1、统计学基础(英文版第7版)教学课件les7e_ppt_02_03Chapter 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 PositionSection 2.3Measures of Central Tendency.Section 2.3 Objectives How to find the mean,median,and
2、mode of a population and of a sample How to find the weighted mean of a data set,and how to estimate the sample 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.Measures of Central TendencyMeasure of centr
3、al tendency A value that represents a typical,or central,entry of a data set.Most common measures of central tendency:Mean Median Mode.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 d
4、ata set.Population mean:Sample mean:xNxxn.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.Solution:Finding a Sample Mean The sum of the weights isx=274+235+223+2
5、68+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 235Measure of Central Tendency:MedianMedian The value that lies in the middle of the data when the data set is o
6、rdered.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 number of entries:median is the mean of the two middle data entries.Example:Finding the MedianFind the median of the weight listed i
7、n the first example.274 235 223 268 290 285 235.Solution:Finding the Median 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.Example:Finding the MedianIn the previous ex
8、ample,the adult weighing 285 pounds decides to not participate in the study.What is the median weight of the remaining adults?223 235 235 268 274 290.Solution:Finding the Median First order the data.223 235 235 268 274 290 There are six entries(an even number),the median is the mean of the two middl
9、e entries.The median weight of the remaining adults is 251.5 pounds.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 occur with the same greatest frequency,each entry is a mode(bimodal).Example
10、:Finding the ModeFind the mode of the weights listed in Example 1.223 235 235 268 274 285 290.Solution:Finding the Mode Ordering 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 pou
11、nds.Example:Finding the ModeAt a political debate a sample of audience members 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.Political P
12、artyFrequency,fDemocrat46Republican34Independent39Other/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.Comparing the Mean,Median,and Mode Al
13、l 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 entries in the data s
14、et).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 of this data set?Ages in a class2020202020202121212122222223232323242
15、465.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 class2020202020202121212122222223232323242465Mode:.Solution:Comparing the Mean,Median,and ModeMean 23.8 years Median=21.5 years Mode=20
16、 years The mean takes every entry into account,but is influenced by the outlier of 65.The median also takes every entry into account,and it is not affected by the outlier.In this case the mode exists,but it doesnt appear to represent a typical entry.Solution:Comparing the Mean,Median,and ModeSometim
17、es a graphical comparison can help you decide which measure of central tendency best represents a data set.In this case,it appears that the median best describes the data set.Weighted MeanWeighted Mean The mean of a data set whose entries have varying weights.The weighted mean is given by where w is
18、 the weight of each entry x.xwxw.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).Solution:Finding a Weighted MeanLast semester,your grade point average wa
19、s 2.5.Mean of Grouped DataMean of a Frequency Distribution Approximated by where x and f are the midpoints and frequencies of a class,respectively.xfxnfn.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
20、 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.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
21、 a recent year.Use the frequency distribution to estimate the mean expense.Using the sample mean formula,the mean expense is$285.50.Compare this with the estimated mean.Solution:Find the Mean of a Frequency Distribution.The mean expense is$287.70.This value is an estimate because it is based on clas
22、s midpoints instead of the original data set.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.The Shape of DistributionsUniform Distribution(rectangular)All entries
23、or classes in the distribution have equal or approximately equal frequencies.Symmetric.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.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.
