Chapter 3 Descriptive Statistics Numerical Summaries

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1 Secto 3.1 Chapter 3 Descrptve Statstcs umercal Summares Measures of Cetral Tedecy 1. Mea (Also called the Arthmetc Mea) The mea of a data set s the sum of the observatos dvded by the umber of observatos. If the data are 1,, 3,,, the Mea = Two otatos for the mea:(a) Sample mea: (read as -bar) (b) Populato Mea: ( Mu ) Thus = where = # of tems the sample data, ad = where = sze of the populato. ote: (sgma) s a Greek symbol that sgfes summato. Eample 1: Fd the mea for ths sample data:, 3, 6, 7, 7, 8, 9, 9, 9, 10 Soluto: = = = 70/10 = 7 Eample : A sample of fve famles Cucumber tow, Iowa showed the followg aual famly comes: $17,00, $3,000, $4,000, $6,000, $30,000 Fd the mea for ths data. = = = 41000/ = $8,100 Etreme Value/Outler: a data value that s too large or too small as compared to most of the data values. ote: The Mea s flueced by outlers. 1

2 . Meda (The meda s the mddle value of the data whe the data has bee arraged ascedg/descedg order.) Eample 3: Fd the meda for the data set 1 ad data set. Data Set 1: 7,, 8,, 9, 4, 7, 8, 6 Data Set : 7,, 8,, 9, 4, 8, 8 Soluto: The meda for data set 1 s 7 The meda for the data set s 7. Eample 4: Fd the meda for the data Eample Soluto: Meda = $4,000 ote: The meda ot affected by etreme values. Thus the presece of etreme values, meda may be a better dcator of the ceter. 3. Mode The most frequetly occurrg data value a set of data s called the mode. That s, the mode s the value that occurs wth greatest frequecy. Eample. Fd the mode for the gve data:, 3, 3,,, 8, 7, 8, 7, 9, 8, 8 Soluto: Mode = 8 Eample 6. Fd the mode for the gve data:, 3, 3,,, 8, 7, 8, 7, 9, 8, 8, Soluto: Mode =? Or 8? ote: Such a dstrbuto s called bmodal. Eample 7. Fd the mode for the gve data:, 3, 8, 7, 9 Soluto: Mode s udefed.

3 ote: Mode s seldom used practce, ecept to aswer the very specal questo that s desged to aswer: a. What s the most watched TV show? b. What s the best sellg automoble? c. What s the most commo cause of death? Dscuss the shapes of the dstrbuto o page 84. Homework-Secto 3.1 Ole - MyStatLab Secto 3. Measures of Dsperso (Sample Stadard Devato) Rage = Largest Value Smallest Value Eample 1: Gve the two data sets below, fd the rage, mea, mode, ad meda. Data Set 1: 99, 91, 84, 84, 80, 80, 80, 76, 76, 69, 61 Data Set : 99, 80, 80, 80, 80, 80, 80, 80, 80, 80, 61 Sol: For all of the data sets, Rage = = 38 ad Mea=Meda= 80 ote: The rage s based o oly two of the tems the data set ad thus s flueced too much by etreme values. Varace: Average Squared Devato from the Mea 3

4 Populato Varace = Sample Varace s = 1 1 ( ) ( X ) 1 = = , ( populato sze)., ( sample sze). Gve the data 46, 4, 4, 46, 3. The mea () for ths data s 44. = 1 X X - (X - ) Total 0 6 ( ) = 6 = 1. 1 X X = = = = = Stadard Devato = Varace Sample Stadard Devato = s = s Populato Stadard Devato = = = Eample : Fd the sample stadard devato for the data below: 9, 11, 16, 14, 1, 1, 10, 9, 9 Soluto: Sample mea, X 11.33, sample varace, S 6.0, sample s.d., S.449 4

5 1 1 % K Based o CHEBYSHEV S THEOREM where K = umber of stadard devatos. At least 7% of the tems must le wth two stadard devatos of the mea; At least 88.89% of the tems must le wth three stadard devatos of the mea; At least 93.7% of the tems must le wth four stadard devatos of the mea. Eample 3: Mdterm scores for 100 studets a college statstcs course had a mea of 70 ad s.d. of. (a) How may studets scored betwee 60 ad 80? (b) How may studets scored betwee 0 ad 90? The Emprcal Rule (For Bell Shaped Dstrbutos) Appromately 68% of the data fall wth 1-stadard devato of the mea; Appromately 9% of the data fall wth -stadard devato of the mea; Appromately 99.7% of the data fall wth 3-stadard devato of the mea. Eample 4: I a class wth 0 studets, the mea score o a test was 60 whle the stadard devato was 1. How may studets (a) scored betwee 48 ad 7? (b) scored betwee 36 ad 84? (c) scored betwee 10 ad 60? (d) scored betwee 48 ad 84? (e) scored betwee 84 ad 96? (f) 9% of the studets scored ths terval? Homework-Secto 3. Ole - MyStatLab

6 Secto 3.4 Measures of Posto ad Outlers Detectg Outlers Sometmes a set of data has oe or more tems wth uusually large or uusually small values. Etreme values such as these are called Outlers. Epereced statstcas take steps to detfy outlers ad the revew each oe carefully. A outler may have bee a tem for whch the value has bee correctly recorded. If so, the value ca be corrected before proceedg wth the aalyss. A outler may also be a tem that was correctly cluded the data set; If so, t ca be removed. Fally, a outler may just be a uusual tem that has bee correctly recorded ad does belog the data set. I such cases, the tem should rema the data set. Z-score = Z-score = X s where where s s the sample s.d. s the populato s.d. Z-score for ay data tem s referred to as ts stadardzed value. It ca be terpreted as a measure of the relatve locato of a tem the data. Eample : If the Z-score of a data tem s, the data value s -stadard devatos above the sample mea. Usg Z-score to detfy outlers: RULE: A value wth a Z Score 3 or Z Score 3wll be treated as a outler. 6

7 Eample 1: Gve the data set below, detfy outlers, f ay, the data. 46, 4, 4, 46, 3 Sol. ote that = 44 ad s= 8 ( - )/s z-score (46-44)/ (4-44)/ (4-44)/ (46-44)/ (3-44)/8-1.0 There are o outlers ths data. Measure of Posto Percetle: A percetle s a umercal measure that also locates values of terest the data set. A percetle provdes formato regardg how the data tems are spread over the terval from the lowest value to the hghest value. Def. The p th percetle of a data set s a value such that at least p percet of the tems take o ths value or less ad at least (100 p) percet of the tems take o ths value or more. Step 1: Sort the data a ascedg order, that s, from the smallest to the largest. P Step : Fd 100 where s the umber of data values. s a locato. s the the umber locato. Step 3: If s ot a teger, roud t up to the et hghest teger, the p th percetle =. If s a teger, the p th percetle = 1 Eample : Gve the data below, fd the 0 th ( 0). P ad 90 th ( P 90) percetles. 6, 4,, 0, 6, 1, 1, 1, 1, 8, 9, 10, 14, 18, 16, 17 Sol: Step 1: Data ascedg order. = = 4,, 6, 8, 9, 10, 1, 14, 1, 1, 1, 16, 17, 18, 0, 6 90 th perc.: = (90/100)16 =14.4; => = 1 = 0; 90 th perc. = 0 0 th perc.: = (0/100)16 =8; sce =8; the 0 th perc= 8 9 ote: The meda ad the 0 th percetle are the same. = 14 1 =14. 7

8 Quartles: It s ofte desred to dvde a data set to four parts wth each part cotag oe-fourth of the data. Q 1 = Frst Quartle = % percetle Q = Secod Quartle = 0% percetle Q3 = Thrd Quartle = 7% percetle Eample 3: For the data gve Eample, fd the frst, secod, ad thrd quartles. Sol. Q 1 = 8., Q = 14., Q3 = 16. Homework-Secto 3.4 ad 3. Ole - MyStatLab Secto 3. The Fve umber Summary ad Boplots The Iterquartle Rage (IQR): IQR = Q 3 - Q 1 ote: The IQR gves the rage of the mddle 0% of the observatos. The Fve-umber Summary The fve umber summary of a data set: M, Q 1, Q, Q3, ad Ma. Eample 1: Fd the fve-umber summary. Sol. The data s 4,, 6, 8, 9, 10, 1, 14, 1, 1, 1, 16, 17, 18, 0, 6 M = 4, Q 1,= 8., Q = 14., Q3 = 16., ad Ma = 6. Boplot : Is Bult to Detect Outlers 1. Fd Q 1, Q, Q3, ad IQR.. Compute Lower Fece ad Upper Fece: Lower Ier Fece=Q 1-1.(IQR), Lower Outer Fece=Q 1-3(IQR), Upper Ier Fece=Q 3+1.(IQR) Upper Outer Fece=Q 3+ 3(IQR) 3. Draw the bo plot dcatg the Lower a Upper feces. 4. Determe whether there are ay outler Eample : Use Eample 1. Buld a boplot ad check for outlers. Is the shape of the data set skewed left, rght, or symmetrc? Aswer: Skewed Left (For help See fgure o page 160 ). Homework-Secto 3.4 ad 3. Ole - MyStatLab 8