.html or to  HYPERLINK "http://faculty.vassar.edu/lowry/tabs.html" \l "z" \t "_blank" http://faculty.vassar.edu/lowry/tabs.html#z and, using the applet found there, answer the following questions about a normally distributed variable with a mean of 50 and a standard deviation of 10: a. What is the z score for a raw score of 72? b. What percent of cases will have scores over 72? c. What percent of cases will have scores between 28 and 72?  For Further Study Brown, James Dean, Skewness and Kurtosis, The JALT Testing & Evaluation SIG Newsletter. April 1997.  HYPERLINK "http://www.jalt.org/test/bro_1.htm" \t "_blank" http://www.jalt.org/test/bro_1.htm. Accessed November 23, 2003. Lane, David M., What is a Normal Distribution? Hyperstat.  HYPERLINK "http://davidmlane.com/hyperstat/normal_distribution.html" \t "_blank" http://davidmlane.com/hyperstat/normal_distribution.html.  [1]  HYPERLINK "http://php.indiana.edu/~wright1/" \t "_blank" http://php.indiana.edu/~wright1/. Accessed July 3, 2007. [2]. Royce Carroll, et al., DW-NOMINATE Scores with Bootstrapped Standard Errors, VoteView.  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STANDARD SCORES AND THE NORMAL DISTRIBUTION           ---  2 n#0    2 n(0   k`--- 2 ` `kSubtopics ---  2 `k  @Symbol---@"Arial------  2 x`k---  2 ~`k --- 2 `kIntroduction   2 `k @Times New Roman---- @ !J- k`'---  2 x`k---  2 ~`k --- 2 `kStandard (z) - @ !M- k`'--- 2 `kscore  2 `ks  2 `k - @ !'- k`'---  2 x`k---  2 ~`k --- 2 `kThe Normal   - @ !K- k`'--- 2  `kDistribution    2 `k - @ !H- k`'---  2 x`k---  2 ~`k --- 2  `kKey Concepts    2 `k   2 `k - @ !Y- k`'---  2 )x`k---  2 )~`k --- 2 ) `kExercises   2 )`k - @ !9+- k`'---  2 ;x`k---  2 ;~`k --- 2 ; `kFor Further  - @ !F=- k`'--- 2 M`kStudy   2 M`k - @ !#O- k`''k--- 2 kSPSS Tools  ---  2 Fk  ---  2 k---  2 k --- )2 )kNew with this topic     2 k  k'@1Courier New------  2 Ako---  2 Ik --- 2 Y kHistogram    2 k - @ !?Y- k'---  2 k---  2 k --- 2 )kReview    2 ^k  k'---  2 Ako---  2 Ik --- "2 YkGetting Started    2 k - @ !`Y- k'---  2 Ako---  2 Ik --- 2 YkCompute    2 k - @ !8Y- k'---  2 Ako---  2 Ik --- 2 Y kDescriptives   2 k - @ !KY- k'---  2 )Ako---  2 )Ik --- 2 )Y kFrequencies   2 )k - @ !I+Y- k'---  2 ;Ako---  2 ;Ik --- 2 ;YkExplore   2 ;k - @ !/=Y- k'---  2 MAko---  2 MIk --- 2 MYkBoxplot   2 Mk - @ !0OY- k'' --- 0,--$`z`||z`z- -'- - @ !z`- - @ !z`-  - @ !nza-  - @ !z-  - @ !z-  - @ !{`-  - @ !{`-  - @ !n{a-  - @ !{- - @ !{-  -'---  2 0   --- 2 `0 In 2 n 0 troduction ---  2 0    2 `^0 There is an old saying that you cant compare apples and oranges. Fortunately, this is not      2 `X0 always the case. Any interval or ratio variable can be converted to a standard unit of      j2 `?0 measurement called a standard score. This is especially conven      52 0 ient whenever variables are     +2 `0 normally distributed.    2 0     0,--$```- -'- - @ !`- - @ !`-  - @ !na-  - @ !-  - @ !-  - @ !`-  - @ !`-  - @ !na-  - @ !- - @ !-  -'---  2 0   --- (2 3`0 Standard (z) scores ---  2 30    b2 Y`:0 Examine the variable descriptions in the codebook for the      2 Y 0 countries  2 Y 0   22 Y0 data. Notice that differe  2 Y0 nt  - @ !6[-  --- 2 k`b0 variables are measured in radically different units of measurement including, among other things,              2 ~`^0 dollars, people, and percentages. Any of these can easily be transformed into a standard (z)      g2 `=0 score which, by definition, will have a mean of 0 and a stand      C2 %0 ard deviation of 1 regardless of the    2 `\0 original unit of measurement. The formula for converting raw scores on any variable X to z             2 ` 0 scores is:  2 0     0,&TNPP'A +Q)T`(Q+CDCDD D D D D DDD0D DDD>0DDDDDDDDDDD<0D@D@ D@ DDDD DDDDDDDDD'A +Q)T`(Q+?Dπ߀߀?aD&TNPP''--- 2 0 , where:   2 0   @Times New Roman- - - - - - ---  2 `0 z- - -  2 f0 i --- =2 m!0 is the standard score for case i,  2 00   ---  2 >`0 X - - -   2 @k0 i---  2 >n0   52 >s0 is the raw score for case i,   2 >0    c`@Times New Roman- - .>2 c`"0  is the mean of the variable, and   .-   2 c40    `@Times New Roman- - .M2 `,0  is the standard deviation of the variable.   .-   2 i0   "Systemwrkaw@dwf k- -   00//..՜.+,D՜.+,p, hp|  h9 CThe POWERMUTT Project: Standard Scores and the Normal Distribution Title 8@ _PID_HLINKSApThttp://voteview.com/!http://php.indiana.edu/~wright1/-T9http://davidmlane.com/hyperstat/normal_distribution.htmlv#http://www.jalt.org/test/bro_1.htmc*http://faculty.vassar.edu/lowry/tabs.htmlzkx;http://psych.colorado.edu/~mcclella/java/normal/normz.html4hhttp://www.vote-smart.org/f*../Codebooks/house.htmlV../Codebooks/states.htmlf*../Codebooks/house.html?W../Tools/select_cases.htmlG../Tools/recode.html+r../Tools/boxplots.htmlF../Codebooks/senate.htmlparty\../Tools/explore.htmlX../Codebooks/senate.htmlunityN../Codebooks/senate.htmldwnom%l../Codebooks/senate.htmlada%k../Codebooks/senate.htmlacuD../Codebooks/senate.html6i../Tools/descriptives.html]../Tools/compute.htmll*../Codebooks/countries.html]../Tools/compute.html6i~../Tools/descriptives.html\{../Tools/explore.html6ix../Tools/descriptives.htmlAu../Tools/frequencies.htmll(r../Tools/histogram.html\o../Tools/explore.htmlAl../Tools/frequencies.htmlziz f skewnessc kurtosis}n`normal] histogramIZlevels_of_measurement.html interval-]Wdescriptive_statistics.html boxplotsAJN figure23 Knote2syBfigure1 ?note1&T9descriptive_statistics.htmlmodeM$6measurement_basics.htmlmedian#U3descriptive_statistics.htmlmean6i0../Tools/descriptives.html]-../Tools/compute.htmll*'../Codebooks/countries.htmlA$../Tools/boxplot.html\!../Tools/explore.htmlA../Tools/frequencies.html6i../Tools/descriptives.html]../Tools/compute.html4\../Tools/getting_started.htmll(../Tools/histogram.html|dfurther  exercises5  key_concepts}nnormalzz introductionGRe WC:\User Files\CP\HTML\jlkorey\POWERMUTT\Topics\normal_distributions_files\image001.gifmX bC:\User Files\CP\HTML\jlkorey\POWERMUTT\Topics\normal_distributions_files\normal_distribution.gift- 0C:\User Files\CP\HTML\jlkorey\POWERMUTT\pup.jpgxfc VC:\User Files\CP\HTML\jlkorey\POWERMUTT\Topics\normal_distributions_files\figure1.gift- 0C:\User Files\CP\HTML\jlkorey\POWERMUTT\pup.jpg{f_VC:\User Files\CP\HTML\jlkorey\POWERMUTT\Topics\normal_distributions_files\figure2.gif  !"#$%&'()*+,-./012456789:;=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]_`abcdefghijklmnopqrstuvwxyz{|}Root Entry F`WData 31Table<CWordDocumentEdSummaryInformation(^?DocumentSummaryInformation8~(CompObjr  F Microsoft Word 97-2003 Document MSWordDocWord.Document.89q