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3.1 Measures of Central Tendency: Mode, Median, and …

Sherlock Holmes spoke these words to hiscolleague Dr. Watson as the two wereunraveling a mystery. The detective wasimplying that if a single member is drawnat random from a population, we cannotpredict exactlywhat that member will looklike. However, there are some average features ofthe entire population that an individual is likely topossess. The degree of certainty with which we wouldexpect to observe such average features in any indi-vidual depends on our knowledge of the variationamong individuals in the population. SherlockHolmes has led us to two of the most important sta-tistical concepts: average and the individual man is an insolvablepuzzle, in the aggregate he becomes amathematical certainty. You can, forexample, never foretell what any one man will do, but you can say with precisionwhat an average number will be up to.

vidual depends on our knowledge of the variation among individuals in the population. Sherlock Holmes has led us to two of the most important sta-tistical concepts: average and variation. While the individual man is an insolvable puzzle, in the aggregate he becomes a mathematical certainty. You can, for example, never foretell what any one man

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Transcription of 3.1 Measures of Central Tendency: Mode, Median, and …

1 Sherlock Holmes spoke these words to hiscolleague Dr. Watson as the two wereunraveling a mystery. The detective wasimplying that if a single member is drawnat random from a population, we cannotpredict exactlywhat that member will looklike. However, there are some average features ofthe entire population that an individual is likely topossess. The degree of certainty with which we wouldexpect to observe such average features in any indi-vidual depends on our knowledge of the variationamong individuals in the population. SherlockHolmes has led us to two of the most important sta-tistical concepts: average and the individual man is an insolvablepuzzle, in the aggregate he becomes amathematical certainty. You can, forexample, never foretell what any one man will do, but you can say with precisionwhat an average number will be up to.

2 Arthur Conan Doyle,The Sign of Measures of Central tendency : Mode, Median, and Measures of Percentiles and Box-and-Whisker Plots74 For on-line student resources, visit follow theStatistics links to the Brase/Brase, Understandable Statistics,9th edition web 7/13/07 4:55 AM Page 74 FOCUS PROBLEMThe Educational AdvantageIs it really worth all the effort to get a college degree? From a philosophicalpoint of view, the love of learning is sufficient reason to get a college , the Census Bureau also makes another relevant , college graduates (bachelor s degree) earn onaverage $23,291 more than high school graduates. Thismeans college graduates earn about more thanhigh school graduates, and according to EducationPays on the next page, the gap in earnings is increas-ing.

3 Furthermore, as the College Board indicates, formost Americans college remains relatively completing this chapter, you will be able toanswer the following questions.(a) Does a college degree guaranteesomeone an in earnings over a high school degree?Remember, we are using only averagesfrom censusdata.(b) Using census data (not shown in EducationPays ), it is estimated that the standard deviationof college-graduate earnings is about $8, a 75% Chebyshev confidence intervalcentered on the mean ($51,206) for bachelor sdegree earnings.(c) How much does college tuition cost? That depends,of course, on where you go to college. Construct aweighted average. Using the data from CollegeAffordable for Most, estimate midpoints for thePREVIEW QUESTIONSWhat are commonly used Measures of Central tendency ?

4 What do they tell you?( )How do variance and standard deviation measure data spread? Why is this important?( )How do you make a box-and-whisker plot, and what does it tell about the spread of the data?( )75 Averages and Variation1020437_Ch03_p074-121 7/13/07 4:55 AM Page 75 Education PaysAverage annual salary gap increases byeducation levelBachelor sdegree$51,206 High schooldiploma$27,915No diploma$18,734$60,000$70,000$50,000$90,0 00$80,000$40,000$30,000$20,000$10,000197 5198519972003 Advanceddegree$88,471$3,000 to$5,9999%9%8%7%21%46%Source:Census BureauSource: The College Boardcost intervals. Say 46% of tuitions cost about $4,500; 21% cost about $7,500;7% cost about $12,000; 8% cost about $18,000; 9% cost about $24,000; and9% cost about $31,000. Compute the weighted average of college tuitioncharged at all colleges.

5 (See Problem 9 in the Chapter Review Problems.)76 Chapter 3 AVERAGES ANDVARIATIONSECTION of Central tendency : Mode, Median, and MeanFOCUS POINTS Compute mean, median, and mode from raw data. Interpret what mean, median, and mode tell you. Explain how mean, median, and mode can be affected by extreme data values. What is a trimmed mean? How do you compute it? Compute a weighted average price of an ounce of gold is $420. The Zippy car averages 39 miles pergallon on the highway. A survey showed the average shoe size for women is size each of the preceding statements, onenumber is used to describe the entiresample or population. Such a number is called an are many waysto compute averages, but we will study only three of the major easiest average to compute is the modeof a data set is the value that occurs most 1 ModeCount the letters in each word of this sentence and give the mode.

6 The numbersof letters in the words of the sentence are5372442483434 Scanning the data, we see that 4 is the mode because more words have 4 lettersthan any other number. For larger data sets, it is useful to order or sort thedata before scanning them for the every data set has a mode. For example, if Professor Fair gives equalnumbers of A s, B s, C s, D s, and F s, then there is no modal grade. In addition,This section can be covered quickly. Gooddiscussion topics include The Story of OldFaithfulin Data Highlights, Problem 1:Linking Concepts, Problem 1; and the tradewinds of Hawaii (Using Technology).1020437_Ch03_p074-121 7/13/07 4:55 AM Page 76the mode is not very stable. Changing just one number in a data set can changethe mode dramatically. However, the mode is a useful average when we want toknow the most frequently occurring data value, such as the most frequentlyrequested shoe average that is useful is the Median, or Central value, of an ordereddistribution.

7 When you are given the median, you know there are an equal num-ber of data values in the ordered distribution that are above it and below of Central tendency : Mode, Median, and Mean77 MedianPROCEDUREHOW TO FIND THE MEDIANThe medianis the Central value of an ordered distribution. To find it,1. Order the data from smallest to For an oddnumber of data values in the distribution,Median Middle data value3. For an evennumber of data values in the distribution,Median Sum of middle two values2 EXAMPLE 2 MedianWhat do barbecue-flavored potato chips cost? According to Consumer Reports,Volume 66, No. 5, the prices per ounce in cents of the rated chips are191927281835(a) To find the median, we first order the data, and then note that there are an evennumber of entries. So the median is constructed using the two middle values(b) According to Consumer Reports,the brand with the lowest overall taste rat-ing costs 35 cents per ounce.

8 Eliminate that brand, and find the medianprice per ounce for the remaining barbecue-flavored chips. Again order thedata. Note that there are an odd number of entries, so the median is simplythe middle middle valueMedian middle value 19 cents(c) One ounce of potato chips is considered a small serving. Is it reasonable tobudget about $ to serve the barbecue-flavored chips to 55 people? Yes, since the median price of the chips is 19 cents per small serving. Thisbudget for chips assumes that there is plenty of other food!Median 19 272 23 centsThe notation x (read xtilde ) is sometimesused to designate the median of a data 7/13/07 4:55 AM Page 77 The median uses the positionrather than the specific value of each data the extreme values of a data set change, the median usually does not is why the median is often used as the average for house prices.

9 If one man-sion costing several million dollars sells in a community of much-lower-pricedhomes, the median selling price for houses in the community would be affectedvery little, if at 3 AVERAGES ANDVARIATIONGUIDED EXERCISE 1 Median and mode(a) Organize the data from smallest to largestnumber of credit hours.(b) Since there are an (odd, even) numberof values, we add the two middle values anddivide by 2 to get the median. What is themedian credit hour load?(c) What is the mode of this distribution? Is itdifferent from the median? If the budgetcommittee is going to fund the school accordingto the average student credit hour load (moremoney for higher loads), which of these twoaverages do you think the college will use?12 12 12 12 12 12 12 12 12 1213 13 13 13 14 14 14 14 15 1515 15 15 15 1616 16 16 17 1717 17 17 18 18 18 19 19 20 20 There are an even number of entries.

10 The twomiddle values are circled in part (a).The mode is 12. It is different from the the median is higher, the school will probablyuse it and indicate that the average being used is 15 152 15 Belleview College must make a report to the budget committee about the average credit hour loada full-time student carries. (A 12-credit-hour load is the minimum requirement for full-time the same tuition, students may take up to 20 credit hours.) A random sample of 40 studentsyielded the following information (in credit hours):171214171316182013121217161514121 2131714151215161218201912151814161715191 2131215 Note:For small ordered data sets, we can easily scan the set to find the loca-tionof the median. However, for large ordered data sets of size n,it is convenientto have a formula to find the middle of the data an ordered data set of size n,For instance, if then the middle value is the or 50th data valuein the ordered data.


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