Chapter 3. Descriptive Statistics Numerical Methods

Transcription

1 Chapter 3 Descrptve Statstcs Numercal Methods

2 Our goal? Numbers to help us answer smple questons. What s a typcal value? How varable are the data? How extreme s a partcular value? Gven data on two varables, how closely do they move together?

3 Measures of Central Tendency Here are three ways to dentfy a typcal observaton Mean the arthmetc average Medan the mddlemost value Mode the most common value

4 There are formulas, but... One confusng thng about the formulas s all the notaton they use. To explan why we need the notaton, and why you need to know t, let me remnd you of an mportant dstncton...

5 Populaton vs. Sample A populaton s the set of all data that characterze some phenomenon, and a number computed from populaton data s called a parameter. A sample s a subset of a populaton, and a number computed from sample data s called a statstc.

6 An Example Populaton - All regstered voters. Parameter The fracton of all regstered voters who prefer John McCan to Barack Obama. Sample 2500 voters surveyed by Gallup. Statstc The fracton of voters n the poll who prefer McCan to Obama.

7 Another Example Populaton - All Duracell AA batteres. Parameter The average lfetme of all Duracell AA batteres n a partcular toy. Sample A hundred batteres beng tested by the manufacturer. Statstc The average lfetme of the 100 tested batteres n the partcular toy.

8 Why s the dstncton mportant? Sample statstcs are very dfferent from populaton parameters. Parameters are fxed numbers. Before the sample s drawn, statstcs depend on the elements that may be selected, and are random. Once a sample s drawn, the numbers themselves are lkely to be dfferent; that s 48% of the populaton but 51% of the sample may prefer Obama. Therefore, our notaton must clearly dstngush sample statstcs from populaton parameters.

9 Now let us return... To those measures of central tendency.

10 The Sample Mean The sample mean s the arthmetc mean of some sample data. The notaton for a sample mean s X-bar. The notaton for sample sze s a lower case n. X = n = 1 n x

11 The Populaton Mean The populaton mean s the arthmetc mean of some populaton data. The notaton for a populaton mean s the Greek letter mu. The notaton for populaton sze s an upper case N. N µ = = 1 N x

12 And THIS s the summaton operator Here t s just tellng us to add the observatons. n = 1 x = x + x + x + x n

13 Don t be ntmdated by the summaton operator It s just shorthand; t saves space. The summaton operator s just the Greek letter Sgma. Sgma s Greek for S, and S stands for Sum. S for Sum, meanng add them all up.

14 Formulas wth summaton operators confuse you? Consult Anderson, Sweeny, and Wllams Appendx C, and memorze the rules lsted there, or Do what I do, whch s fgure t out as you go along. For example...

15 Is ths a vald operaton? n? n ax = a x = 1 = 1

16 Just undo the shorthand to see! n = 1 So n = ax = ax + ax + ax + ax n a x = a x + x + x + x ( ) = 1 = n ax = a x n n

17 How about ths? Is t ok? n n n ( )? ax + by = a x + b y = 1 = 1 = 1

18 Undo the shorthand to check n = 1 n ( ) ax + by = ax + by + ax + by + ax + by + ax + by n n ( ) ( ) ( ) n n = 1 = 1 So, yes, n a x + b y = a x + x + x x + b y + y + y y n n n ax + by = a x + b y = 1 = 1 = 1

19 Let s go back to our formulas Suppose you are gven the followng data 2,3, 3, 4, 6, 7, 8,11,12,13,15,16,17

20 Sample or populaton data? It depends on the context. Suppose ths data came from askng 13 people the number of computer games they own. If you are nvestgatng the number of games owned by these partcular 13 people, then ths s populaton data If you are nvestgatng the number of games owned by a larger group, and these 13 people are members of that group, then ths s sample data. In homework problems, the default s sample data.

21 The mean n ths example X x = 1 X = n = = 13 n 9

22 The medan n ths example Snce the medan s the mddlemost observaton, one way to fnd t s to order the observatons and throw away observatons one at a tme from each end, untl one s left n the mddle n ths case, 8. 2,3, 3, 4, 6, 7, 8,11,12,13,15,16,17 2,3,3, 4, 6,7,8,11,12,13,15,16,17

23 Suppose you have an even number of observatons! In that case, you wll be left wth two numbers n the mddle when you have fnshed elmnatng numbers from both the top and bottom. The medan s found by addng those two numbers and dvdng by two.

24 The mode n ths example Here s our data. Note that the only observaton that appears twce s 3 makng t the mode, or most common observaton. But 3 s not a very typcal observaton, whch s why the mode s hardly ever used. 2,3, 3, 4, 6, 7, 8,11,12,13,15,16,17

25 Mean vs. Medan In 1983, the average startng salary of Rhetorc and Communcatons majors at the Unversty of Vrgna was approxmately $35,000 a year, far more than that of other majors n the college of Arts and Scence. Can you guess why?

26 Ralph Sampson, a Rhetorc and Communcatons major, was the frst pck n the NBA draft. The Houston Rockets pad hm $2,000,000 a year. Here s your answer!

27 Robust Statstcs A statstc s sad to be robust f t s not dramatcally affected by a small number of extreme observatons. The medan s robust, the mean s not. Therefore the medan s usually a better ndcaton of a typcal value.

28 How the mean and medan dffer If a dstrbuton s not symmetrc... And there are a handful of extremely large or small values... The mean wll be pulled n the drecton of the extreme values. The Ralph Sampson story llustrates the problem.

29 Look at the ncome dstrbuton n the USA n 1992

30 Asymmetrc, skewed to the rght The medan ncome s marked on the graph, at about $22,000 a year. The mean s not reported, but t appears to be about $30,000 a year.

31 Many people use statstcs the way a drunk uses a lamp post For support... Not for Illumnaton.

32 And they play games wth the mean and medan An ncumbent poltcan wll boast of how well the economy s dong, and use mean ncome numbers as evdence. The challenger wll complan of how badly the economy s dong, and use medan ncome numbers as evdence. Confusng voters.

33 Measures of Varablty Range Interquartle Range Varance Standard Devaton Coeffcent of Varaton

34 The Range The Range of a data set s just the dfference between the bggest and smallest observaton. The Range s easy to compute, but t s not robust, and therefore may be msleadng. As n: Startng salares for Rhetorc and Communcatons majors range from $18,000 to $2,000,000 a year.

35 An Example Lets use the earler data to llustrate. 2,3, 3, 4, 6, 7, 8,11,12,13,15,16,17 The Range s 17 2 = 15

36 Interquartle Range (IQR) Ths s the spread of the mddle 50% of the observatons. It s defned as Q3 Q1 Q3 s the thrd quartle, or 75 th percentle. 75% of all observatons are smaller than Q3. Q1 s the frst quartle, or 25 th percentle. 25% of all observatons are smaller than Q1. (Q2 s the second quartle, or medan.)

37 How do you fnd quartles? Bascally, to fnd Q3, the 75 th percentle, order the data and throw away 3 observatons from the bottom for every one from the top. Here, Q3 s 13. 2,3, 3, 4, 6, 7, 8,11,12,13,15,16,17 2,3,3,4,6,7,8,11,12,13, 15, 16,17

38 Q1 works the same way To fnd Q1, the 25 th percentle, order the data and throw away 1 observaton from the bottom for every 3 from the top. Here Q1 s 4, so IQR = Q3 - Q1= 13 4 = 9. 2,3, 3, 4, 6, 7, 8,11,12,13,15,16,17 2,3,3,4, 6,7,8,11,12,13,15,16,17

39 I cheated a bt to make t smple Wth 13 observatons, elmnatng observatons n ths way leaves you wth just one observaton remanng. If the number of observatons you have s not equal to 4n+1 for some n, there wll be two, three, or four observatons remanng. Then you must round or nterpolate.

40 A, S, & W propose ths soluton Arrange data n ascendng order. Compute, where p s the percentle you seek and n s the sample sze. If s an nteger, average the th and +1 st observatons. If s not nteger, round up. p = 100 n

41 An Example fndng Q3 Here p = 75, n = 6. Whch gves = 4.5. Whch s not nteger. So round up to 5. The 5 th observaton s 9, so Q3 = 9. 2, 3, 6, 7, 9,10 75 = 6 =

42 An Example fndng Q1 Here p = 25, n=8 Whch gves = 2. Whch IS an nteger. So average the second and thrd observatons. To get (5+6)/2 = 5.5 So Q1 = 5.5 2, 5, 6, 6, 7, 8, 9,10 25 = 8 = 2 100

43 But ths s an arbtrary conventon Mntab uses a dfferent rule. In our frst example, where we got Q3 = 9, Mntab gets Q3 = In our second example, where we got Q1 = 5.5, Mntab gets Q1 = 5.25.

44 Varance of a Populaton The varance s the average sze of a squared devaton about the mean. Lower-case sgma squared s populaton varance. Note the use of mu and N n the formula: all these are populaton parameters. N σ 2 = 1 = ( x µ ) 2 N

45 Varance of a Sample Lower-case s-squared denotes sample varance. Note the use of X-bar and n n the formula: these are sample statstcs. Also note the funky denomnator, n-1, where you would expect to see n. s n 2 = 1 = ( x x) 2 n 1

46 Why use n-1 wth sample data? A sophstcated explanaton s comng n Chapter 7, but thnk of t as a fudge factor. Havng to compute squared devatons around the sample mean nstead of the true populaton mean makes the numerator too small. Dvdng by n-1 corrects for ths.

47 Example of the Calculaton The heart of the calculaton s evaluatng the numerator. Here s our example. Remember, the mean s 9. n = 1 ( x ) x = ( 2 9) + ( 3 9) + ( 3 9) n = 1 2,3, 3, 4, 6, 7, 8,11,12,13,15,16,17 ( x ) x = = 338

48 Fnshng the Varance calculaton Gven the sum of squared devatons from the mean, the calculaton s as follows: Dvde by n-1 for sample data. Dvde by N for populaton data. n ( x x) 2 2 = s = = = n 1 12 N σ = ( x µ ) = = = N 13 26

49 The Standard Devaton The varance measures varablty n nonsense unts; n ths case, number of computer games squared. To correct ths, we ntroduce the standard devaton, whch s just the square root of the varance. The standard devaton can be thought of as the sze of a typcal devaton from the mean. s = s 2 s = = 5.31 σ = 2 σ σ = 26 = 5.099

50 Coeffcent of Varaton Seldom used n ths course. Answers: The standard devaton s what percent of the average? Why s ths useful? An nch more or less n the heght of a skyscraper s meanngless. An nch more or less n the length of your nose s a bg deal. s coeffcent of varaton = 100 x s = = 59 x 9

51 Can you show us a use for the standard devaton? Many real world data sets have an approxmate bell shape, as you no doubt have been told. 0.2 The Famous Bell Curve Otherwse known as the Normal Dstrbuton C C2

52 A Rule for such varables 68% of all observatons are found wthn one standard devaton of the mean. 95.5% of all observatons are found wthn two standard devatons of the mean. 99.7% of all observatons are found wthn three standard devatons of the mean. So any observaton more than 2s from the mean s unusual, and one more than 3s from the mean s very unusual.

53 The z-score Ths measures how many standard devatons above or below the mean a partcular observaton s. Postve values are above the mean, negatve ones below. Any z greater than 2 n absolute value s a mld outler. z = x s x Any z greater than 3 n absolute value s a substantal outler.

54 Why should we care about outlers? Depends on the crcumstances, but outlers often requre nvestgaton. An outler may sgnal a data entry error! An outler may dentfy a partcularly poor outcome that needs to be corrected. An outler may dentfy a partcularly good outcome that needs to be emulated.

55 Measures of Assocaton between two varables Often we want to know whether varables are postvely or negatvely assocated, and f so, how strong the assocaton s. Some examples wll llustrate what I mean.

56 Postve assocaton

57 No assocaton

58 Negatve assocaton

59 No lnear assocaton, but...

60 Covarance A measure of the degree of lnear assocaton. It s an average sze of a cross product. Frst formula s the populaton covarance. Second formula s the sample covarance. N = 1 σ xy = s xy = ( x )( ) µ x y µ y n = 1 ( )( ) N x x y y n 1

61 Why ths cross product? ( x x)( y y) When x and y are smultaneously bgger than ther means, both terms are postve, contrbutng a postve term to the sum. When x and y are both smultaneously smaller than ther means, both terms are negatve, once agan contrbutng a postve term to the sum. Postvely related varables wll have many + terms n the sum.

62 Conversely... ( x x)( y y) If a bg x and small y are pared, the frst term s postve,whle the second s negatve, contrbutng a negatve term to the sum. If a small x and bg y are pared, the frst term s negatve,whle the second s postve, contrbutng a negatve term to the sum. Negatvely related varables wll have many negatve terms n the sum.

63 And f x and y are unrelated The sum wll consst of offsettng postve and negatve terms. Therefore: A postve covarance means postve assocaton A negatve covarance means negatve assocaton A zero covarance means no (lnear) assocaton.

64 An example Here are some data: on casual observaton they seem postvely assocated. Frst we need the mean of each of the two varables. The mean of x s 16. The mean of y s 10. X Y

65 Computng the numerator n = 1 x x y y ( )( ) = ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) = ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) = = 106

66 Completng the calculaton The numerator s the same for both the sample and populaton covarance. The only dfference s n the denomnator, because of the n-1 dvsor. N = 1 σ xy = s xy ( x )( ) µ x y µ y N 106 = = = n = = = 4 ( x x)( y y) n

67 Hmmm... We can see that the relatonshp s postve because the covarance s postve, but what are we to make of 26.5? Is that bg or small?

68 The Covarance s Flawed There are really TWO problems. The covarance lacks a scale, so we have no way to judge ts sze. The covarance depends on unts. Measurng x and y n nches we d get one answer. Someone measurng x and y n centmeters would get a dfferent answer even though the degree of assocaton s exactly the same!

69 Correlaton s superor Top formula defnes the populaton correlaton. Bottom formula defnes the sample correlaton. Correlaton s unt free and always between 1 and +1. ρ r xy xy = = s x σ x xy s s y xy σ σ y

70 Interpretng the correlaton A postve correlaton mples a postve assocaton, and conversely, snce the correlaton has the same sgn as the covarance. A zero correlaton mples no lnear assocaton. A correlaton near one n absolute value s a very strong relatonshp; one near zero, weak.

71 For Example...

72 Our second example...

73 Our thrd example...

74 Remember, correlaton measures lnear assocaton!

75 Returnng to the example... Here s our data. To compute a correlaton coeffcent, we need to frst compute standard devatons of both X and Y. X Y

76 Standard devaton of X n = 1 ( x x) = ( 6 16) + ( 11 16) ( 15 16) ( 21 16) ( 27 16) = = = 272 n 2 ( ) ( ) s = x x n 1 = = x N ( x ) x x = 1 2 σ = µ N = = 7.376

77 Standard devaton of Y n = 1 ( y y) = ( 6 10) + ( 9 10) ( 6 10) ( 17 10) ( 12 10) = = = 86 n 2 ( ) ( ) s = y y n 1 = 86 4 = y N ( y ) y y = 1 2 σ = µ N = 86 5 = 4.147

78 Completng the computaton r xy ρ xy sxy 26.5 = = =.693 ss x y ( )( ) σ xy 21.2 = = =.693 σσ x y ( )( )

79 A few comments It s not an accdent that the numbers are the same. The only dfference n the populaton and sample formulas s n dvsors: n-1 vs. N. Those dfferent dvsors appear n both numerator and denomnator and cancel out. The concluson, a correlaton of.693, mples a moderately strong postve assocaton

80 Grouped Data Here s a problem usng grouped data. Orgnal observatons lost though groupng. Treat ths as follows: 4 observatons of 5, 7 observatons of 10, 9 observatons of 15, and 5 observatons of 20. Class Mdpont ASW, #53, p. 119 Freq

81 Exstng formulas work but are tedous! X n x = 1 = = n 4 tmes 7 tmes 9 tmes 5 tmes X n x = 1 = = = n

82 Ths s why one multples The top formula s for sample data. The bottom formula s for populaton data. M sub- s mdpont of the th category. f sub- s the frequency or count of the th category. X = µ = f M n f M N

83 So the calculaton works ths way Just take each mdpont Mdpont Frequency f M Multply by the correspondng frequency. Add up the products, and you have the numerator. 325 fm =

84 The last step Just the same for the populaton mean. Warnng: the sample sze s 25 (the sum of the frequences), not 4 (the number of categores). X X = fm 325 = = n

85 Varance formulas for grouped data Same ratonale as formulas for the mean; whle prevous formulas work, these are easer, because multplcaton replaces addton. Top formula s for sample data; bottom for populaton data. s 2 = 2 σ = f M x f ( ) n 1 M 2 ( ) x N µ 2

86 Computng the numerator Mdpont Frequency ( M x ) 2 f ( ) 2 M x ( ) 2 f M x = 600

87 Completng the calculaton Calculaton of the numerator s dentcal for both the populaton and sample varance. Only dfference s for sample measure, dvde by n-1; populaton measure dvde by N. s s 2 2 σ σ 2 = s = ( ) f M x 600 = = = 25 n ( M µ ) f 600 = = = 24 N 25 2 = = = σ

88 That s t for today!