Business Statistics A DecisionMaking Approach, 7th …：业务统计决策方法第七…

1、business statistics: a decision-making approach, 7e 2008 prentice-hall, inc.chap 3-1 business statistics: a decision-making approach 7th edition chapter 3 describing data using numerical measures business statistics: a decision-making approach, 7e 2008 prentice-hall, inc.chap 3-2 after completing th

2、is chapter, you should be able to: ncompute and interpret the mean, median, and mode for a set of data ncompute the range, variance, and standard deviation and know what these values mean nconstruct and interpret a box and whisker graph ncompute and explain the coefficient of variation and z scores

3、nuse numerical measures along with graphs, charts, and tables to describe data chapter goals business statistics: a decision-making approach, 7e 2008 prentice-hall, inc.chap 3-3 chapter topics nmeasures of center and location nmean, median, mode nother measures of location nweighted mean, percentile

4、s, quartiles nmeasures of variation nrange, interquartile range, variance and standard deviation, coefficient of variation nusing the mean and standard deviation together ncoefficient of variation, z-scores business statistics: a decision-making approach, 7e 2008 prentice-hall, inc.chap 3-4 summary

5、measures center and location mean median mode other measures of location weighted mean describing data numerically variation variance standard deviation coefficient of variation range percentiles interquartile range quartiles business statistics: a decision-making approach, 7e 2008 prentice-hall, in

6、c.chap 3-5 measures of center and location center and location meanmedianmodeweighted mean n x n x x n i i n i i 1 1 i ii w i ii w w xw w xw x overview business statistics: a decision-making approach, 7e 2008 prentice-hall, inc.chap 3-6 mean (arithmetic average) nthe mean is the arithmetic average o

7、f data values npopulation mean nsample mean n = sample size n = population size n xxx n x x n n i i 211 n xxx n x n n i i 211 business statistics: a decision-making approach, 7e 2008 prentice-hall, inc.chap 3-7 mean (arithmetic average) nthe most common measure of central tendency nmean = sum of val

8、ues divided by the number of values naffected by extreme values (outliers) (continued) 0 1 2 3 4 5 6 7 8 9 10 mean = 3 0 1 2 3 4 5 6 7 8 9 10 mean = 4 3 5 15 5 54321 4 5 20 5 104321 business statistics: a decision-making approach, 7e 2008 prentice-hall, inc.chap 3-8 median nin an ordered array, the

9、median is the “middle” number, i.e., the number that splits the distribution in half nthe median is not affected by extreme values 0 1 2 3 4 5 6 7 8 9 10 median = 3 0 1 2 3 4 5 6 7 8 9 10 median = 3 business statistics: a decision-making approach, 7e 2008 prentice-hall, inc.chap 3-9 median nto find

10、the median, sort the n data values from low to high (sorted data is called a data array) nfind the value in the i = (1/2)n position nthe ith position is called the median index point nif i is not an integer, round up to next highest integer (continued) business statistics: a decision-making approach

11、, 7e 2008 prentice-hall, inc.chap 3-10 median example nnote that n = 13 nfind the i = (1/2)n position: i = (1/2)(13) = 6.5 nsince 6.5 is not an integer, round up to 7 nthe median is the value in the 7th position: md = 12 (continued) data array: 4, 4, 5, 5, 9, 11, 12, 14, 16, 19, 22, 23, 24 business

12、statistics: a decision-making approach, 7e 2008 prentice-hall, inc.chap 3-11 shape of a distribution ndescribes how data is distributed nsymmetric or skewed mean = median mean median median mean right-skewed left-skewed symmetric (longer tail extends to left)(longer tail extends to right) business s

13、tatistics: a decision-making approach, 7e 2008 prentice-hall, inc.chap 3-12 mode na measure of location nthe value that occurs most often nnot affected by extreme values nused for either numerical or categorical data nthere may be no mode nthere may be several modes 0 1 2 3 4 5 6 7 8 9 10 11 12 13 1

14、4 mode = 5 0 1 2 3 4 5 6 no mode business statistics: a decision-making approach, 7e 2008 prentice-hall, inc.chap 3-13 weighted mean nused when values are grouped by frequency or relative importance days to complete frequency 54 612 78 82 example: sample of 26 repair projects weighted mean days to c

15、omplete: days 6.31 26 164 28124 8)(27)(86)(125)(4 w xw x i ii w business statistics: a decision-making approach, 7e 2008 prentice-hall, inc.chap 3-14 nfive houses on a hill by the beach review example $2,000 k $500 k $300 k $100 k $100 k house prices: $2,000,000 500,000 300,000 100,000 100,000 busin

16、ess statistics: a decision-making approach, 7e 2008 prentice-hall, inc.chap 3-15 summary statistics nmean: ($3,000,000/5) = $600,000 nmedian: middle value of ranked data = $300,000 nmode: most frequent value = $100,000 house prices: $2,000,000 500,000 300,000 100,000 100,000 sum 3,000,000 business s

17、tatistics: a decision-making approach, 7e 2008 prentice-hall, inc.chap 3-16 nmean is generally used, unless extreme values (outliers) exist nthen median is often used, since the median is not sensitive to extreme values. nexample: median home prices may be reported for a region less sensitive to out

18、liers which measure of location is the “best”? business statistics: a decision-making approach, 7e 2008 prentice-hall, inc.chap 3-17 other location measures other measures of location percentilesquartiles n1st quartile = 25th percentile n2nd quartile = 50th percentile = median n3rd quartile = 75th p

19、ercentile the pth percentile in a data array: np% are less than or equal to this value n(100 p)% are greater than or equal to this value (where 0 p 100) business statistics: a decision-making approach, 7e 2008 prentice-hall, inc.chap 3-18 percentiles nthe pth percentile in an ordered array of n valu

20、es is the value in ith position, where nexample: find the 60th percentile in an ordered array of 19 values. (n) 100 p i 11.4(19) 100 60 (n) 100 p i if i is not an integer, round up to the next higher integer value so use value in the i = 12th position business statistics: a decision-making approach,

21、 7e 2008 prentice-hall, inc.chap 3-19 quartiles nquartiles split the ranked data into 4 equal groups: nnote that the second quartile (the 50th percentile) is the median 25%25%25%25% q1q2q3 business statistics: a decision-making approach, 7e 2008 prentice-hall, inc.chap 3-20 quartiles sample data in

22、ordered array: 11 12 13 16 16 17 18 21 22 nexample: find the first quartile (n = 9) q1 = 25th percentile, so find i : i = (9) = 2.25 so round up and use the value in the 3rd position: q1 = 13 25 100 business statistics: a decision-making approach, 7e 2008 prentice-hall, inc.chap 3-21 box and whisker

23、 plot na graphical display of data using a central “box” and extended “whiskers”: example: 25% 25% 25% 25% outliers lower 1st median 3rd upper limit quartile quartile limit * business statistics: a decision-making approach, 7e 2008 prentice-hall, inc.chap 3-22 constructing the box and whisker plot o

24、utliers lower 1st median 3rd upper limit quartile quartile limit * the lower limit is q1 1.5 (q3 q1) the upper limit is q3 + 1.5 (q3 q1) nthe center box extends from q1 to q3 nthe line within the box is the median nthe whiskers extend to the smallest and largest values within the calculated limits n

25、outliers are plotted outside the calculated limits business statistics: a decision-making approach, 7e 2008 prentice-hall, inc.chap 3-23 shape of box and whisker plots nthe box and central line are centered between the endpoints if data is symmetric around the median n(a box and whisker plot can be

26、shown in either vertical or horizontal format) business statistics: a decision-making approach, 7e 2008 prentice-hall, inc.chap 3-24 distribution shape and box and whisker plot right-skewedleft-skewedsymmetric q1q2q3q1q2q3 q1 q2 q3 business statistics: a decision-making approach, 7e 2008 prentice-ha

27、ll, inc.chap 3-25 box-and-whisker plot example nbelow is a box-and-whisker plot for the following data: 0 2 2 2 3 3 4 5 6 11 27 nthis data is right skewed, as the plot depicts 0 2 3 6 12 27 min q1 q2 q3 max * upper limit = q3 + 1.5 (q3 q1) = 6 + 1.5 (6 2) = 12 27 is above the upper limit so is shown

28、 as an outlier business statistics: a decision-making approach, 7e 2008 prentice-hall, inc.chap 3-26 measures of variation variation variancestandard deviationcoefficient of variation population variance sample variance population standard deviation sample standard deviation range interquartile rang

29、e business statistics: a decision-making approach, 7e 2008 prentice-hall, inc.chap 3-27 nmeasures of variation give information on the spread or variability of the data values. variation same center, different variation business statistics: a decision-making approach, 7e 2008 prentice-hall, inc.chap

30、 3-28 range nsimplest measure of variation ndifference between the largest and the smallest observations: range = xmaximum xminimum 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 range = 14 - 1 = 13 example: business statistics: a decision-making approach, 7e 2008 prentice-hall, inc.chap 3-29 nignores the way i

31、n which data are distributed nsensitive to outliers 7 8 9 10 11 12 range = 12 - 7 = 5 7 8 9 10 11 12 range = 12 - 7 = 5 disadvantages of the range 1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,5 1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,120 range = 5 - 1 = 4 range = 120 - 1 = 119 business st

32、atistics: a decision-making approach, 7e 2008 prentice-hall, inc.chap 3-30 interquartile range ncan eliminate some outlier problems by using the interquartile range neliminate some high-and low-valued observations and calculate the range from the remaining values. ninterquartile range = 3rd quartile

33、 1st quartile business statistics: a decision-making approach, 7e 2008 prentice-hall, inc.chap 3-31 interquartile range example median (q2) x maximum x minimum q1q3 example: 25% 25% 25% 25% 12 30 45 57 70 interquartile range = 57 30 = 27 business statistics: a decision-making approach, 7e 2008 prent

34、ice-hall, inc.chap 3-32 naverage of squared deviations of values from the mean npopulation variance: nsample variance: variance n )(x n 1i 2 i 2 1- n )x(x s n 1i 2 i 2 business statistics: a decision-making approach, 7e 2008 prentice-hall, inc.chap 3-33 standard deviation nmost commonly used measure

35、 of variation nshows variation about the mean nhas the same units as the original data npopulation standard deviation: nsample standard deviation: n )(x n 1i 2 i 1-n )x(x s n 1i 2 i business statistics: a decision-making approach, 7e 2008 prentice-hall, inc.chap 3-34 calculation example: sample stan

36、dard deviation sample data (xi) : 10 12 14 15 17 18 18 24 n = 8 mean = x = 16 4.3095 7 130 18 16)(2416)(1416)(1216)(10 1n )x(24)x(14)x(12)x(10 s 2222 2222 business statistics: a decision-making approach, 7e 2008 prentice-hall, inc.chap 3-35 comparing standard deviations mean = 15.5 s = 3.338 11 12 1

37、3 14 15 16 17 18 19 20 21 11 12 13 14 15 16 17 18 19 20 21 data b data a mean = 15.5 s = .9258 11 12 13 14 15 16 17 18 19 20 21 mean = 15.5 s = 4.57 data c same mean, but different standard deviations: business statistics: a decision-making approach, 7e 2008 prentice-hall, inc.chap 3-36 coefficient

38、of variation nmeasures relative variation nalways in percentage (%) nshows variation relative to mean nis used to compare two or more sets of data measured in different units 100% x s cv 100% cv population sample business statistics: a decision-making approach, 7e 2008 prentice-hall, inc.chap 3-37 c

39、omparing coefficients of variation nstock a: naverage price last year = $50 nstandard deviation = $5 nstock b: naverage price last year = $100 nstandard deviation = $5 both stocks have the same standard deviation, but stock b is less variable relative to its price 10%100% $50 $5 100% x s cv a 5%100%

40、 $100 $5 100% x s cv b business statistics: a decision-making approach, 7e 2008 prentice-hall, inc.chap 3-38 nif the data distribution is bell-shaped, then the interval: ncontains about 68% of the values in the population or the sample the empirical rule 1 68% 1 business statistics: a decision-makin

41、g approach, 7e 2008 prentice-hall, inc.chap 3-39 ncontains about 95% of the values in the population or the sample ncontains about 99.7% of the values in the population or the sample the empirical rule 2 3 3 99.7%95% 2 business statistics: a decision-making approach, 7e 2008 prentice-hall, inc.chap

42、3-40 nregardless of how the data are distributed, at least (1 - 1/k2) of the values will fall within k standard deviations of the mean n examples: (1 - 1/12) = 0% . k=1 ( 1) (1 - 1/22) = 75% . k=2 ( 2) (1 - 1/32) = 89% . k=3 ( 3) tchebysheffs theorem withinat least business statistics: a decision-ma

43、king approach, 7e 2008 prentice-hall, inc.chap 3-41 na standardized data value refers to the number of standard deviations a value is from the mean nstandardized data values are sometimes referred to as z-scores standardized data values business statistics: a decision-making approach, 7e 2008 prenti

44、ce-hall, inc.chap 3-42 where: nx = original data value n = population mean n = population standard deviation nz = standard score (number of standard deviations x is from ) standardized population values x z business statistics: a decision-making approach, 7e 2008 prentice-hall, inc.chap 3-43 where:

45、nx = original data value nx = sample mean ns = sample standard deviation nz = standard score (number of standard deviations x is from ) standardized sample values s xx z business statistics: a decision-making approach, 7e 2008 prentice-hall, inc.chap 3-44 niq scores in a large population have a bell- shaped distribution with mean = 100 and standard deviation = 15 find the standardized score (z-score) for a person with an iq of 121. someone with an iq of 121 is 1.