1 Lesson 6: Measure of Variation

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1 1 Lesso 6: Measure of Variatio 1.1 The rage As we have see, there are several viable coteders for the best measure of the cetral tedecy of data. The mea, the mode ad the media each have certai advatages ad certai disadvatages. I ay speci c situatio, ayoe of these could provide the best ituitive value for the ceter. Oce a ceter has bee established, the ext questio is, how much does the data vary from this ceter? As it turs out, there are very few alteratives i mathematics for this measure. The rst measure of the variatio i a data set is the rage. The rage of umerical data set is simply the di erece betwee the highest value ad the lowest. Let us cosider our familiar example of studet grades: Name Test 1 Test 2 Test 3 April Barry Cidy David Eilee Frak Gea Harry Ivy Jacob Keri Larry Mary Norm The rage of the rst test comes from subtractig April s score of 55 from David s 97. The rage is 42. O test 2 the rage is 45, ad o test 3, it is 37: The rage is a rather crude measure of variability of data, but it is evertheless rather a importat oe whe lookig for a graphical represetatio of the data. We have see how the iteractio betwee the scale used i a chart ad the actual rage of the data i the chart ca chage the visual implicatios of a chart. Chart scales that are close to the rage ted to emphasize di ereces i the data, while larger scales have the opposite e ect. 1.2 The Variace The ext possible measure of variability i data begis with a failure of sorts. A reasoable rst guess might be to d the average distace betwee data poits 1

2 ad the ceter, say as measured by the mea. For the rst test i our class, the,mea was Usig this, we would compute the various distaces from that mea. Name Test 1 Distace April 55 23:29 Barry 63 15:29 Cidy 88 9:71 David 97 18:71 Eilee 58 20:29 Frak 90 11:71 Gea 88 9:71 Harry 71 7:29 Ivy 65 13:29 Jacob 77 1:29 Keri 75 3:29 Larry 88 9:71 Mary 95 16:71 Norm 86 7:71 Average 79:29 0:00 However, that is exactly what we were expectig. We have already see that the best property that the mea has goig for it is that the average distace from the average will always be 0. Oe idea for xig this is to exaggerate the distace from the ceter. We could try doublig it, but that will ot work because it exaggerates all the distaces uiformly. We eed to pealize data for beig further away from the ceter. We do this by squarig the distace. That way a distace of 1 is left aloe, but a distace of 2 gets boosted to 4. Ad a distace of 5 gets couted as a whoppig 25: The average square distace is called the variace. I our example, 2

3 Name Test 1 Distace Distace 2 April 55 23:29 542:22 Barry 63 15:29 233:65 Cidy 88 9:71 94:37 David 97 18:71 350:22 Eilee 58 20:29 411:51 Frak 90 11:71 137:22 Gea 88 9:71 94:37 Harry 71 7:29 53:08 Ivy 65 13:29 176:51 Jacob 77 1:29 1:65 Keri 75 3:29 10:8 Larry 88 9:71 94:37 Mary 95 16:71 279:37 Norm 86 7:71 59:51 Average 79:29 0:00 181:35 The variace is a bit strage, but it is a good measure of variatio. It has woderful mathematical properties that allow mathematicias ad statisticias to study it i great detail. Still it does seem a bit odd. Oe reaso for this is the uits. The distaces from the mea i the example above are measured i poits. Whe we square these digits, the uits are squared as well. That meas that the variace is 181:35 poits 2 : Squared poits is ot the most atural uit of aythig except variace. For ow, we will try to live with it; later it will become far less of a problem. So what exactly is the variace of a set of data? Numerically we have said it is the average squared distace from the mea. Algebraically this is just as easy to see, although perhaps a little frighteig. Suppose our data is The average of this is d 1 ; d 2 ; d 3 ; ::::d 1 ; d : The distaces from the mea are a d 1 + d 2 + d 3 + ::::d 1 + d : The square distaces are (d 1 a) ; (d 2 a) ; (d 3 a) ; :::: (d 1 a) ; (d : a) : (d 1 a) 2 ; (d 2 a) 2 ; (d 3 a) 2 ; :::: (d 1 a) 2 ; (d : a) 2 : So the variace must be v (d 1 a) 2 + (d 2 a) 2 + (d 3 a) 2 + : : : + (d : a) 2 : 3

4 However, we ca take this a bit further. v (d 1 a) 2 + (d 2 a) 2 + (d 3 a) 2 + : : : + (d : a) 2 d 2 1 2d 1 a + a 2 + d 2 2 2d 2 a + a 2 + : : : + d 2 2d a + a 2 d d 2 2; + : : : d 2 (2d 1 a + 2d 2 a + : : : + 2d a) + a 2 + a 2 + : : : a 2 d d 2 2; + : : : d 2 2a (d 1 + d 2 + : : : + d ) + a 2 d d 2 2; + : : : d 2 2a (d 1 + d 2 + : : : + d ) + a2 d d 2 2; + : : : d 2 2a (d 1 + d 2 + : : : + d ) + a 2 : But otice that d 1 + d 2 + ::: + d appears i this last statemet, ad it is just the average. v d d 2 2; + : : : d 2 d d 2 2; + : : : d 2 d d 2 2; + : : : d 2 d d 2 2; + : : : d 2 So we have 2a (d 1 + d 2 + : : : + d ) 2a a + a 2 a 2 2 d1 + d 2 + : : : + d : The oly reaso we did this algebra is that, very ofte, this is the de itio of variace oe ds i math books or computer programs. It looks a lot di eret tha "the average square distace from the mea," but that is just what it is. Notice the two parts of this formula: d d 2 2; + : : : d 2 is the average of the squares of the data. Now d1 + d 2 + : : : + d 2 is the square of the average of the data. equivalet ways of describig the variace: Thus we have two algebraically The variace is the average squared distace from the mea. 4

5 The variace is the mea of the squares mius the square of the mea. The rst descriptio illustrates the reaso it measures variatio from the ceter i squared poits. The secod descriptio gives a formula that makes the variace easier to compute. 1.3 The Stadard Deviatio The biggest problem with the variace, util you get used to it, is that it is measured i square uits. I our test data, the variace i o rst test is 181:35 poits 2 : If we wat to brig these uits back to ormal, we ca take the square root. I this case q 181:35 poits 2 ' 13:47 poits. The square root of the variace is the stadard deviatio. Thus o test 1 of our example, the stadard deviatio is 13:47 poits. O test 2 the variace is 160:27 poits 2 ; so that makes its stadard deviatio 12:66 poits. O test 3 the variace is 135:37 poits 2 ; so that makes its stadard deviatio 11:63 poits. It looks like the class grades are becomig less varied through the three tests. The stadard deviatio is the most commo measure of variatio i data. The variace has better mathematical properties tha the stadard deviatio, but they are so closely related that it hardly matters. What makes the stadard deviatio preferred is that it is measured i the atural uits of the data. As the ame suggests, the stadard deviatio is also used as a measure i its ow right. The stadard deviatio works as a good uit of measure whe comparig the relative positio of a datum withi a set. Cosider the grades o test 1 above, ad distaces of those grades from the mea: Name Test 1 Distace April 55 23:29 Barry 63 15:29 Cidy 88 9:71 David 97 18:71 Eilee 58 20:29 Frak 90 11:71 Gea 88 9:71 Harry 71 7:29 Ivy 65 13:29 Jacob 77 1:29 Keri 75 3:29 Larry 88 9:71 Mary 95 16:71 Norm 86 7:71 5

6 Frak had a score of 90%. If the purpose of the test was to measure Frak s kowledge of the material covered out of a theoretical 100%, the Frak s grade was quite good. Learig 90% of the material is quite a accomplishmet. Frak s performace should be judged solely o the fact that he got 90% out of 100%. If the oly poit is to lear the material, Frak has a good claim to have doe that. But Frak had aother accomplishmet of which he ca be proud. Frak s 90% was the third highest grade i the class. I a competitio betwee studets, this is the importat thig. If the poit is to lear the material, all that matters is the grade. If the poit is to outscore as may people i the class as possible, the rakig of your score is importat: Name Test 1 Rakig Distace April :29 Barry :29 Cidy 88 4 tie 9:71 David :71 Eilee :29 Frak :71 Gea 88 4 tie 9:71 Harry :29 Ivy :29 Jacob :29 Keri :29 Larry 88 4 tie 9:71 Mary :71 Norm :71 Aother way to compare Frak to the rest of the class is to otice that he scored almost 12 poits above the class average. That meas that, i a race to the highest total score at the ed of the course, he has a 12 poit lead over a lot of studets i the class. If the poit is to establish a lead over as may people i the class as possible, the distace from the mea is the importat measure. But has Frak s achievemet really distiguished him as better tha the rest of the class; is a 90% a extraordiary score o this test relative to the results i the class. Here is where usig a measure of stadard deviatios ca be very useful. Frak scored 11:71 poits above the mea o a test with a stadard deviatio of 13:47. Measured i a di eret uit, this is 11:71 13:47 0:86934 stadard deviatios above the mea. Notice that we are usig "stadard deviatios" as a uit of stadard measure. We are comparig Frak to the rest of the class usig a more objective measure tha the umber of poits. I geeral, a distace of 1 stadard deviatio or less is ot cosider particularly special. So Frak still did quite well, but so far, othig of extra ote compared to others i the class. If the poit is to see how remarkable a test score is objectively, the distace from the mea i stadard deviatios is the importat measure. Look at April. Clearly April did poorly. If the purpose is to lear the 6

7 material, the April has a way to go. She had the lowest grade i the class, ad so is far from the top i that competitio. If she hopes to catch up, her distace from the mea of 23:29 is quite tellig. However, how bad was her performace o this test? After 55% is more tha half. I stadard deviatios, April s score was 23:29 13:47 1:729 below the mea. This is almost 2 stadard deviatios below the average. Two stadard deviatios is de itely quite a bit o, ad a teacher who uderstads this way of measurig a studet s place relative to the rest of the class will de itely be alarmed. April is de itely ot learig the material as well as the other studets. Certaily the fact that she is 23 poits below the average shows this. The importace of the value 23, however, depeds o the test, the way it was graded, the scale used, ad eve the umber of studets i the class. However i a more objective measure, she is 1:7 stadard deviatios below the mea. I ay class of ay size ad uder ay gradig scheme, this is very low. We ca measure the stadigs of all the studets i the class i stadard deviatios: Name Test 1 Rakig Pts Distace S.D. Distace April :29 1:73 Barry :29 1:14 Cidy 88 4 tie 9:71 0:72 David :71 1:39 Eilee :29 1:51 Frak :71 0:87 Gea 88 4 tie 9:71 0:72 Harry :29 0:54 Ivy :29 0:99 Jacob :29 0:1 Keri :29 0:24 Larry 88 4 tie 9:71 0:72 Mary :71 1:24 Norm :71 0:57 We always have a choice betwee measurig distace from the mea i origial uits or i stadard deviatios. I geeral, keepig the origial uits is best whe makig comparisos withi the data set; while usig stadard deviatios works best whe comparig di eret data sets. We will say more about this later. So while stadard deviatio is, o the oe had, a sigle measure of the variatio of a collectio of data, it ca also be used as a uit to measure the positio of a idividual datum withi the data set. 1.4 Quartiles Now the variace ad the stadard deviatio are measures of variatio that treat the mea as the ceter of the data. This meas that they are good 7

8 measures of variatio whe the mea is a good measure of the ceter. We have see, however, that this is ot always the case. There are data sets where the media is a better measure of the ceter. I these cases there are alterate measures of the variatio as well. The media is the half way poit i the data of the set. To compute the media of a data set, we eed to rak the data i order. Usig our familiar test 1: Name Test 1 Rakig April Barry Cidy 88 4 tie David 97 1 Eilee Frak 90 3 Gea 88 4 tie Harry 71 9 Ivy 65 8 Jacob 77 6 Keri 75 7 Larry 88 4 tie Mary 95 2 Norm 86 5 It would be best to rearrage this data i the order of rak: Name Test 1 Rakig David 97 1 Mary 95 2 Frak 90 3 Cidy 88 4 tie Gea 88 4 tie Larry 88 4 tie Norm 86 7 Jacob 77 8 Keri 75 9 Ivy Harry Barry Eilee April There are scores, so the media is the average of the 6-th ad 7-th scores: 2 87: The media is the score where half the class is above that score ad half the class is below it. The media divides the class i equal halves. If we divide each of those halves ito their ow equal halves, we get quartiles. There are 8

9 6 2 3 scores i each half, so the quartile breaks half way betwee the 3-rd ad 4-th grade i each half. That puts the top quartile at , ad the bottom quartile is put at : The results of test oe ca be summarized by the followig statistics: Miimum: 55 Bottom Quartile: 67 Media: 87 Top Quartile: 89 Maximum: 97. There is a ice diagram that ca be used to display this summary called a "Box ad Whiskers" graph. The diagram is draw o a umber lie. The box i the diagram is a rectagle where the left side is labeled with the bottom quartile ad the right side labeled with top quartile. The media is located i the box properly placed i the iterval. Some people just ote its locatio with a *, but a lie parallel to the sides is more commo. Protrudig from the two sides of the box are lies extedig to the miimum ad the maximum of the data; So our test would be summarized as Notice how the box ad whisker diagram clearly shows that most of the class did quite well. Eve though the low grades did ot really stadout as outliers, the media grade of 87 is a better idicator of the class performace tha the mea 79: The box ad whisker diagram gives a pretty good idea of how the grades came out. 1.5 Percetile The al way to measure variatio we will cosider is percetiles. This is ot a sigle measure of the variatio like variace or stadard deviatio. Rather it is a measure of the variatio idividual data. Computig the percetiles of each data poit begis by placig the data i raked order, with the "best" results 9

10 placed highest. I our test Name Test 1 Rakig David 97 1 Mary 95 2 Frak 90 3 Cidy 88 4 tie Gea 88 4 tie Larry 88 4 tie Norm 86 7 Jacob 77 8 Keri 75 9 Ivy Harry Barry Eilee April The percetile of each score is the percetage of the rak i the total umber of studets. For David, this is %: We would say that David scored i the 7-th percetile of the class. Cidy, Gea ad Larry tied for 4-th; so they would all have the same percetile: %: That puts them all i the 29-th percetile. Actually this is ot a good example of how percetile ratigs are used. Typically percetiles are used i very large raked data sets. Stadardized test ofte report grades as both raw scores ad percetiles. If you kow the scale of a test, the a raw score of 129 gives you iformatio. However, kowig that that score puts a studet i the 85-th percetile, gives you a better idea of the meaig of the score without ay additioal iformatio about the test. The studet with that score is i the top quartile ad Thus percetile measuremet is the approach used to measure the distace from the ceter as measured be the media, which is the 50-th percetile. This correspods to measurig distace of a datum from the mea i stadard deviatios. Prepared by: Daiel Madde ad Alyssa Keri: May

Median and IQR The median is the value which divides the ordered data values in half.

Median and IQR The median is the value which divides the ordered data values in half. STA 666 Fall 2007 Web-based Course Notes 4: Describig Distributios Numerically Numerical summaries for quatitative variables media ad iterquartile rage (IQR) 5-umber summary mea ad stadard deviatio Media