1 Lesson 5 Measures of Central Tendencies

Transcription

1 1 Lesso 5 Measures of Cetral Tedecies 1.1 Itroductio We have outlied the correct process for dealig with data. The steps are 1. Orgaize the data. 2. Arrage the data. 3. Preset the data. We have spet some time o step 3, but have cocetrated o presetig the data graphically. I doig so, we have used foud that it is ofte a good idea to rearrage the data by performig a bit of arithmetic. To illustrate various data sets, we created totals, computed a few percetages, ad eve created a ew measure (deviatio from par.) Now we are goig to look at other arithmetic processes that ca help summarize ad preset large data sets. This is our rst real look at the ideas of Statistics. Oe of the rst ad most commo uses of descriptive statistics is the use of averages to describe the cetral tedecies of a data set. For example, whe lookig at a data set made up of umbers, it is a atural to ask thigs like: What is the ceter? Where is the data balaced? What is a typical umber of the data set? Ayoe of these questios is libel to get us thikig about takig the average of the umbers. However, o closer ispectio thigs we realize that are a bit more complicated tha that. Cosider our class of studets; suppose we give them oe more surprise test ad record the grades: Name Test April 80 Barry Cidy 78 David 80 Eilee 30 Frak 80 Gea 75 Harry 55 Ivy 80 Jacob 27 Keri Larry 80 Mary 100 Norm 70 1

2 The results are rather disappoitig. There are some pretty low grades; oe high oe, ad two studets did ot take the test. Well what was the test average? It is 69:6 if you do ot cout the two abseces, but oly 59:6 if you do. Either oe of these might be someoe s idea of the ceter of the data. Also o oe would miss all the scores of 80 i the list, so that might make it the typical score. Also if we say that of the 12 results i this test, half the studets (6) have 80 or above, ad half (6) have 80 or below. That make 80 aother kid of ceter of these scores. If we say all 14 cout i the results, half the studets (7) have 80 or above, ad half (6) have 78 or below. That makes somethig betwee 78 ad 80 a way to measure the ceter of these scores. So eve i a simple example, there are several ways to measure cetral tedecies of data. Most commo such measures have mathematical ames like: the arithmetic mea, the geometric mea, the harmoic mea, a trucated mea, weighted meas, the media, the mode. All of these ca be reasoably used to deote the cetral value of a data set made up of umbers measurig speci c thigs. Depedig o the situatio, oe of these might be superior to the others, ad oe or two might be at out wrog as a measure of ceter. 1.2 Mea, Mode ad Media. I this sectio we will study the three most commo measures of cetral tedecy: the mea, the media, ad the mode. Actually we have just see all three. I our ew test results, we saw two examples of the mea: 69:6 ad 59:6. The mea is the arithmetic average of the grade scores: the sum of the scores divided by the umber of scores. The oly reaso there are two meas because we have a (o-mathematical) choice betwee coutig the missig scores as 0 or igorig them. We oticed that may studets got the same score of 80. This is called the mode of the data. It is the umber that occurs most ofte i the data. We also oted that 80 is the media whe we said that half the studets have 80 or above, ad half have 80 or below. To compute the media of a list of umbers we place the umbers i order, ascedig or descedig, ad idetify the middle of the set. If there are a odd umber of data poits, say = 2k+1, 2

3 the the media is the umber k + 1 from the top. That meas that at least k of the umbers are larger tha or equal to the media. But the we also kow that at least k of the umbers are smaller tha or equal to the media. If there are a eve umber of data poits, say = 2k, the the media is the average of the umber k from the top, ad the umber k from the bottom. This also guaratees that at least k of the umbers are larger tha or equal to the media ad at least k of the umbers are smaller tha or equal to the media. We otice that, like the media, we eed to choose how may studets to cout to compute the media. If we use the umber of studets who took the test, 12 the media is 80. If we use the umber of studets i the class, 14, the media is = 79: Why might there be three di eret measures of cetral tedecies like these? Each has it s advatages; each has its disadvatages. However, we will see later that, i large data sets of the right type, these three measures ca be expected to be close to each other. So i cases of large data sets the three di eret computatios yield umbers that are ot that di eret. For ow, we will cocetrate o the di ereces betwee these di eret computatios. Cosider the teacher whose class produced these grades. She wats to represet the results of the test usig oe umber that measures the ceter of the scores. First, we would expect her to decide ot to cout those studets who did ot take the test. That is oe problem solved. Next it is sort of clear how to aswer, What was the typical grade? This has to be 80. After all 5 out of 12 studets received this score. Suppose she is asked What was average? Most people are familiar with the calculatio that gives the average, ad that it exactly how we calculate the mea of a data set. The aswer is 69:6: Notice, however, that this is ot a very accurate re ectio of the class performace. Nie of the twelve studet did better tha this average. The mea has exaggerated the importace of the low scores, particularly the 30 ad the 27: This would be far worse if the etire class were couted, the additio of two zeros pulls the mea dow 10 full poits. It is i this respect that the media is better tha the mea. The media really does split the class i half. If oe or two umbers are much larger or smaller tha the others, their actual values have less impact o the media tha they do o the mea. I this example addig two scores to the mix moved the media less tha oe poit. Let us cosider a test from a larger class. The test is made up of 20 questios each give a value of 5 poits. Suppose that the teacher is asked how the class did o the test. After the test is graded, the teacher could quickly tabulate the scores usig a stem ad leaf plot usig the two digits of the umbers. She could the quickly chage this ito a class ad frequecy table: Score Frequecy If she does ot have a calculator or a computer close by, how ca she determie the a cetral grade o the test? With the frequecy table above it 3

4 should be easy to d the mode. Historically before powerful calculators ad computers, data sets very ofte oly appeared as frequecy tables like this. Cosequetly the mode was a more commo measure of the ceter tha it is today. It was easy to compute by had, ad that was eough for may applicatios. The most logical ad practical place to start aalyzig a frequecy table was to look for the datum or data i the set that occurred most frequetly. You ca easily look at our frequecy table ad see that oe mode occurs at 55% (F) ad aother occurs at 90% (A). I this data set, there is more tha oe mode. If you tell the studets that the ceter grade o the test was both a A ad a F most studets would wat more iformatio; although this mode iformatio is tellig i itself. The mode is easy to compute, but it may ot yield oe uique umber. Typically, the mode or modes do covey some iformatio about the data, but sometimes they do ot. The ext most logical ad practical idea is to write the test scores i icreasig order ad d the middle score to produce the media. Although sortig large sets of data is time cosumig, that time was already spet i costructig the stem ad leaf plot. With a frequecy table, the media is easy to d. There are ie di eret grades, but of course, this is ot the importat umber for dig the media. There are 40 di eret scores recorded i this table. This is the umber we care about, ad 40 = We eed to d the 20-th score from the bottom. To illustrate exactly what we are doig, we will add a row to the table accumulatig the umber of tests up to that poit Score Frequecy Scores The score 20 from the bottom is 80. The 20-th from the top will also be 80. So the media grade o this test is 80. However, the teacher may still be usure whether the modes or the media provide the best accout of the studets scores. There do ot seem to be ay high or low scores to pull the average o -ceter. I this case, the arithmetic mea should provide good iformatio, accoutig for every datum i the data set. The mea (or the arithmetic mea) is the quotiet of the sum of all of the data ad the umber of the data i the set. Agai it is ot importat that there are oly ie di eret scores i the table; the poit is that the table gives us a total of 40 studet scores. Agai we make the calculatio explicit by addig some arithmetic to the table; this time a row of total poits ad a colum of totals. Score Frequecy Poits Eared Thus 40 studets eared a total of 3025 poits for a average of 76 (C). The (arithmetic) mea of the data is = 76: The mea is the most di cult to compute of the three possibility we have cosidered. However, that is because our data is i a prited frequecy table, 4

5 ad our calculatios are doe by had. But ow a days, data is more likely to be stored i a calculator or computer. I this situatio, it ca actually take the machie loger to d a media or a mode tha it does to compute a mea. Although i all but the very largest data sets, a computer ca appear to do ayoe of them istataeously. As we will see i large data sets of the proper type, the three measures of cetral tedecies ca be expected to be fairly close to each other. Whe this is true, dig the mode would give approximately the same iformatio as dig the mea or the media. Each measure of cetral tedecy has its advatages ad disadvatages, but i geeral we will focus o the arithmetic mea because most of its advatages are mathematical rather tha simply practical. These properties ca be used to predict ukow values for a large populatio usig methods of samplig. 1.3 Comparig measures of cetral tedecy We have just see a example of data where there was more tha oe mode, ad further these modes were su cietly separated that either would be a good measure of the ceter of the data. So certaily oe disadvatage of usig the mode to determie the ceter of data is that it occasioally does ot work as such. So the questio arises, what are the advatages of the mode that keeps aroud? We have promised that as samples sizes become large, the mea, the media, ad the mode will all approach the same value, but that may ot be eough to bother computig modes. Still the mode is the fastest ad easiest measure of cetral tedecy to calculate by had for large data sets sice it ofte falls out of the rst attempt to orgaize data ito a maageable form, like a stem ad leaf plot. O some data sets, the media ad the mea are iappropriate measures of ceter simply some data sets are ot made up of umbers. For example, imagie a teachig evaluatio. I several areas such as pedagogy, egagemet, cotet kowledge, etc., teachers are evaluated as "eeds improvemet", "meets the stadards", or "exceeds the stadards" perhaps by several di eret evaluators. So how ca we represet a overall summary of such a evaluatio. There is o mea to be take. Oh, oe could assig umbers to the three categories; say 3; 2; ad 1 respectively. But oe could just as well accetuate the importace of a ratig of "eeds improvemet" by assigig the umbers 4; 2 ad 1. This would allow use to compute a mea, but the meaig of this mea would deped more o the assigmet of umbers tha the outcome of the evaluatio. To compute a mea, you eed umbers. The mode, however, ca be used eve whe there are o umbers. If a teacher were to receives 5 ratig of "eeds improvemet", 12 ratigs of "meets the stadards", ad 8 ratigs of "exceeds the stadards," a reasoable summary of the results is that the teacher "meets the stadards" because that is the mode of the data. The largest umber of ratigs were "meets the stadard." Here the mode is de itely a appropriate way to describe performace. Now these three categories do have a atural order. I thik that all would 5

6 agree that, i mathematical otatio: "eeds improvemet" < "meets the stadards" < "exceeds the stadards." Eve thought the data poits are still phrases, the implied order i those phrases meas that we ca probably d a media the set. The teacher who received 5 ratig of "eeds improvemet", 12 ratigs of "meets the stadards", ad 8 ratigs of "exceeds the stadards" has 25 ratigs. With 25 = data poits, we pick out the 12-th data poit from the bottom usig the implied orderig. This gives a media of "meets the stadards." This agrees with the mode. I this particular example, it would take a rather bizarre set of ratigs, for the mode ad media to tur out to be di eret. We could certaily costruct such a set of ratigs. However, i this particular example, the media ad the mode will ofte coicide ad make a strog case that their value is the ceter value of the ratigs. What about assigig umbers to the ratigs ad computig a mea? If we say "eeds improvemet" is a 3, "meets the stadards" is a 2, ad "exceeds the stadards" is a 1, the a teacher who receives 5 ratig of "eeds improvemet", 12 ratigs of "meets the stadards", ad 8 ratigs of "exceeds the stadards" ears = 45 poits ad has a average of 45=25 = 1:75: The lower the average the better the teacher. We might chage the order of these poits so that higher ratigs get the higher umbers. After all the umbers we assiged have o atural associatio to the ratigs, so why ot make them more ituitive? We may wat to reward ratigs of "exceeds the stadards" by givig them more value. Thus if we say "eeds improvemet" is a 1, "meets the stadards" is a 2, ad "exceeds the stadards" is a 4, each "exceeds" will cacel out the value of two "eeds." All these possibilities are examples of weighted meas. The assigmet of umber values to data classes is "weightig" ad the values are the "weights." I weighted data, the mea of the data is a measure of cetral tedecy, but its umerical value ad its meaig as such are directly related to the weightig used. There is aother o-mathematical poit to be made about the example above. The measures used o the teacher above, he did ot seem to get su ciet credit for havig more "exceeds stadards" tha "eeds improvemet," ad that could de itely be see as ufair. However, the data i the ratigs above must have come from the subjective opiios of the evaluators. It is possible to desig a ratig system that limits the impact of the atural subjectivity of evaluators. All of these are ecessarily part of process that produces the data: multiple evaluators; ratig check lists; pre-evaluatio coordiatio. After that, there will still be some subjectivity i the data, but oe ca hope that the precautios take do provide limits. Oce the data is collected, however, o amout of mathematics ca possibly remove the subjectivity from the data. Assigig umbers to the ratigs ad computig a mea to six decimal places is o more precise a measuremet tha the mode. Subjective i meas subjective out. But ow, back to cetral tedecies. The media is very useful for data sets with outliers, or extreme values i the data set. The media has the ice 6

7 property that it measures oly the relative size of the data, ad the actual size. For example, suppose a charity receives doatios of $2, $5, $7, $10, $15, $17, $25, $30, $35, $100, $1,000, ad $1,000. The charity would certaily be misleadig people if they were to claim that the mode of $1,000 had aythig at all to say about a typical doatio. These two are ot oly atypical, but they are outliers i size compared to the other doatios. These two extreme values ot oly form a orepresetative mode, but the pull the mea of ceter as well. The average doatio may arithmetically be $187.17, but that is o more a represetative of the ormal doatio tha the mode. But we have 12 = 2 6 doatios were made, the media will be the average of the 6th ad 7th doatio oce the doatios are writte i icreasig order. The media doatio of $21 is closer to a typical doatio to the charity. The media is a measure of cetral tedecy that is resistat to outliers. I small samples of data such as home prices or icomes, extreme outliers are a commo problem. Thus the media typically is used to miimize the e ect of multimillio dollar masios, or icomes of persos like Bill Gates. This property of the media, ad the fact that it is geerally easy to compute, make it almost just as useful as a measure of cetral tedecy as the mea. The mea has oe advatage over all others measures of cetral tedecy that make it the most commo measure ayoe uses. The arithmetic mea has mathematical properties that make it much easier to aalyze tha ay other measure. we will see most of these properties i use later, but we ca poit oe out right away. Cosider the teacher with the large set of test results: Score Frequecy We foud that the mea, or average, of these scores is 76, or more precisely 75:625: Now 10 studets scored 55 poits which was 20:625 poits below the mea. Two studets scored 95 poits, 20:625 poits above the mea. If we cosider all the scores this way, we costruct a table S c o r e A w ay from m ea Fr e q u e c y We expad the table to compute the total umber of poits away from the mea S c o r e T o t a l A w ay from m ea Fr e q u e c y Total aw ay Thus we see that the average distace from the mea is 0 40 = 0: This is just as easy to see algebraically as umerically, although perhaps a little frighteig. Suppose our data is d 1 ; d 2 ; d 3 ; ::::d 1 ; d : 7

8 The average of this is a = d 1 + d 2 + d 3 + ::::d 1 + d : The distace of each data poit from the average is The average of these is (d 1 a) ; (d 2 a) ; (d 3 a) ; :::; (d 1 a) ; (d a) : (d 1 a) + (d 2 a) + (d 3 a) + ::: (d 1 a) + (d a) = (d 1 + d 2 + d 3 + :::d 1 + d ) a = (d 1 + d 2 + d 3 + :::d 1 + d ) = (d 1 + d 2 + d 3 + :::d 1 + d ) = 0: a (d 1 + d 2 + d 3 + :::d 1 + d ) The mea accouts for every piece of data i the set ad ds a mathematical ceter that ca be very pleasig. For example, cosider the bar chart below for the umber of blue M&Ms that ve studets collected from a umber of bags: The mathematical mea of the umber of blue M&Ms the studets foud is 10. If we ask the studets to redistribute the cady so that each has the arithmetic mea. Now Steve ad Jill are both 3 below the mea; while Sarah is 2 above ad Lace is 4 above. As we saw algebraically, it is ot a accidet that the total below is the same as the total above. To redistribute the cady, Belem would stay the same; Sarah would give two, to say Steve; Lace would give oe more to Steve, ad three to Jill. Because the mea was a whole umber, it allowed a equal distributio of the M&Ms to a cetral value. Oe adaptatio of the mea that is commoly used is a weighted average. We have already see oe example of this i our teacher rakigs. We had 8

9 teachers evaluated as "eeds improvemet", "meets the stadards", or "exceeds the stadards." WE poited out that data collected this way would have o mea. To compute a mea, you eed umbers. Oe way aroud this is to assig umbers, or weights, to the three categories; say 3; 2; ad 1 respectively. The teacher who received 5 ratig of "eeds improvemet", 12 ratigs of "meets the stadards", ad 8 ratigs of "exceeds the stadards" has received a total of 47 weight-poits i 25 ratigs. This gives a weighted average of = 1:75: Remember, the lower a score, the better the ratig. This weighted average is 0:12 below 2, ad therefore slightly higher tha weightig of the ceter "meets stadards" ratig. Now weights like this ca be assiged ayway oe likes. We could reverse the weights above so that higher umbers are better. If the three categories are weighted as 1; 2,ad 3, the the weighted average is = 2:12: This is, ot coicidetly, exactly 0:12 above 2; ad so the weighted average is agai slightly higher tha weightig of "meets stadards." As we chage the weights, the weighted average will chage. But, as log as our three weights are equally spaced, the results will also give a ratig slightly above the ceter. Weightig ca also be used to add importace to oe particular ratig. A admiistrator might choose to give more weight to "eeds improvemet" by assigig the umbers 0; 2 ad 3. Notice this keeps the same spacig betwee "meets" ad "exceeds,", but stretches the gap betwee "eeds" ad "meets." This time the weighted mea is = 1:96, which drops the teacher below the 2 level. May teachers use weighted averages i a di eret way o umerical data the data i a set are ot iheretly worth the same importace. For example, studets receive grades of three types: tests, quizzes, ad al exam. The grade i each category is computed as a percetage of the category, givig three pieces of data like: Tests Quizzes Fial 84% 91% 77% Rather tha simply average these three compoets to determie a summary grade, the teacher may choose to weight the di eret compoets. She might wat the quizzes ad the al to cout equally, but the test average should cout 3 times as much. Choosig weights i these ratios will do the trick: Tests Quizzes Fial Scores 84% 91% 77% Weights The weighted average is computed by multiplyig the scores times the weights, dividig the sum of the total weights scored, ad ally dividig by the total of the weights used. Tests Quizzes Fial Total Scores 84% 91% 77% - Weights Scored Weights

10 The weighted average is = 84%. This distributio of weights meas that the test score couts as three scores, while the quizzes ad the al cout as oe test each. That meas that the test average couts as 3 5 of the total score; the quiz average couts as 1 5 of the total, ad the al as 1 5 :Thus the exact same calculatio could appear as Tests Quizzes Fial Total Scores 84% 91% 77% Weights Scored Weights 50:4 18:2 15:4 84:0 For that matter, 3 5 is also 60%; ad 1 5 calculatio is is 20%: So aother versio of the same Tests Quizzes Fial Total Scores 84% 91% 77% - Weights 60% 20% 20% 100% Scored Weights Agai the weighted average is = 84%. Prepared by: Daiel Madde ad Alyssa Keri: May