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1、Chapter,Descriptive Statistics,2,Chapter Outline,2.1 Frequency Distributions and Their Graphs2.2 More Graphs and Displays2.3 Measures of Central Tendency2.4 Measures of Variation2.5 Measures of Position,Section 2.3,Measures of Central Tendency,.,Section 2.3 Objectives,How to find the mean, median, a
2、nd mode of a population and of a sampleHow to find the weighted mean of a data set, and how to estimate the sample mean of grouped dataHow 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 Tendency,Measure o
3、f central tendencyA value that represents a typical, or central, entry of a data set.Most common measures of central tendency:MeanMedianMode,.,Measure of Central Tendency: Mean,Mean (average)The sum of all the data entries divided by the number of entries.Sigma notation: x = add all of the data entr
4、ies (x) in the data set.Population mean:Sample mean:,.,Example: Finding a Sample Mean,The 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 weig
5、hts isx = 274 + 235 + 223 + 268 + 290 + 285 + 235 = 1810To 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.,., = = 1810 7 258.6,274 235 223 268 290 285 235,Measure of Central Tendency: Median,MedianThe val
6、ue 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 anodd number of entries: median is the middle data entry.even number of entries: median is the mean of the two middle data entries.,.
7、,Example: Finding the Median,Find the median of the weight listed in the first example. 274 235 223 268 290 285 235,.,Solution: Finding the Median,First, order the data.223 235 235 268 274 285 290There are seven entries (an odd number), the median is the middle, or fourth, data entry.,The median wei
8、ght of the adults is 268 pounds.,.,Example: Finding the Median,In the previous example, 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
9、 274 290There are six entries (an even number), the median is the mean of the two middle entries.,The median weight of the remaining adults is 251.5 pounds.,.,Median= 235 +268 2 =251.5,Measure of Central Tendency: Mode,ModeThe data entry that occurs with the greatest frequency.If no entry is repeate
10、d the data set has no mode.If two entries occur with the same greatest frequency, each entry is a mode (bimodal).,.,Example: Finding the Mode,Find 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
11、 268 274 285 290The entry of 235 occurs twice, whereas the other data entries occur only once.,The mode of the weights is 235 pounds.,.,Example: Finding the Mode,At a political debate a sample of audience members was asked to name the political party to which they belong. Their responses are shown i
12、n the table. What is the mode of the responses?,.,Solution: Finding the Mode,The 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,All thr
13、ee 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 set).
14、,.,Example: Comparing the Mean, Median, and Mode,The 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?,.,Solution: Comparing the Mean, Median, and
15、 Mode,Mean:,Median:,20 years (the entry occurring with thegreatest frequency),Mode:,.,Solution: Comparing the Mean, Median, and Mode,Mean 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
16、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 Mode,Sometimes a graphical comparison can help you decide which measure of central tendency best represents a data set.,In
17、this case, it appears that the median best describes the data set.,.,Weighted Mean,Weighted MeanThe 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.,.,Example: Finding a Weighted Mean,Your grades from last semester are in the
18、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 Mean,Last semester, your grade point average was 2.5.,., = = 40 16 = 2.5,Mean of Grouped Data,Mean of a Frequency DistributionApp
19、roximated by where x and f are the midpoints and frequencies of a class, respectively.,.,Finding the Mean of a Frequency Distribution,In Words In Symbols,Find the midpoint of each class.Find the sum of the products of the midpoints and the frequencies.Find the sum of the frequencies.Find the mean of
20、 the frequency distribution.,.,Example: Find the Mean of a Frequency Distribution,The 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 sample mean for
21、mula, the mean expense is $285.50. Compare this with the estimated mean.,.,Solution: Find the Mean of a Frequency Distribution,., = = 8631 30 = 287.7,The mean expense is $287.70. This value is an estimate because it is based on class midpoints instead of the original data set.,The Shape of Distribut
22、ions,Symmetric 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 Distributions,Uniform Distribution (rectangular)All entries or classes in the distribution have equal or approximately eq
23、ual frequencies.Symmetric.,.,The Shape of Distributions,Skewed 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 Distributions,Skewed Right Distribution (positively skewed)The “tail” of the graph elongates more to the right.The mean is to the right of the median.,.,
