1、1Copyright 2019,2015,2012,Pearson Education,Inc.ChapterDescriptive Statistics22Copyright 2019,2015,2012,Pearson Education,Inc.Chapter Outline 2.1 Frequency Distributions and Their Graphs 2.2 More Graphs and Displays 2.3 Measures of Central Tendency 2.4 Measures of Variation 2.5 Measures of Position3
2、Copyright 2019,2015,2012,Pearson Education,Inc.Section 2.3Measures of Central Tendency.4Copyright 2019,2015,2012,Pearson Education,Inc.Section 2.3 Objectives How to find the mean,median,and mode of a population and of a sample How to find the weighted mean of a data set,and how to estimate the sampl
3、e mean of grouped data How to describe the shape of a distribution as symmetric,uniform,or skewed and how to compare the mean and median for each.5Copyright 2019,2015,2012,Pearson Education,Inc.Measures of Central TendencyMeasure of central tendency A value that represents a typical,or central,entry
4、 of a data set.Most common measures of central tendency:Mean Median Mode.6Copyright 2019,2015,2012,Pearson Education,Inc.Measure of Central Tendency:MeanMean(average)The sum of all the data entries divided by the number of entries.Sigma notation:x=add all of the data entries(x)in the data set.Popula
5、tion mean:Sample mean:xNxxn.7Copyright 2019,2015,2012,Pearson Education,Inc.Example:Finding a Sample MeanThe weights(in pounds)for a sample of adults before starting a weight-loss study are listed.What is the mean weight of the adults?274 235 223 268 290 285 235.8Copyright 2019,2015,2012,Pearson Edu
6、cation,Inc.Solution:Finding a Sample Mean The sum of the weights isx=274+235+223+268+290+285+235=1810 To find the mean weight,divide the sum of the weights by the number of adults in the sample.The mean weight of the adults is about 258.6 pounds.274 235 223 268 290 285 2359Copyright 2019,2015,2012,P
7、earson Education,Inc.Measure of Central Tendency:MedianMedian The value that lies in the middle of the data when the data set is ordered.Measures the center of an ordered data set by dividing it into two equal parts.If the data set has an odd number of entries:median is the middle data entry.even nu
8、mber of entries:median is the mean of the two middle data entries.10Copyright 2019,2015,2012,Pearson Education,Inc.Example:Finding the MedianFind the median of the weight listed in the first example.274 235 223 268 290 285 235.11Copyright 2019,2015,2012,Pearson Education,Inc.Solution:Finding the Med
9、ian First,order the data.223 235 235 268 274 285 290 There are seven entries(an odd number),the median is the middle,or fourth,data entry.The median weight of the adults is 268 pounds.12Copyright 2019,2015,2012,Pearson Education,Inc.Example:Finding the MedianIn the previous example,the adult weighin
10、g 285 pounds decides to not participate in the study.What is the median weight of the remaining adults?223 235 235 268 274 290.13Copyright 2019,2015,2012,Pearson Education,Inc.Solution:Finding the Median First order the data.223 235 235 268 274 290 There are six entries(an even number),the median is
11、 the mean of the two middle entries.The median weight of the remaining adults is 251.5 pounds.14Copyright 2019,2015,2012,Pearson Education,Inc.Measure of Central Tendency:ModeMode The data entry that occurs with the greatest frequency.If no entry is repeated the data set has no mode.If two entries o
12、ccur with the same greatest frequency,each entry is a mode(bimodal).15Copyright 2019,2015,2012,Pearson Education,Inc.Example:Finding the ModeFind the mode of the weights listed in Example 1.223 235 235 268 274 285 290.16Copyright 2019,2015,2012,Pearson Education,Inc.Solution:Finding the Mode Orderin
13、g the data helps to find the mode.223 235 235 268 274 285 290 The entry of 235 occurs twice,whereas the other data entries occur only once.The mode of the weights is 235 pounds.17Copyright 2019,2015,2012,Pearson Education,Inc.Example:Finding the ModeAt a political debate a sample of audience members
14、 was asked to name the political party to which they belong.Their responses are shown in the table.What is the mode of the responses?Political PartyFrequency,fDemocrat46Republican34Independent39Other/dont know5.18Copyright 2019,2015,2012,Pearson Education,Inc.Political PartyFrequency,fDemocrat46Repu
15、blican34Independent39Other/dont know5Solution:Finding the ModeThe response occurring with the greatest frequency is Democrat.So,the mode is Democrat.In this sample,there were more Democrats than people of any other single affiliation.19Copyright 2019,2015,2012,Pearson Education,Inc.Comparing the Mea
16、n,Median,and Mode All three measures describe a typical entry of a data set.Advantage of using the mean:The mean is a reliable measure because it takes into account every entry of a data set.Disadvantage of using the mean:Greatly affected by outliers(a data entry that is far removed from the other e
17、ntries in the data set).20Copyright 2019,2015,2012,Pearson Education,Inc.Example:Comparing the Mean,Median,and ModeThe table shows the sample ages of students in a class.Find the mean,median,and mode of the ages.Are there any outliers?Which measure of central tendency best describes a typical entry
18、of this data set?Ages in a class2020202020202121212122222223232323242465.21Copyright 2019,2015,2012,Pearson Education,Inc.Solution:Comparing the Mean,Median,and ModeMean:2020.246523.8 years20 xxnMedian:212221.5 years220 years(the entry occurring with thegreatest frequency)Ages in a class202020202020
19、2121212122222223232323242465Mode:.22Copyright 2019,2015,2012,Pearson Education,Inc.Solution:Comparing the Mean,Median,and ModeMean 23.8 years Median=21.5 years Mode=20 years The mean takes every entry into account,but is influenced by the outlier of 65.The median also takes every entry into account,
20、and it is not affected by the outlier.In this case the mode exists,but it doesnt appear to represent a typical entry.23Copyright 2019,2015,2012,Pearson Education,Inc.Solution:Comparing the Mean,Median,and ModeSometimes a graphical comparison can help you decide which measure of central tendency best
21、 represents a data set.In this case,it appears that the median best describes the data set.24Copyright 2019,2015,2012,Pearson Education,Inc.Weighted MeanWeighted Mean The mean of a data set whose entries have varying weights.The weighted mean is given by where w is the weight of each entry x.xwxw.25
22、Copyright 2019,2015,2012,Pearson Education,Inc.Example:Finding a Weighted MeanYour grades from last semester are in the table.The grading system assigns points as follows:A=4,B=3,C=2,D=1,F=0.Determine your grade point average(weighted mean).26Copyright 2019,2015,2012,Pearson Education,Inc.Solution:F
23、inding a Weighted MeanLast semester,your grade point average was 2.5.27Copyright 2019,2015,2012,Pearson Education,Inc.Mean of Grouped DataMean of a Frequency Distribution Approximated by where x and f are the midpoints and frequencies of a class,respectively.xfxnfn.28Copyright 2019,2015,2012,Pearson
24、 Education,Inc.Finding the Mean of a Frequency DistributionIn Words In Symbolsxfxn(Lower limit)+(Upper limit)2x xfnf 1.Find the midpoint of each class.2.Find the sum of the products of the midpoints and the frequencies.3.Find the sum of the frequencies.4.Find the mean of the frequency distribution.2
25、9Copyright 2019,2015,2012,Pearson Education,Inc.Example:Find the Mean of a Frequency DistributionThe frequency distribution shows the out-of-pocket prescription medicine expenses(in dollars)for 30 U.S.adults in a recent year.Use the frequency distribution to estimate the mean expense.Using the sampl
26、e mean formula,the mean expense is$285.50.Compare this with the estimated mean.30Copyright 2019,2015,2012,Pearson Education,Inc.Solution:Find the Mean of a Frequency Distribution.The mean expense is$287.70.This value is an estimate because it is based on class midpoints instead of the original data
27、set.31Copyright 2019,2015,2012,Pearson Education,Inc.The Shape of DistributionsSymmetric DistributionA vertical line can be drawn through the middle of a graph of the distribution and the resulting halves are approximately mirror images.32Copyright 2019,2015,2012,Pearson Education,Inc.The Shape of D
28、istributionsUniform Distribution(rectangular)All entries or classes in the distribution have equal or approximately equal frequencies.Symmetric.33Copyright 2019,2015,2012,Pearson Education,Inc.The Shape of DistributionsSkewed Left Distribution(negatively skewed)The“tail”of the graph elongates more to the left.The mean is to the left of the median.34Copyright 2019,2015,2012,Pearson Education,Inc.The Shape of DistributionsSkewed Right Distribution(positively skewed)The“tail”of the graph elongates more to the right.The mean is to the right of the median.
