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Chapter 3 Measures of Central Tendency 1 © McGraw-Hill, Bluman, 5 th ed, Chapter 3

Central Tendency

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Chapter 3

Chapter 3Measures of Central Tendency1 McGraw-Hill, Bluman, 5th ed, Chapter 31Chapter 3 Overview Introduction3-1 Measures of Central Tendency3-2 Measures of Variation2Bluman, Chapter 32Chapter 3 ObjectivesSummarize data using measures of central tendency.Describe data using measures of variation.3Bluman, Chapter 33IntroductionTraditional StatisticsAverageVariation4Bluman, Chapter 343.1 Measures of Central TendencyA statistic is a characteristic or measure obtained by using the data values from a sample.A parameter is a characteristic or measure obtained by using all the data values for a specific population.5Bluman, Chapter 35Numbers that describe what the average of the distribution is

You can think of this value as where the middle of a distribution lies.What is a measure of Central Tendency?Measures of Central TendencyGeneral Rounding RuleThe basic rounding rule is that rounding should not be done until the final answer is calculated. Use of parentheses on calculators or use of spreadsheets help to avoid early rounding error.

7Bluman, Chapter 37Measures of Central Tendency: MeanThe mean is the quotient of the sum of the values and the total number of values. The symbol is used for sample mean.

For a population, the Greek letter (mu) is used for the mean.

8Bluman, Chapter 38Example 3-1: Days Off per YearThe data represent the number of days off per year for a sample of individuals selected from nine different countries. Find the mean.20, 26, 40, 36, 23, 42, 35, 24, 309Bluman, Chapter 3

The mean number of days off is ___ years.9Measures of Central Tendency: Mean for Grouped Data fidi X = AM + N i

Where:AM= Assumed mean (Xmid of the class interval assumed to be containing the mean)F= frequency of a class intervald= number of deviations of the class intervals from the assumed meani= size of the class intervalsN= total number of scores 10Bluman, Chapter 310Measures of Central Tendency: Mean for Grouped Data fidi X = AM + N iwhere xi - xAd = i11Bluman, Chapter 311Measures of Central Tendency: Mean for Grouped DataThe mean for grouped data is calculated by multiplying the frequencies and midpoints of the classes.

12Bluman, Chapter 3

12Example 3-3: Miles RunClass BoundariesFrequency 5.5 - 10.510.5 - 15.515.5 - 20.520.5 - 25.525.5 - 30.530.5 - 35.535.5 - 40.51235432Below is a frequency distribution of miles run per week. Find the mean.f = 2013Bluman, Chapter 313Example 3-3: Miles RunClass IntervalsFrequency, fMidpoint, Xm6-1011-1516-2021-2526-3031-3536-4012354328131823283338f = 2082654115112997614Bluman, Chapter 3f Xm f Xm =14Measures of Central Tendency: MedianThe median is the midpoint of the data array. The symbol for the median is MD.The median will be one of the data values if there is an odd number of values.The median will be the average of two data values if there is an even number of values.15Bluman, Chapter 315Example 3-4: Hotel RoomsThe number of rooms in the seven hotels in downtown Pittsburgh is 713, 300, 618, 595, 311, 401, and 292. Find the median.

Sort in ascending order.292, 300, 311, 401, 596, 618, 713

Select the middle value.MD = 401

16Bluman, Chapter 3The median is 401 rooms.16Example 3-6: Tornadoes in the U.S.The number of tornadoes that have occurred in the United States over an 8-year period follows. Find the median.684, 764, 656, 702, 856, 1133, 1132, 1303

Find the average of the two middle values.656, 684, 702, 764, 856, 1132, 1133, 130317Bluman, Chapter 3The median number of tornadoes is 810.

17Median for grouped dataBluman, Chapter 318

Where:

L = lower real limit of the median class N = total number of scores in the distributionCf = cumulative frequency of the class interval before reaching the median class starting from the lowest class intervalf = frequency of the class interval containing the mediani = size of the class intervalMeasures of Central Tendency: ModeThe mode is the value that occurs most often in a data set.It is sometimes said to be the most typical case.There may be no mode, one mode (unimodal), two modes (bimodal), or many modes (multimodal). 19Bluman, Chapter 319Example 3-9: NFL Signing BonusesFind the mode of the signing bonuses of eight NFL players for a specific year. The bonuses in millions of dollars are18.0, 14.0, 34.5, 10, 11.3, 10, 12.4, 10

You may find it easier to sort first.10, 10, 10, 11.3, 12.4, 14.0, 18.0, 34.5

Select the value that occurs the most.20Bluman, Chapter 3The mode is 10 million dollars.20Example 3-10: Coal Employees in PAFind the mode for the number of coal employees per county for 10 selected counties in southwestern Pennsylvania.110, 731, 1031, 84, 20, 118, 1162, 1977, 103, 752

No value occurs more than once.21Bluman, Chapter 3There is no mode.21Example 3-11: Licensed Nuclear ReactorsThe data show the number of licensed nuclear reactors in the United States for a recent 15-year period. Find the mode.104 104 104 104 104 107 109 109 109 110109 111 112 111 109

104 and 109 both occur the most. The data set is said to be bimodal.22Bluman, Chapter 3The modes are 104 and 109.104 104 104 104 104 107 109 109 109 110109 111 112 111 109

22Example 3-12: Miles Run per WeekFind the modal class for the frequency distribution of miles that 20 runners ran in one week.23Bluman, Chapter 3The modal class is20.5 25.5.ClassFrequency5.5 10.5110.5 15.5215.5 20.5320.5 25.5525.5 30.5430.5 35.5335.5 40.52The mode, the midpointof the modal class, is 23 miles per week.23Mode for Grouped Data d1 Mo = Lmo + i d1 + d2Where : Mo =mode L = lower class boundary of the modal class d1= difference between the frequency of the modal class and the frequency of the class next lower in value d2= difference between the frequency of the modal class and the frequency of the class next higher in value i= class widthBluman, Chapter 324The following is the distribution of length of service in years of 50 employees of United Laboratories Inc.Compute for the mean, median and mode.Bluman, Chapter 325Length of Service(Years)Number of Employees1-556-10711-151216-201321-25626-30431-353Answers:x = 810/50 ; 18 + [(-18/50) 5] = 16.2

Mdn = 15.5 + {[(50/2)-24] / 13}5 = 15.88

Mo = 15.5 + {[1/ (1+7)] 5} = 16.13Bluman, Chapter 326The following is a frequency distribution of an entrance examination. Compute the measures of central tendency.Bluman, Chapter 327ScoresNumber of Employees21-30831-401141-501551-601861-702071-801281-90991-1007Answers:x = 5930 / 100 = 59.3

Mdn = 50.5 + {[(100/2) -34] / 18}10 = 59.39

Mo = 60.5 + [2/ (2+8)]10 = 62.5Bluman, Chapter 328Measures of Central Tendency: Weighted MeanFind the weighted mean of a variable by multiplying each value by its corresponding weight and dividing the sum of the products by the sum of the weights.

29Bluman, Chapter 329Example 3-17: Grade Point AverageA student received the following grades. Find the corresponding GPA.30Bluman, Chapter 3The grade point average is 2.7.

CourseCredits, wGrade, XEnglish Composition3 A (4 points)Introduction to Psychology3 C (2 points)Biology4 B (3 points)Physical Education2 D (1 point)

30Properties of the MeanUses all data values.Varies less than the median or mode Used in computing other statistics, such as the varianceUnique, usually not one of the data valuesCannot be used with open-ended classesAffected by extremely high or low values, called outliers31Bluman, Chapter 331Properties of the MedianGives the midpointUsed when it is necessary to find out whether the data values fall into the upper half or lower half of the distribution.Affected less than the mean by extremely high or extremely low values.32Bluman, Chapter 332Properties of the ModeUsed when the most typical case is desiredEasiest to computeCan be used with nominal dataNot always unique or may not exist33Bluman, Chapter 333Considerations for Choosing a Measure of Central TendencyFor a nominal variable, the mode is the only measure that can be used.For ordinal variables, the mode and the median may be used. The median provides more information (taking into account the ranking of categories.)For interval-ratio variables, the mode, median, and mean may all be calculated. The mean provides the most information about the distribution, but the median is preferred if the distribution is skewed.Bluman, Chapter 335Measure of Central TendencyAdvantagesDisadvantagesMeanAll the data are used to find the central tendencyCan be affected by outliersMedianVery big and very small values dont affect itTakes a long time to calculate for a very large set of dataModeThe only measure of central tendency to be used for non-numerical dataThere may be more than one modeThere may be no mode at allIt may not accurately represent the dataThe values of 19 houses in San Marcelino Street are as follows:

Find the mean and median values of these houses.Which measure of central tendency is most suitable for this set? Why?

Bluman, Chapter 336Value per houseNumber of Houses Php 300,0001Php 700,0008Php 900,0006Php 1,100,0004Bluman, Chapter 337

Bluman, Chapter 338

Bluman, Chapter 339

Distributions40Bluman, Chapter 3

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546

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Assignment:Quiz tomorrow:Measures of Central tendencyBluman, Chapter 351