Reference for Ch. 4 Quest. 12

# Reference for Ch. 4 Quest. 12 - Other Measures of...

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Unformatted text preview: Other Measures of Dispersion The standard deuiation is the most widely used measure oi dispersion. Howener. there are other ways oi describing the satiation or spread in a set of data. One ntethod is to determine the location oi yalues that diuide a set oi obseruations into equal parts. These measures include quartiles, deciles, artd peroenllles. Quartiles diuide a set ol obseruations into Tour equal parts. To enplain lurther. think oi any set oi yalues arranged lrom smallest to largest. in Chapter 3 we called the mid- dle ualue ol aset oi data arranged imm smallest to largest the median. That is. so per- cent oi the obseryations are larger than the median and 50 percent are smaller. The median is a measure oi location because it pinpoints the center ol the data. In a sim- ilar fashion quartiles diuide a set oi obseruations into [our equal parts. The iirst quar- tile. usually labeled Til. is the rains below which 25 percent oi the obseryations occur. and the third quartile. usually labeled 03. is the rains below which is percent of the Descrlhlnu Data: Dlsplavtng and Exploltng Data 10? observations occur. Logically. Qa is the median. O. can be thought of as the "median" of the lower half of the data and OS the “median” of the upper half of the data. In a similar fashion deciles divide a set of observations into 10 equal parts and percentiles into 100 equal parts. So if y'ou found that y'our GPA was in the 81h decile at your university. you could conclude that 80 percent of the students had a GPA lower than yours and 20 percent had a higher GPA. A GPA in the 33rd percentile means that 33 percent of the students have a lower GPA and 6? percent have a higher GPA. Percentile scores are frequently used to report results on such national standardized tests as the SAT. ACT. GMAT [used to judge entry into many master of business administration programs]. and LSAT [used to judge entry into law school). Quartiles. Deciles. and Percentiles To formalize the computational procedure. let LP refer to the location of a desired percentile. So if we want to find the 33rd percentile we would use Leo and if we wanted the median. the 50th percentile. then L50. The number of observations is n. so if we want to locate the median. its position is at [n + 102. or we could write this as [n + 1iiPx'100]. where P is the desired percentile. .r' ' \ ic-Ivnnl'. m n Pl 141:! fill. LP = (:1 +11% [4-1] ..H J) An example will help to explain further. Listed below are the commissions earned last month by a sample of 15 brokers at Salomon Smith Barney's Oakland. California office. Salomon Smith Barney is an investment company with offices located throughout the United States. \$2.038 1 .940 \$1.?58 2.311 \$1.?21 2.054 \$1 .63? 2.406 \$2.09? 1.4?1 \$2.04? 1.460 \$2.205 \$1 .?8? \$2. 28? Locatethe median. the first quartile. and the third quartile for the commissions earned. The first step is to sort the data from the smallest commission to the largest. \$1 .460 2.04? \$1 . 4?1 2. 054 \$1.63? 2.09? \$1 .?21 2.205 \$1 .?58 2.28? \$1.?8? 2.311 \$1.040 2.406 \$2.038 The median value is the observation in the center. The center value or 1.50 is located at to + 1] (50x’1001. where n is the num— ber of observations. In this case that is position number 8. found by [15 + 1:11:50r'100). The eighth largest commission is \$2.038. So we conclude this is the median and that half the brokers earned commissions more than \$2.038 and half earned less than \$2.038. Flecall the definition of a quartile. Ouartiles divide a set of observations into four equal parts. Hence 25 percent of the obser- vations will be less than the first quartile. Seventy—five percent of the observations will be less than the third quartile. To locate the first quartile. we use formula [4—1]. where n = 15 and P = 25: P 25 L25= if] +Tlm=l15+1]—= 4 100 and to locate the third quartile. n = 15 and P = F5: . _ P . _ r5 _ LE.— I.n + 1] 100 — 1.15 +11 100 — 12 Therefore. the first and third quartile values are located at positions 4 and 12. respectively. The fourth value in the ordered array is \$1 .?21 and the twelfth is \$2.205. These are the first and third quartiles. 108 Chapter 4 In the above example the location formula yielded awhole number. That is. we wanted to find the first quartile and there were 15 observations. so the location for— mula indicated we should find the fourth ordered value. What if there were 20 obser- vations in the sample. that is n = 20. and we wanted to locate the first quartile? From the location formula (4—1]: F 25 L25=fn+11ﬁ=120+11m=525 We would locate the fifth value in the ordered arraji,r and then move .25 of the dis— tance between the fifth and sixth values and report that as the first quartile. Like the median. the quartile does not need to be one of the actual values in the data set. To explain further. suppose a data set contained the six values: 91. 1'5. 61. 101. 43. and 10-11. We want to locate the first quartile. We order the values from small- est to largest: £13. 61. 7’5. 91. 101. and 104. The first quartile is located at P 2:: Lzs=1n+0ﬁ=15+llﬁ The position formula tells us that the first quartile is located between the first and the second value and that it is .15 of the distance between the first and the sec- ond values. The first value is 43 and the second is E51. So the distance between these two values is 18. To locate the first quartile. we need to move .F"5 of the dis— tance between the first and second values. so .?5(18]I = 13.5. To complete the pro- cedure. we add 13.5 to the first value and report that the first quartile is 56.5. We can extend the idea to include both deciles and percentiles. To locate the 23rd percentile in a sample of 80 observations. we would lookfor the 18.03 position. P 23 L = +1—= 80+1—=18.B3 33 [n ll100 E 1|100 To find the value corresponding to the 23rd percentile. we would locate the 18th value and the 19th value and determine the distance between the two values. Next. we would multiva this difference by 0.53 and add the result to the smaller value. The result would be the 23rd percentile. With a statistical software package. it is quite easy to sort the data from smallest to largest and to locate percentiles and deciles. Both MINTI'AB and Excel output sum- mary statistics. Usted below is the lvllNlTAB output. The data are reported in \$000. It includes the first and third quartiles. as well as the mean. median. and standard devi- ation for the Whither Autoplex data [see Table 2—4). We conclude that 25 percent of the vehicles sold for less than \$20.0?‘a1 and that T5 percent sold for less than \$25305. =1.T5 lump"- IIIﬂllll'ﬂIIﬂln 'I_.jl'l 'I h. Ill" h_ H ILL- )I III— r....:w._ I. nu: at" 1:2. 11'. :.-_m. :9'» run _I Describing Data: Displaying and Exploring Data 109 The following Excel output includes the same information regarding the mean. median, and standard deviation. it will also output the quartiles: but the method of calculation is not as precise. To find the quartiles. we multiplyI the sample she by the desired percentile and report the integer of that value. To explain: in the Whit- ner Autoplea data there are fit] observations. and we wish to locate the 25th per- centile. We multiply,r n + i = 80 + i = 81 by .25: the result is 2025. Excel will not allow us to enter a fractional value. so we use 20 and request the location of the largest 2i] 1.ralues and the smallest 20 values. The result is a good approximation of the 25th and ?5th percentiles. m: .tatai uuﬂhwwwwhwnuamn- -n pattiwuiiﬂlldim-H vn Irwin-loam: .Iu-a-I llallﬂlllﬁll‘la'ﬂﬂlln'. - I 111'! II I In MINE a mu II I mm a mum 'II'i Hill a I Inn um I ! .mr 'i I In: mu F4 :3 m I mm 1W“ [i li'tﬂ it t Mvﬂtﬂl tlittttlttl a Ill-"I n I ama- am ‘I lull I 1 mum Ira-1m "I no: u I lug. man 'a i new . I Mum Ian .1". i! I. I Ian-II. 533 3g 1an 'l I an rum: "‘1! ram -. 1 ﬁll-I II 'I : an!" .a 1 LII-m nan -" ' m" H ' M ‘I 211:1 H 1 1L mu :1 1 . I II 'I Inf - -aa’ai-.‘omr I Li] .__1!_!.!- HI I! ...
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## This note was uploaded on 11/18/2010 for the course QRB QRB 501 taught by Professor Conrad during the Spring '10 term at University of Phoenix.

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Reference for Ch. 4 Quest. 12 - Other Measures of...

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