Bài giảng Statistical Techniques in Business and Economics - Chapter 3 Describing Data: Measures of Central Tendency

Chapter 3Describing Data:1.Calculate the arithmetic mean, the weighted mean, the median, the mode, and the geometric mean of a given data set.2.Identify the relative positions of the arithmetic mean, median and mode for both symmetric and skewed distributions.Chapter GoalsWhen you have completed this chapter, you will be able to:Explain your choice of the measure of central tendency of data.3.Point out the proper uses and common misuses of each measure. 4.5.Explain the result of your analysis.Five Measures of Central Tendency medianarithmetic meanweighted meanmodegeometric meanExamplesAverage price of a house in Ottawa (2000) was $126 000The average income of two parent families with children in Canada was $65,847 in 1995 and $72,910 in 1999. (StatCan)The average price of a house in Toronto in 1996 was $238,511 (StatCan)My grade point average for last semester was 4.0CharacteristicsArithmetic MeanAll values are used It is unique The sum of the deviations from the mean is 0 It is calculated by summing the values and dividing by the number of values It requires the interval scale is the most widely used measure of location.Population MeanFormula is the population mean (pronounced mu)where is the total number of observations is a particular value indicates the operation of adding (sigma)mNxTerminologyParameteris a measurable characteristic of a PopulationStatisticis a measurable characteristic of a Sample= 48 500 The Kiers family owns four cars. The following is the current mileage on each of the four cars:QFind the mean mileage for the cars.Population Mean56,000 23,000 42,000 73,000 56000 + 23000 + 42000 + 73000 4=Formula Sample Meanis the sample mean (read “x bar”)where is the number of sample observations is a particular value indicates the operation of adding (sigma)nxFormula A sample of five executives received the following bonuses last year ($000):14.0 15.0 17.0 16.0 15.0Determine the average bonus given last year:A 14 + 15 + 17 + 16 + 15 5== 15.4Question The average bonus given last year was $15 400= 77 / 5Formula Properties of an Arithmetic MeanEvery set of interval-level and ratio-level data has a mean All the values are included in computing the meanA set of data has a unique meanThe arithmetic mean is the only measure of central tendency where the sum of the deviations of each value from the mean is zero! The mean is affected by unusually large or small data values= 5 Arithmetic Mean as a Balance PointIllustrate the mean of the values 3, 8 and 4.= 15 / 3251139Determining the Mean in ExcelSeeUsingClick on ToolsClick on DATA ANALYSISSeeHighlight DESCRIPTIVE STATISTICS Click OKUsingSeeSeeUsingSee INPUT NEEDSA3:A42Click on OKSeeUsingSee SolutionAlternate solutionUsingCLICK ON PASTE FUCTIONSeeCLICK ONSee UsingSee SCROLL DOWN TO STATISTICALUsingClick on OK HIGHLIGHT AVERAGE IN RIGHT MENUSee SeeUsingSee The mean (average) is placed in the cell on the worksheet where your cursor was when you began.Weighted MeanThe weighted mean of a set of numbers x1, x2, ... xn, with corresponding weights w1, w2, ...,wn, is computed from the following formula: Example89.0$=5050.44$=)50.0($51515155+++)15.1($15+)90.0($15+)75.0($15+=wμExampleDuring a one hour period on a hot Saturday afternoon cabana boy Chris served fifty drinks. He sold: five drinks for $0.50 fifteen for $0.75fifteen for $0.90 fifteen for $1.10 Compute: - the weighted mean of the price of the drinks -The Median is the midpoint of the values after they have been ordered from the smallest to the largestThere are as many values above the median as below it in the data array The MedianNoteFor an even set of values, the median will be the arithmetic average of the two middle numbers The ages for a sample of five college students are:21, 25, 19, 20, 22The heights of four basketball players, in inches, are: 76, 73, 80, 75ExamplesThus the median is 21Thus the median is 75.5 A.B.Arranging the data in ascending order gives: 19, 20, 21, 22, 25Arranging the data in ascending order gives: 73, 75, 76, 80Properties of the Median There is a unique median for each data set It is not affected by extremely large or small values and is therefore a valuable measure of central tendency when such values occur It can be computed for ratio-level, interval-level, and ordinal-level data It can be computed for an open-ended frequency distribution if the median does not lie in an open-ended class The ModeThe Mode is the value of the observation that appears most frequently used ExampleThe exam scores for ten students are: 81, 93, 84, 75, 68, 87, 81, 75, 81, 87The score of 81 occurs the most often it is the Mode!The Geometric Mean (GM) of a set of n numbers is defined as the nth root of the product of the n numbers. The geometric mean is used to average percents, indexes, and relatives. The formula is: Geometric MeanGMxxxxnn=()()()...()123The interest rate on three bonds was 5, 21, and 4 percentThe Geometric Mean is:Geometric MeanExample49.7)4)(21)(5(3==GMThe arithmetic mean is (5+21+4)/3 =10.0 The GM gives a more conservative profit figure because it is not heavily weighted by the rate of 21percent Another use of the geometric mean is to determine the percent increase in sales, production or other business or economic series from one time period to another. Geometric MeanExamples continuedThe formula is:- 1nGM =(Value at end of period)(Value at beginning of period)The total number of females enrolled in American colleges increased from 755,000 in 1992 to 835,000 in 2000. Geometric MeanExamples continuedi.e. the Geometric Mean rate of increase is 1.27%.Determining the Median, Mode or Geometric Mean in ExcelUsingClick on ToolsSeeClick DATA ANALYSISUsingHighlight DESCRIPTIVE STATISTICS SUMMARY STATISTICS INPUT NEEDSSee SOLUTIONUsingSolutionAlternate solutionNoteThe geometric mean doesn’t show up in summary statistics!UsingCLICK ON PASTE FUCTIONSeeCLICK ONSee Using SCROLL DOWN to STATISTICALSeeUsingOR HIGHLIGHT MEDIAN IN RIGHT MENUSee UsingSee OR HIGHLIGHT GEOMETRIC MEAN IN RIGHT MENUUsingSee See HIGHLIGHT MODE IN RIGHT MENUThe calculated values are placed in the cell on the worksheet where your cursor was when you beganUsing NewCountries VisitedExpenditures ($Cdn millions)Australia227Cuba265Dominican Rep.122France506Germany183Hong Kong138Ireland114Italy283Japan150Mexico557Netherlands107Spain105Switzerland91United Kingdom1009United States8401 The following table shows the expenditures of Canadians in 15 countries they visited in 1999Source: Statistics Canada, Tourism and the Centre for Education StatisticsIs the mean or median expenditure a more accurate reflection of the “average” Canadian out-of-country expenditure?What happens to the values of the mean and median when you remove the United States expenditures from the sample?if you remove both the UK and US from the sample?UsingAThe mean is strongly affected by the inclusion of these two OUTLIERS therefore, the median is a more appropriate measure of “average” in this caseThe Mean of Grouped DataThe mean of a sample of data organized in a frequency distribution is computed by the following formula:NfxS=A sample of ten movie theatres in a metropolitan area tallied the total number of movies showing last week. Compute the mean number of movies showing per theatre. The Mean of Grouped DataExampleExampleContinued6610Total301039 to under 118817 to under 918635 to under 78423 to under 52211 to under 3(f)(x)Class MidpointFrequencyfMovies Showing The Mean of Grouped Data NfxS== 6.61066=AExampleContinued(f)(x)Class MidpointFrequencyfMovies Showing6610TotalFormula nXfS= The Mean of Grouped Data NfxS= Example 26106532.5137.527.513522.521017.562.512.530Total230 to under 35525 to under 30620 to under 251215 to under 20510 to under 15(f)(x)Class MidpointFrequencyfHours StudyingDetermine the average student study time The Mean of Grouped Data NfxS== 20.3330610=Formula NfxS=Finding the Median of Grouped Data1. Construct a cumulative frequency distributionTo determine the median class for Grouped Data:2. Divide the total number of data values by 23. Determine which class will contain this value E.g. If n = 50, 50/2 = 25, then determine which class will contain the 25th value)(2-ifCFNL + Median =where is the lower limit of the median class is the cumulative frequency as you enter the median class is the frequency of the median class is the class interval or sizeLCFfiFinding the Median of Grouped DataEstimate the median value within chosen classCumulative f10Total1039 to under 11717 to under 9635 to under 7323 to under 5111 to under 3FrequencyfMovies ShowingUsing earlier example...2- 310= 5 + = 6.3332)(2-ifCFNL + CFMedian classLfi = 2Finding the Median of Grouped DataThe Mode of Grouped DataThe mode for grouped data is approximated by the midpoint of the class with the largest class frequency1039 to under 11817 to under 9635 to under 7423 to under 5211 to under 3Class MidpointFrequencyfMovies ShowingThis is considered to be BiModalApproximate the Mode of this distributionThe Mode of Grouped Data32.527.522.517.512.530Total230 to under 35525 to under 30620 to under 251215 to under 20510 to under 15Class MidpointFrequencyfHours StudyingThe modal class is 15 to under 20, approximately 17.5 zero skewness mode = median = meanSymmetric DistributionRight Skewed DistributionMean and Median are to the right of the Mode Skewed RightPositively skewedMode<Median<MeanLeft Skewed DistributionMean and Median are to the left of the Mode Negatively skewedSkewed left< Mode< MedianMeanTest your learning www.mcgrawhill.ca/college/lindClick onOnline Learning Centrefor quizzesextra contentdata setssearchable glossaryaccess to Statistics Canada’s E-Stat dataand much more!This completes Chapter 3