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1、Chapter 4 Landmark Summaries:Interpreting Typical Values and Percentiles,Practical Business Statistics,,Chapter Topics,Measures of Central Tendency Mean, Median, Mode Midrange, Quartiles Exploratory Data Analysis Five-Number Summary Box Plot,,,,,Summary Measures,Central Tendency,,Mean,Median,Mode,Mi

2、drange,Interquartile Range,Midhinge,,,,,,,,,,,,Summary Measures,,,,Variation,,Variance,Standard Deviation,Coefficient of Variation,,,,,Range,,,,,,Measures of Central Tendency,,Central Tendency,,Mean,Median,Mode,Midrange,Midhinge,,,,,,,,,,,,,,,,Sample Mean,,,,Population Mean,The Arithmetic Average of

3、 data values:,,The Mean (Arithmetic Average),,Sample Mean,Population Mean,Sample Size,Population Size,,,The Most Common Measure of Central Tendency Affected by Extreme Values (Outliers),,,The Mean (continued),,,0 1 2 3 4 5 6 7 8 9 10,0 1 2 3 4 5 6 7 8 9 10 12 14,,,,,,,,,,,,,Mean = 5,Mean = 6,,The

4、Median,Important Measure of Central Tendency In an ordered array, the median is the “middle” number. If n is odd, the median is the middle number. If n is even, the median is the average of the 2 middle numbers.,,,The Median (continued),,,0 1 2 3 4 5 6 7 8 9 10,0 1 2 3 4 5 6 7 8 9 10 12 14,,,,,,,,,

5、,,,,Median = 5,Median = 5,Not Affected by Extreme Values For skewed data, represents the “typical case” better than the average does,,The Mode,,A Measure of Central Tendency Value that Occurs Most Often Not Affected by Extreme Values,,,,,,,,,,Mode = 8,,,,,,,,0 1 2 3 4 5 6 7 8 9 10 11 12 13,,The Mod

6、e (continued),,There May Not be a Mode There May be Several Modes Used for Either Numerical or Categorical Data,,0 1 2 3 4 5 6,,,,,,,,No Mode,,0 1 2 3 4 5 6,,,,,,,,Two Modes,,,,,,Midrange,,A Measure of Central Tendency Average of Smallest and Largest Observation:,Midrange,,Midrange (continued),,A

7、ffected by Extreme Value,,,0 1 2 3 4 5 6 7 8 9 10,0 1 2 3 4 5 6 7 8 9 10,,,,,,,,,Midrange = 5,Midrange = 3,,,,,,,Which summary to use?,Average Best for normal data Preserves totals Median Good for skewed data or data with outliers, provided you do not need to preserve or estimate total amounts Mode

8、Best for categories (nominal data). The mode is the only summary computable for nominal data!,Quartiles,Not a measure of central tendency Split ordered data into 4 quarters,,,,,25%,25%,25%,25%,Q1,Q2,Q3,,,Selected landmarks to represent entire data set Median = 50th percentile Quartiles LQ = Lower Qu

9、artile = 25th percentile Rank = UQ = Upper Quartile = 75th percentile Rank is n+1–[rank of lower quartile] Extremes Smallest = 0th percentile Largest = 100th percentile,Five-Number Summary,Five-Number Summary (continued),Provides information about Central summary Range of the data “Middle half” of t

10、he data Skewness,Exploratory Data Analysis,Box Plot Graphical display of data using 5-number summary,,,,Median(Q2),,,4,6,8,10,12,,,Q,3,,Q,1,,X,largest,,X,smallest,,Spending rank ordered from smallest to largest 0.3, 0.6, 0.9, 1.1, 1.4, 2.8, 3.8, 5.5 1 2 3 4 5 6 7 8 LQ is (0.6+0.9)/2 = 0.75

11、 UQ is (2.8+3.8)/2 = 3.3,Example: Spending,Example: Spending (continued),Five-number summary 0.3, 0.75, 1.25, 3.3, 5.5 Box plot Shows some skewness (lack of symmetry),,,Exercise,,A systems manager in charge of a company’s network keeps track of the number of server failures that occur in a day.

12、The following data represent the number of server failures in a day for the past two weeks. 3 0 3 26 2 7 4 0 2 3 3 6 3 Obtain the mode for these data,,Solution,,The ordered array for these data is: 0 0 1 2 2 3 3 3 3 3 4 6 7 26 The most typical value ,or mode, is 3. Thus, the systems manager can say

13、that the most common occurrence is to have three server failures in a day. Note that for this data set the median is equal to 3 and the arithmetic mean is equal to 4.5. The value 26 is an outlier;thus the median and the mode is a better description of central tendency than the mean.,,Exercise,,決策者一旦

14、信奉某種無效的行動方針，常會使自己所犯錯誤逐步升級。組織行為學家和社會心理學家們對這一逐步升級過程產(chǎn)生強烈興趣。諸如“沉沒成本”效應，“陷進泥沼”效應，以及“投入過多，難以自拔”效應，均屬這種現(xiàn)象。不過大多數(shù)人則把此種現(xiàn)象看作是“落入陷阱”。今有52名初學心理學的大學生參加一項實驗室實驗，旨在探究將先出現(xiàn)的結(jié)果視作自我同一性（主觀與客觀的一致性）體現(xiàn)的個人傾向，是否會加強上述落入陷阱效應（Administrative Science Quarterly, May. 1986）。整個實驗由30項試驗組成，試驗中根據(jù)學生判斷不同形狀幾何圖形的準確性打分，每項試驗的總得分見表。計算這個數(shù)據(jù)集的平均值、中位數(shù)和眾數(shù)（類）這幾個集中趨勢量度是否出現(xiàn)在數(shù)據(jù)分布的中心？,,Exercise,,5 4 7 24 6 12 11 15 11 10 23 4 20 5 4 5 6 6 15 5 15 10 13 9 4 6,,Solution,,mean=9.7 Median=7 Mode=5 b. The answer is yes,