Chapter 5: Why Is Variability Important? The Importance Of .

Chapter 5:The Importance of MeasuringVariability Measures of Central Tendency -Numbers that describe what is typical or“central” in a variable’s distribution (e.g.,mean, mode, median). Measures of Variability - Numbers thatdescribe diversity or variability in a variable’sdistribution (e.g., range, interquartile range,variance, standard deviation).Why is Variability important?Example: Suppose you wanted to know howsatisfied students are with their livingarrangements and you found that the meananswer was “3” on a five point scale where:1 very unsatisfied, 2 satisfied, 3 neutral,4 satisfied, 5 very satisfiedWhat would you conclude?Would knowing the variability of the answershelp you to understand how satisfied studentsare with their living arrangements?Answer: It would help you to see whetherthe average score of “3” means that themajority of students are neutral abouttheir jobsorthat there is a split with students eitherfeeling very satisfied (score of 5) orunsatisfied (score of 1) with their livingarrangements (average of 1’s and 5’s 3).Another example.The RangeTable 1: Level of Ethnic Diversity (IQV) in the StateWhat is the range for these diversity scores?Steps to determine: subtract the lowest score from thehighest to obtain the range of IQV scores . Range – A measure of variation in intervalratio variables. It is the difference between the highest(maximum) and the lowest (minimum)scores in the distribution.Range highest score - lowest score

What is the range for these diversity scores?What is the range for these diversity scores?Steps to determine: subtract the lowest score .06 fromthe highest to obtain the range of IQV scores .Steps to determine: subtract the lowest score .06 fromthe highest .80 to obtain the range of IQV scores .What is the range for these diversity scores?Steps to determine: subtract the lowest score .06 from thehighest .80 to obtain the range of IQV scores .74 .Another example.Inter-quartile RangeInter-quartile Range Inter-quartile range (IQR) – The width of themiddle 50 percent of the distribution. Inter-quartile range (IQR) – The width ofthe middle 50 percent of the distribution. The IQR helps us to get a better picture of thevariation in the data than the range because itfocuses on the width of the middle 50% ratherthan extreme scores in the distribution. It is defined as the difference betweenthe lower and upper quartiles (Q1 and Q3.) The shortcoming of the range is that an “outlying”case at the top or bottom can increase the rangesubstantially. IQR q3 – q1(e.g., 75th percentile – 25th percentile)

What is the IQR for theseDiversity Scores?What is the IQR for theDiversity Scores?Steps to determine the IQR (Q3 – Q1):1. Order the categories from highest to lowest (or vice versa)2. To obtain Q1, begin by dividing N (total number of categories orstates) by 4 (or alternatively multiply N by .25). Thisequals ?3. We now know that Q1 falls between the 12th and 13th categoryor, in this case, states.4. To find the exact number for Q1, determine the midpointbetween the 12th and 13th states or between .59 and .57)5. Q1 (Steps are provided on the next slides)What is the IQR for theDiversity Scores?What is the IQR for theDiversity Scores?Steps to determine the IQR (Q3 – Q1):Steps to determine the IQR (Q3 – Q1):1. Order the categories from highest to lowest (or viceversa)2. To obtain Q1, begin by dividing N (total number ofcategories or states) by 4 (or alternatively multiply Nby .25). This equals 12.5 ?3. We now know that Q1 falls between the 12th and 13thcategory or, in this case, states.4. The diversity score between these two states is:between .59 and .57 or .585. To obtain Q3, multiply the quarter figure (12.5) by 3 and then locate this category (the 37thand 38th states).1. Order the categories from highest to lowest (or viceversa)2. To obtain Q1, begin by dividing N (total number ofcategories or states) by 4 (or alternatively multiply Nby .25). This equals 12.5 ?3. We now know that Q1 falls between the 12th and 13thcategory or, in this case, states.4. The diversity score between these two states is:between .59 and .57 or Q1 .585. To obtain Q3, multiply the quarter figure (12.5) by 3 37.5 and then locate this category (the 37th and38th states).The difference between theRange and IQRWhat is the IQR for theDiversity Scores?Steps to determine the IQR (Q3 – Q1):6. Based on this number (37.5), Q3 falls between the37th and 38th states.7. To find the exact number for Q3 determine themidpoint between the 37th and 38th states orQ3 .248. This tells us that 50% of the cases fall between theIQR scores of .58 and .24.9. The IQR .58 – .24 .34Thesevalues falltogethercloselyYet theranges areequal!ShowsgreatervariabilityImportanceof the IQR

The Box Plot The Box Plot is a graphic device that visuallypresents the following elements: the range, theIQR, the median, the quartiles, amount anddirection of skewness, the minimum (lowestvalue,) and the maximum (highest value.)Procedures forCreating Box Plots for Variables Open SPSS Click “graphs” Click “legacy dialogs” Click “box plot” Click “simple” and “summaries for separatevariables” Click “define” Select desired variable and put in “BoxesRepresent” Procedures forCreating Box Plots for Groups (for example,Males and Females by Income) Open SPSSClick “graphs”Click “legacy dialogs”Click “box plot”Click “simple” and “summaries for groups ofcases”Click “define”Select desired dependent variable (such asincome) and put in “Variable Box”Move desired grouping variable (such assex) into “Category Axis”Click “okay”Measures of Variability:the Variance The variance allows us to account for thetotal amount of variation. The variance is an important statistic that isused in most other sophisticated statistics.Therefore, it is important for you to give itparticular attention.Be sure to read the sections of the chapter onvariability and standard deviation very carefully.Click “okay”Measures of Variability:Shortcomings of the Range and IQR The range is based on only two categories(the highest and lowest) Likewise, only two categories are used tocalculate the inter-quartile range. Neither allows us to know how muchvariation there is among all the categories.Determining the Variance in the “Percentage Increase” in theNursing Home Population, 1980-1990Nine Regions of U.S.PacificWest North CentralNew EnglandEast North CentralWest South CentralMiddle AtlanticEast South CentralMountainSouth 71.7What statistics have we learned so far to describe the variation above?Is there a lot of variation between the categories (regions of U.S.)?Range, Inter-Quartile Range (IQR)There appears to be a lot of variation between regions.

First Step in Calculating the Variation:Determine the “Average” Between Regions for the percentchange in the Nursing Home Population, 1980-1990Nine Regions of U.S.PercentagePacificWest North CentralNew EnglandEast North CentralWest South CentralMiddle AtlanticEast South CentralMountainSouth AtlanticNine Regions of U.S.(mean 31.5)15.716.217.623.224.328.538.047.971.7Y 15.716.217.623.224.328.538.047.971.7283.1Y PercentagePacificWest North CentralNew EnglandEast North CentralWest South CentralMiddle AtlanticEast South CentralMountainSouth AtlanticPercentagePacificWest North CentralNew EnglandEast North CentralWest South CentralMiddle AtlanticEast South CentralMountainSouth ining the Variation in the Percentage Changein the Nursing Home Population, 1980-1990Nine Regions of U.S.The “average” percentage increase in theNursing Home Population, 3.1-31.531.531.531.531.531.531.531.531.5 (Y – Y) 0mean 31.45Percentage Change in the Nursing Home Population,1980-1990Nine Regions of U.S.-15.8-15.3-13.9- 8.3- 7.2- 3.06.516.440.2Average “% increase”PercentagePacificWest North CentralNew EnglandEast North CentralWest South CentralMiddle AtlanticEast South CentralMountainSouth an 31.5)Y 31.531.531.531.531.531.531.5 -15.8-15.3-13.9- 8.3- 7.2- 3.06.516.440.2(Y – Y) 0Next, we can determine the distance between (1) each region and (2) theaverage (31.5), in order to get the amount of variation from the mean for eachregion. Then, we can add up the variation scores for each region to get the“total” variation of the scores (but this is not the actual “VARIANCE”).Problem: when you add up the distances you end up with zero rather than thetotal variation from all the categories. Why is this?Percentage Change in the Nursing Home Population,1980-1990Percentage Change in the Nursing HomePopulation, 1980-1990Nine Regions of U.S.PercentagePacificWest North CentralNew EnglandEast North CentralWest South CentralMiddle AtlanticEast South CentralMountainSouth Atlantic(mean 31.5)15.716.217.623.224.328.538.047.971.7Y 1.531.531.531.531.531.531.531.5 Nine Regions of U.S. Percentage-15.8-15.3-13.9- 8.3- 7.2- 3.06.516.440.2(Y – Y) 0 One solution would be to add up the absolute values for each number (ignorethe minus signs), or 126.6 and then divide by the number of regions (9) 14.1).Unfortunately, absolute values are very difficult to work with mathematically. Fortunately, there is another alternative.PacificWest North CentralNew EnglandEast North CentralWest South CentralMiddle AtlanticEast South CentralMountainSouth Atlantic(mean 31.5)15.716.217.623.224.328.538.047.971.7Y 531.531.531.531.531.531.531.531.5 ( Y – Y)2(squared deviations)-15.8249.64-15.3234.09-13.9193.21- 8.368.89- 7.251.84- 3.09.006.542.2516.4268.9640.21616.04(Y – Y)2 2733.92 The best solution is to square the differences before adding them up (whentwo negative numbers are multiplied the resulting product is a positive number).This eliminates the problem of adding negative and positive numbers.

Measures of Variability:the VarianceThe Variance is the average of the squared deviations fromthe mean.In our example we would take the sum of the squareddeviations (2733.92) and divide this number by the totalnumber of cases minus one (9 – 1 8). This would give us341.74 or the variance for the Percent Increase inthe Nursing Home population by region.Measures of Variability:Standard Deviation One problem with the variance is that the final numberobtained is in a squared form(that is, we squared all the deviations from the meanand so the final number is still “inflated” in this waymaking it difficult to interpret) One solution is to take the square root of the varianceso that the number is no longer in a squared form (or“inflated”) and it is back to its original form. The squareroot of the variance is called the Standard Deviation.In SumThe Standard Deviation is a measure ofvariation for interval-ratio variables; it isequal to the square root of the variance.Measures of Variability:The VarianceTo Sum Up:The Variance is the average of thesquared deviations from the mean.The Variance is a measure of variabilityfor interval-ratio variables.Measures of Variability:Standard Deviation To obtain the square root of the variance simply enterthe number (variance) into your calculator and then pushthe square root button. If the variance is 341.74 the standard deviation would be18.49 . This tells us that the percent of changein the nursing home population for the nine regions iswidely dispersed around the mean (mean 31.45). Thus, the standard deviation is a measure of the averageamount of variation (or deviation) around the mean.Considerations for Choosing aMeasure of Variability For ordinal variables, you can calculate theIQR (range and inter-quartile range.) For interval-ratio variables, you can usethe range, the IQR, the variance or thestandard deviation. The variance andstandard deviation provide the mostinformation, since they use all of the valuesin the distribution in their calculations.

Fini

Numbers that describe what is typical or “central” in a variable’s distribution (e.g., mean, mode, median). Measures of Variability - Numbers that describe diversity or variability in a variable’s distribution (e.g., range, interquartile range, variance, standard deviation). Why is Variability important?