Example Consider the following set of data, showing the number of times a sample of 5 students check their per day:


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1 Sectio 82: Measures of cetral tedecy Whe thikig about questios such as: how may calories do I eat per day? or how much time do I sped talkig per day?, we quickly realize that the aswer will vary from day to day ad ofte modify our questio to somethig like how may calories do I cosume o a typical day? or o average, how much time do I sped talkig per day? I this sectio we will study three ways of measurig cetral tedecy i data, the mea, the media ad the mode Each measure give us a sigle value (the mode may give more tha oe) that might be cosidered typical As we will see however, ay oe of these values ca give us a skewed picture if the give data has certai characteristics A populatio of books, cars, people, polar bears, all games played by Babe Ruth throughout his career etc is the etire collectio of those objects For ay give variable uder cosideratio, each member of the populatio has a particular value of the variable associated to them, for example the umber of home rus scored by Babe Ruth for each game played by him durig his career These values are called data ad we ca apply our measures of cetral tedecy to the etire populatio, to get a sigle value (maybe more tha oe for the mode) measurig cetral tedecy for the etire populatio Whe we calculate the mea, media ad mode usig the data from the etire populatio, we call the results the populatio mea, the populatio media ad the populatio mode A sample is a subset of the populatio, for example, we might collect the data o the umber of home rus scored i a radom sample of 20 games played by Babe Ruth If we calculate the mea, media ad mode usig the data from a sample, the results are called the sample mea, sample media ad sample mode The Mea: The populatio mea of m umbers x 1, x 2,, x m (the data for every member of a populatio of sixe m) is deoted by µ ad is computed as follows: µ = x 1 + x x m m The sample mea of the umbers x 1, x 2,, x (data for a sample of size from the populatio) is deoted by x ad is computed similarly: x = x 1 + x x Example Cosider the followig set of data, showig the umber of times a sample of 5 studets check their per day: 1, 3, 5, 5, 3 Here = 5 ad x 1 = 1, x 2 = 3, x 3 = 5, x 4 = 5 ad x 5 = 3 Calculate the sample mea x 1
2 Example The followig data shows the results for the umber of books that a radom sample of 20 studets were carryig i their book bags: 0, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 4 The the mea of the sample is the average umber of books carried per studet: x = Not that the mea here caot be a observatio i our data = 25 Calculatig the mea more efficietly: We ca calculate the mea above more efficietly here by usig frequecies We ca see from the calculatio above that x = The frequecy distributio for the data is: 0 + (1 2) + (2 8) + (3 4) + (4 5) 20 # Books Frequecy # Books Frequecy = x = Sum 20 The geeral case ca be dealt with as follows: If our frequecy/relative frequecy table for our sample of size, looks like the oe below, (where the observatios are deoted 0 i, the correspodig frequecies, f i ad the relative frequecies f i /): Observatio Frequecy Relative Frequecy 0 i f i f i / 0 1 f 1 f 1 / 0 2 f 2 f 2 / 0 3 f 3 f 3 / 0 R f R f R / the, x = 0 1 f f R f R = 0 1 f f f3 + 0 R fr We ca also use our table with a ew colum to calculate: 2
3 Outcome Frequecy Outcome Frequecy 0 i f i 0 i f i 0 1 f f f f f f 3 0 R f R 0 R f R SUM = x Alteratively we ca use the relative frequcies, istead of dividig by the at the ed Outcome Frequecy Relative Frequecy Outcome Relative Frequecy 0 i f i f i / 0 i f i / 0 1 f 1 f 1 / 0 1 f 1 / 0 2 f 2 f 2 / 0 2 f 2 / 0 3 f 3 f 3 / 0 3 f 3 / 0 R f R f R / 0 R f R / SUM = x You ca of course choose your favorite method for calculatio from the three methods listed above Example The umber of goals scored by the 32 teams i the 2014 world cup are show below: 18, 15, 12, 11, 10, 8, 7, 7, 6, 6, 6, 5, 5, 5, 4, 4, 4, 4, 4, 4, 3, 3, 3, 3, 3, 2, 2, 2, 2, 1, 1, 1 Make a frequecy table for the data ad takig the soccer teams who played i the world cup as a populatio, calculate the populatio mea, µ 3
4 Estimatig the mea from a histogram If we are give a histogram (showig frequecies) or a frequecy table where the data is already grouped ito categories ad do ot have access to the origial data, we ca estimate the mea usig the midpoits of the itervals which serve as categories for the data Suppose there are k categories (show as the bases of the rectagles) with midpoits m 1, m 2,, m k respectively ad the frequecies of the correspodig itervals are f 1, f 2,, f k, the the mea of the data set is approximately where = f 1 + f f k m 1 f 1 + m 2 f m k f k Example Approximate the mea for the set of data used to make the followig histogram, showig the time (i secods) spet waitig by a sample of customers at Grigotts Wizardig bak midpoits: approximatio of mea: Time spet waitig (i secods) The Media The Media of a set of quatitative data is the middle umber whe the measuremets are arraged i ascedig order To Calculate the Media: of the data by M 1 If is odd, M is the middle umber 2 If is eve, M is the average of the two middle umbers Arrage the measuremets i ascedig order We deote the media Example The umber of goals scored by the 32 teams i the 2014 world cup are show below: 18, 15, 12, 11, 10, 8, 7, 7, 6, 6, 6, 5, 5, 5, 4, 4, 4, 4, 4, 4, 3, 3, 3, 3, 3, 2, 2, 2, 2, 1, 1, 1 Fid the media of the above set of data 4
5 Example A sample of 5 studets were asked how much moey they were carryig ad the results are show below: $75, $2, $5, $0, $5 Fid the mea ad media of the above set of data Notice that the media gives us a more represetative picture here, sice the mea is skewed by the outlier $75 The Mode Defiitio The mode of a set of measuremets is the most frequetly occurrig value; it is the value havig the highest frequecy amog the measuremets Example Fid the mode of the data collected o the amout of moey carried by the 5 studets i the example above: $75, $2, $5, $0, $5 You fid that i some cases the mode is ot uique: Example What is the mode of the data o the umber of goals scored by each team i the world cup of 2006? 13, 12, 11, 10, 9, 9, 7, 5, 5, 5, 5, 5, 5, 4, 4, 3, 3, 3, 3, 3, 3, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 0 Note The mode ca be computed for qualitative data The mode is ot ofte used as a measure of ceter for quatitative data The Histogram ad the mea, media ad mode With large sets of data ad arrow class widths, the histogram looks roughly like a smooth curve The mea, media ad mode, have a graphical iterpretatio i this case The mea is thep balace 315 poit of the histogram of the data, whereas the media is the poit o the xaxis such that half of the area uder the histogram lies to the right of the media ad half of the area lies to its left The mode occurs at the data poit where the graph reaches its highest poit This of course524 may ot Which be uique Measure of Ceter to Use? Bellshaped, Symmetric Bimodal mea=media=mode mea=media two modes Skewed Right 5 Skewed Left
6 mea=media=mode two modes Skewed Right mode mea media Skewed Left Skewed Left mode mea media mea mode media mea mode media Skewed Data Defiitio A data set is said to be skewed if oe tail of the distributio has more extreme observatios tha the other tail Mea, Media, ad Mode The mea is sesitive to extreme observatios, but the media is ot(check out the example below) Example Cosider the data from the above example The most cocerig commo themeasure amout of ceter moeyis carried the mea, by the which loca five studets i the sample balacig poit of the distributio The mea equals the sum Mea, Media, ad Mode $75, $2, $5, observatios, $0, $5 divided by how may there are The mea is a We have already calculated the mea ad the media extreme of the observatios data, which (outliers we foud ad values to be : which mea are = far i the $174, media The most = commo $5 distributio that is skewed) So the mea teds to be a good measure of ceter is the mea, locatig which the ceter locates of the a distributio that is uimodal ad roug Now cosider balacig the same poit set of the of data distributio with thethe largest mea amout with equals o of outliers the moey sum of replaced the by $5,000, that is suppose our dataobservatios, was divided by how may there are The mea is also affected by extreme observatios (outliers ad $5, 000, values $2, which are far i the tail of a The $5, media $0, $5 is a more robust measure of ceter, that is, it is distributio that is skewed) So the mea teds to be a good choice for What is by extreme values The media is the middle observatio wh locatig the ewthe mea ceter adof media? a distributio that is uimodal ad roughly symmetric, with o outliers We ca The see from media theis histograms, a more robust that measure for data of skewed ceter, that to the is, right, it is ot the iflueced mea is larger tha the media ad for by data extreme skewedvalues to left, The themedia mea is is less the tha middle the observatio media whe the data are Differet Measures Ca Give Differet Impressios The famous trio, the mea, the media, ad the mode, represet three differet methods for fidig a socalled ceter value These three values may be the same for a set of data but it is very likely that they will have three differet values Whe they are differet, they ca lead to differet iterpretatios of the data beig summarized Cosider the aual icomes of five families i a eighborhood: What is the typical icome for this group? $12, 000 $12, 000 $30, 000 $51, 000 $100, 000 The mea icome is: $41,000, The media icome is: $30,000, The modal icome is: $12,000 If you were tryig to promote that this is a affluet eighborhood, you might prefer to report the mea icome If you were a Sociologist, tryig to report a typical icome for the area, you might report the media icome If you were tryig to argue agaist a tax icrease, you might argue that icome is too low to afford a tax icrease ad report the mode 6
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