SticiGui “Chapter 4: Measures of Location and Spread ... · SticiGui “Chapter 4: Measures of Location and Spread” Philip Stark (2013) Summarizing data can help us understand

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SticiGui “Chapter 4: Measures of Location and Spread”Philip Stark (2013)

Summarizing data can help us understand them, especially when the number of data islarge. This chapter presents several ways to summarize quantitative data by a typicalvalue (a measure of location, such as the mean, median, or mode) and a measure ofhow well the typical value represents the list (a measure of spread, such as the range,inter-quartile range, or standard deviation). Markov’s and Chebychev's inequalities showthat these summary measures can contain a surprisingly large amount of informationabout the data.

Measures of Location

The farthest one can reduce a set of data, and still retain any information at all, is tosummarize the data with a single value. Measures of location do just that: They try tocapture with a single number what is typical of the data. What single number is mostrepresentative of an entire list of numbers? We cannot say without defining"representative" more precisely. We will study three common measures of location: themean, the median, and the mode. The mean, median and mode are all "mostrepresentative," but for different, related notions of representativeness.

o We saw the median in Chapter 3, Statistics. The median is the number thatdivides the (ordered) data in half—the smallest number that is at least as big ashalf the data. At least half the data are equal to or smaller than the median, andat least half the data are equal to or greater than the median.

o The mode of a set of data (as opposed to the mode of a histogram) is the mostcommon value among the data. It is rare that several data coincide exactly,unless the variable is discrete, or the measurements are reported with lowprecision.

o The mean (more precisely, the arithmetic mean) is commonly called the average.It is the sum of the data, divided by the number of data:

sum of data totalmean = ---------------------- = ------------------- .

number of data number of data

For qualitative and categorical data, the mode makes sense, but the mean andmedian do not. It is hard to see the connection between the mean, median, and modefrom their definitions.



However, the mean, the median, and the mode are "as close as possible" to all thedata: For each of these three measures of location, the sum of the distances betweeneach datum and the measure of location is as small as it can be. The differences amongthe three measures of location are in how "distance" is defined.

o For the mean, the distance between two numbers is defined to be the square oftheir difference. That is, the sum of the squares of the differences between thedata and the mean is smaller than the sum of squares of the differences betweenthe data and any other number. (Equivalently, the RMS or root mean square ofthe differences from the mean is smaller than the rms of the list of differencesfrom any other number—the rms is defined and discussed below.)

o For the median, the distance between two numbers is defined to be the absolutevalue of their difference. That is, the sum of the absolute values of thedifferences between a median and the data is no larger than the sum of theabsolute values of the differences between any other number and the data.

o For the mode, the distance between two numbers is defined to be zero if thenumbers are equal, and one if they are not equal. That is, the number of datathat differ from a mode is no larger than the number of data that differ from anyother value. Equivalently, a mode is a number from which the fewest possibledata differ: a "most common" value.

All three of these measures of location are examples of statistics (with a lowercase"s"): numbers computed from data.

The mean, median, and mode can be related (approximately) to the histogram: looselyspeaking, the mode is the highest bump, the median is where half the area is to theright and half is to the left, and the mean is where the histogram would balance, were ita solid object cut out of a uniform block of metal. (All these heuristics are approximate,and depend on the class intervals.)

Example 4-1: Calculating the mean, median and mode of a list.For illustration, let's compute the mean, median and mode, from the hypotheticaldata in Table 4-1.

Table 4-1: Random data to illustrate calculating measures of location.

data 4 0 5 -2 -3 -5



Table 4-2: Sorted random data to illustrate calculating measures of location.This table shows these random data sorted into increasing order, which makes iteasier to calculate the median.

data -5 -3 -2 0 4 5

Half the data are less than or equal to every number between -2 (inclusive) and 0(exclusive). By our definition, the median is the smallest such number, namely, -2. For these hypothetical data, every value in the list is a mode: each valueoccurs exactly once, so all are "most common."

Computing the mean is familiar:

4 + 0 + 5 + (-2) + (-3) + (-5)--------------------------------- = -0.167.

6

In general, the mean and the median need not be close together. If the data have asymmetric distribution, the mean and median are exactly equal, but if the distribution ofthe data is skewed, the difference between mean and the median can be large. This isbecause data in the tails of the distribution have a lot of leverage on the mean, just as alight person can balance a much heavier one on a teeter-totter if she sits much fartherfrom the fulcrum than the heavier person does. The median is smaller than the mean ifthe data are skewed to the right, and larger than the mean if the data are skewed to theleft. Because the mean is (essentially) the balance point of the histogram, a smallnumber of data can affect it a great deal, if they are very large (positive or negative).Corrupting just one datum can make the mean arbitrarily large or small.

The median is affected much less by small subsets of the data. To make the medianarbitrarily large or small, one must corrupt half the data. Corrupting just one datumchanges the median by a limited amount, and not at all if one of the observations abovethe median is made larger, or one of the observations below the median is madesmaller. Statistics that are not affected too much by small subsets of the data areresistant. The median is resistant; the mean is not.

Which measure of location is the most appropriate depends on the intended use of thesummary. If we are interested in a total, the mean tends to be the most relevant,



because the mean is equal to the total divided by the number of data. For example, themean income of the individuals in a family indicates how much the family can spend oneach family member's necessities of life. On the other hand, the median can be muchmore informative in other situations.

Suppose we want to know how much money a family can afford to spend on housing.That depends on the total family income, which is the mean income of the familymembers, times the number of family members. For a family of five, consisting of twoparents who work and three children with no income, the mean income, times five, isthe total amount of money the family makes each year. The median income of thesefive family members is zero, because more than half of them make nothing.

On the other hand, suppose we want to decide whether a country is affluent. At issue, insome sense, is whether most of the citizens have a high income. The mean familyincome could be quite high even if most families earn essentially nothing—if income ishighly concentrated in a few very wealthy families. Then the median family incomewould be a more meaningful measure: At least half the families make no more than themedian, and at least half make at least as much as the median.

Similarly, suppose you are applying for a job as an architect at several large firms, andyou want to get an idea of how much money you might expect to be earning in fiveyears if you join a particular firm. Consider the salaries of architects in each firm fiveyears after they are hired. Just one very high salary could make the mean salary high,so the mean might not reflect what is typical. On the other hand, half the architectsmake the median salary or less, and half make the median salary or more, so themedian would give you a better idea of a typical salary.

Choosing a measure of location favorable to one's point of view is a common way tomislead people with statistics. For example, suppose you are the CEO of a companythat makes gizmos and gadgets. It might be in your interest to claim to your customersthat you have lowered your prices, and to claim to your shareholders that you haveraised your prices. Suppose that last year, you sold 100,000 gizmos at $10 each, and1,000 gadgets at $1000 each. This year, you sold 100,000 gizmos at $8 each, and1,000 gadgets at $1200 each (see Table 4-3).

Table 4-3: Quantities and Prices for Two Years of Gizmo and Gadget Sales.



Item Quantity Each Year Price Last Year Price This YearGizmo 100,000 $10 $8Gadget 1,000 $1,000 $1,200

The median price of the 101,000 items sold last year is $10, because more thanhalf of the items sold were gizmos. The median price of the 101,000 items soldthis year is $8. The mean price on the price list (without regard for the number ofitems sold) was $505 last year and $604 this year. The mean price of the101,000 items sold last year is

! (!1 !0 !0 !,!0 !0 !0 ! !x! !$ !1 !0 ! !+! !1 !,!0 !0 !0 ! !x! !$ !1 !,!0 !0 !0 !)!/!1 !0 !1 !,!0 !0 !0 ! !=! !$ !1 !9 !.!8 !0

!w !h !i !l !e ! !t!h !i !s! !y!e !a !r! !i !t! !i !s!!! (!1 !0 !0 !,!0 !0 !0 ! !x !$ !8 ! !+! !1 !,!0 !0 !0 ! !x! !$ !1 !,!2 !0 !0 !)!/!1 !0 !1 !,!0 !0 !0 ! !=! !$ !1 !9 !.!8 !0 !.

The mean price per item sold is the same in both years: the total revenue wasthe same, and the number of items sold was the same. The moral is that one canmake data appear to tell conflicting stories by choosing a measure of locationdisingenuously.

The following exercises check your ability to compute and to use the mean,median, and mode.

Exercise 4-1. Consider the following list:

data -10 -2 -2 5 -5 -2

1. What is the median of the list?

2. What is the mode of the list?

3. What is the mean of the list?

SOLUTIONS: 1. –2, 2. –2 3. -2.667

Exercise 4-2. Homes in a certain area have a mean price of $400,000, but amedian price of "only" $250,000. How might you explain this best?



a. A small percentage of very inexpensive homes makes the mediansmall, but does not affect the mean much.

b. A small percentage of very expensive homes makes the mean large,but does not affect the median much.

c. There must be an error in the computation.d. More than half of the home prices are less than $250,000.

SOLUTION: b

Exercise 4-3. TRUE or FALSE: Two countries have the same mean per capitapersonal income. The total personal income in the larger country is larger thanthe total personal income in the smaller country.

SOLUTION: True. The total personal income is the mean personal income timesthe number of people, so if the means are the same, the total is larger for thelarger country.

Exercise 4-4. TRUE or FALSE: Two countries have the same median per capitapersonal income. The total personal income in the larger country is larger thanthe total personal income in the smaller country.

SOLUTION: False. The median could be larger or smaller than the mean, so thetotal could be larger or smaller than the median times the number of people, andwe do not have enough information in this problem to tell which country has thelarger total personal income. Typically, income distributions are skewed to theright, so the mean income is generally larger than the median income; however,even if that is true for both countries, it need not be larger by the same amount inboth countries.

Exercise 4-5. Consider the following game. You pick a number (not necessarilyan integer). I roll a fair die, and pay you $10, minus the square of the differencebetween your guess and the number the die lands on. We play over and overagain. To win the most money in the long run, you should pick

a. 3b. 3.5c. 4d. 5e. It doesn’t matter



SOLUTION: b. !T!o ! !m!a !k!e ! !y!o !u !r! !w !i !n !n !i !n !g !s! !l !a !r!g !e !,! !y!o !u ! !w !a !n !t! !t!o ! !m!a !k!e ! !t!h !e ! !a !v!e !r!a !g !e ! !o !f!!t!h !e ! !s!q !u !a !r!e !d ! !d !i !f!f!e !r!e !n !c!e ! !b !e !t!w !e !e !n ! !y!o !u !r! !g !u !e !s!s! !a !n !d ! !t!h !e ! !o !u !t!c!o !m!e ! !o !f! !t!h !e ! !r!o !l !l ! !s!m!a !l !l !.!!T!h !e ! !m!e !a !n ! !m!i !n !i !m!i !z!e !s! !t!h !e ! !a !v!e !r!a !g !e ! !o !f! !t!h !e ! !s!q !u !a !r!e !d ! !d !e !v!i !a !t!i !o !n !s!,! !s!o ! !y!o !u ! !s!h !o !u !l !d ! !p !i !c!k!!t!h !e ! !m!e !a !n ! !o !f! !t!h !e ! !p !o !s!s!i !b !l !e ! !o !u !t!c!o !m!e !s!,! !3 !.5.

Measures of Location Review

o Measures of location summarize a list of numbers by a "typical" value.

o The three most common measures of location are the mean, the median, and themode.

o The mean is the sum of the values, divided by the number of values. It has thesmallest possible sum of squared differences from members of the list.

o The median is the middle value in the sorted list. It is the smallest number that isat least as big as at least half the values in the list. It has the smallest possiblesum of absolute differences from members of the list.

o The mode is the most frequent value in the list (or one of the most frequentvalues, if there are more than one). It differs from the fewest possible membersof the list.

Spread or Variability

Measures of location summarize what is typical of elements of a list, but not everyelement is typical. Are all the elements close to each other? Are most of the elementsclose to each other? What is the biggest difference between elements? On the average,how far are the elements from each other? Measures of spread or variability tell us.

The Importance of Variability

Consider three mechanical golfers (this example is from Hooke, 1983). In golf, theobject is to get a low score—to take fewer strokes to complete the course. Suppose thegolfers play as shown in Table 4-4.

Table 4-4: Performance of mechanical golfers.



Golfer Score 1 Frequency 1 Score 2 Frequency 2 Average score1 72 100% 722 69 25% 73 75% 723 70 50% 74 50% 72

!T!h !e ! !g !o !l !f!e !r!s!'! !a !v!e !r!a !g !e ! !s!c!o !r!e !s! are equal—nominally, t!h !e !y! !a !r!e ! !e !q !u !a !l !l !y! !s!k!i !l !l !e !d !.! !H !o !w !e !v!e !r!,!!c!o !n !s!i !d !e !r! !w !h !a !t happens! !w !h !e !n ! !t!h !e !y! !p !l !a !y! !e !a !c!h ! !o !t!h !e !r!.! !G!o !l !f!e !r! !1 ! !b !e !a !t!s! !g !o !l !f!e !r! !2 ! !w !h !e !n ! !g !o !l !f!e !r! !2 !!s!c!o !r!e !s! !7 !3 !,! !w !h !i !c!h happens 7 !5 !% ! !o !f! !t!h !e ! !t!i !m!e !.! !G!o !l !f!e !r! !2 ! !b !e !a !t!s! !g !o !l !f!e !r! !3 ! !w !h !e !n ! !g !o !l !f!e !r! !3 ! !s!c!o !r!e !s!!7 !4 !,! !a !n !d ! !w !h !e !n ! !g !o !l !f!e !r! !3 ! !s!c!o !r!e !s! !7 !0 ! !a !n !d ! !g !o !l !f!e !r! !2 ! !s!c!o !r!e !s! !6 !9 !.! !T!h !e ! !f!i !r!s!t! !o !c!c!u !r!s! !h !a !l !f! !t!h !e ! !t!i !m!e !,!!a !n !d !,! !a !s!s!u !m!i !n !g ! !t!h !a !t! !t!h !e ! !p !l !a !y!e !r!s!'! !s!c!o !r!e !s! !a !r!e! !i!n!d!e !p!e !n!d!e !n!t! !(!w !e !'!l !l ! !g !e !t! !t!o ! !t!h !a !t! !n !o !t!i !o !n ! !i !n !!C !h !a !p !t!e !r! !1 !7 !,! !P!r!o !b !a !b !i !l !i !t!y!:! !A!x!i !o !m!s! !a !n !d ! !F!u !n !d !a !m!e !n !t!s!)!,! !t!h !e ! !s!e !c!o !n !d ! !o !c!c!u !r!s! !5 !0 !% ! !x! !2 !5 !% ! !=!!1 !2 !.!5 !% ! !o !f! !t!h !e ! !t!i !m!e !,! !s!o ! !g !o !l !f!e !r! !2 ! !b !e !a !t!s! !g !o !l !f!e !r! !3 ! !6 !2 !.!5 !% ! !o !f! !t!h !e ! !t!i !m!e !.! !F!i !n !a !l !l !y!,! !g !o !l !f!e !r! !3 ! !b !e !a !t!s!!g !o !l !f!e !r! !1 ! !w !h !e !n ! !g !o !l !f!e !r! !3 ! !s!c!o !r!e !s! !7 !0 !,! !n !a !m!e !l !y!,! !5 !0 !% ! !o !f! !t!h !e ! !t!i !m!e ! !(!t!h !e !y! !p !l !a !y! !e !v!e !n !l !y!)!.! !T!h !e !i !r!!a !v!e !r!a !g !e ! !s!c!o !r!e !s! !a !r!e ! !e !q !u !a !l !,! !b !u !t! !1 ! !b !e !a !t!s! !2 ! !m!o !r!e ! !o !f!t!e !n ! !t!h !a !n ! !n !o !t!,! !2 ! !b !e !a !t!s! !3 ! !m!o !r!e ! !o !f!t!e !n ! !t!h !a !n !!n !o !t!,! !a !n !d ! !3 ! !p !l !a !y!s! !1 ! !e !v!e !n !.! !T!h !i !s! !s!h !o !w !s! !t!h !a !t! !t!h !e !r!e ! !i !s! !m!o !r!e ! !g !o !i !n !g ! !o !n ! !t!h !a !n ! !t!h !e ! !a !v!e !r!a !g !e !!s!c!o !r!e !s! !i !n !d !i !c!a !t!e !:! !v!a !r!i !a !b !i !l !i !t!y! !m!a !t!t!e !r!s! !t!o !o !.

Here is another example of the importance of variability. The average number ofchildren under 18 per family in the US was 0.89 according to the 1990 census, so theaverage family size is about 2.9 people (is this logic sound? what is a family?). If youwere in the construction business that might suggest to you that a two-bedroom home isthe right size to build for the average American family (two parents sharing a room, andanother room for the 0.89 children). However, family sizes vary over quite a large range;indeed, the same report shows that the average number of children for families thathave children is 1.86, so families that have children would tend to need a three bedroomhome, rather than a two bedroom home, if the children are to have their own rooms.

Much information is lost in reducing a list of numbers to a single summary number, suchas the mean or median. Measures of location alone are not very informative.

For Java applet Figures 4-1, 4-2 and 4-3 (histograms of different data sets withmeans equal to zero), please visit:http://www.stat.berkeley.edu/~stark/SticiGui/Text/location.htm

We need more than just the mean or median to tell these distributions apart. In Figure4-1 the data cluster both in the middle and at the ends. In Figure 4-2 the data are moreconcentrated near the middle—there is much less spread than in the first. Figure 4-3 isextremely concentrated: The data are much closer to each other than in the other two



examples. Measures of spread or variability summarize with a single number whetherthe observations tend to cluster near the center of the distribution, or how spread outthey are. If the spread is small, most of the data are nearly equal; if the spread is large,there are large differences among the data.

The Range, IQR and SD

The three most common measures of spread or variability are the range, theinterquartile range (IQR), and the standard deviation (SD).

The range of a list is the largest value minus the smallest value. It is the width of thesmallest interval that contains all the data, so it measures spread. It is not resistant,because changing just one datum can make it arbitrarily large.

The IQR is the upper quartile (75th percentile), minus the lower quartile (25thpercentile). It is the width of the interval that contains the middle 50% of the data—andthus is a measure of spread. It is insensitive to the most extreme values of the data(assuming that there are more than four data). The IQR is resistant: changing just onedatum has a limited effect on it. Note that neither the range nor the IQR is a range ofnumbers, despite their names—each is a single number.

The RMS (root mean square) of a list measures the average size of its entries. It isdefined as follows:

RMS !=! !s!q !u !a !r!e !-!r!o !o !t!(! !(!s!u !m! !o !f! !t!h !e ! !s!q !u !a !r!e !s! !o !f! !t!h !e ! !e !n !t!r!i !e !s!)!/!(!# o !f! !e !n !t!r!i !e !s!)! !)!!

! = ![! !(!s!u !m! !o !f! !s!q !u !a !r!e !s! !o !f! !t!h !e ! !e !n !t!r!i !e !s!)!/!(!n !u !m!b !e !r! !o !f! !e !n !t!r!i !e !s!)! !]!1/2.

(Recall that a number raised to the one-half power is the square-root of the number; thisis the notation we shall use from now on.)

In computing the RMS, we divide by the number of entries before taking the square-root. What difference does it make to square the entries? Squaring them makes everyterm in the sum positive, so positive and negative entries do not cancel. If we ignoredthe square and the square root, we would just have the mean of the list, which could bezero, even if all the numbers were large in magnitude, because positive and negativeentries could cancel. Squaring the entries before averaging them preventscancellations.



The RMS is not the only measure of the average size of the elements of a list; forexample, the average absolute value of the terms is another measure of the typical sizeof elements in a list. The RMS is used more often. Example 4-2 illustrates calculatingthe RMS of a list.

Example 4-2: Calculating the RMS of a list.

data 2 0 -2 4 -4

The average of this list is:

2 + 0 + (-2) + 4 + (-4) 0-------------------------- = ------ = 0

5 5

Nonetheless, the typical "size" of elements of the list is about 2.8.

The RMS of the list is

! (! !(!2 !2! !+! !0 !2! !+! !(!-!2 !)!2! !+! !4 !2! !+! !(!-!4 !)!2!)!/!5 !)!1/2

!! =! !(! !(!4 ! !+! !0 ! !+! !4 ! !+! !1 !6 ! !+! !1 !6 ! !)!/!5 !)!1/2 ! !=! !(! !(!4 !0 !)!/!5 !) !1/2

!!w !h !i !c!h ! !i !s! !a !p !p !r!o !x!i !m!a !t!e !l !y! !2 !.!8 !.

Example 4-2 makes it clear that the mean of the squares of the elements of a list is notgenerally equal to the square of the mean of the elements of the list: the square of themean is 0, but the mean of the squares is not.

The RMS of a list is zero if and only if all the entries in the list are zero.

The standard deviation (SD) of a list is the "typical size" of the difference betweenelements of the list and the mean of the list, measured by the RMS. The SD measureshow spread out the data are around their mean. To find the SD, we first find the mean ofthe list, then make a list of deviations from the mean:

!d !e !v!i !a !t!i !o !n ! !o !f! !v!a !l !u !e ! !=! !v!a !l !u !e ! -! !m!e !a !n ! !o !f! !l !i !s!t!,



and finally, find the RMS of the list of deviations from the mean (the square-root of theaverage of the squares of the deviations). In the example just given, the mean is zero,so the SD is equal to the RMS.

Example 4-3: Calculating the SD of a list.

data 4 6 1 3 5 2

The mean of the list is

( 4 + 6 + 1 + 3 + 5 + 2)/6 = 3.5.

!T!h !e ! !l !i !s!t! !o !f! !d!e !v !i!a !t!i!o!n!s ! !f!r!o !m! !t!h !e ! !m !e !a !n! !i !s!!! {!(!4 ! - !3 !.!5 !)!,! !(!6 ! - 3 !.!5 !)!,! !(!1 ! - !3 !.!5 !)!,! !(!3 ! - 3 !.!5 !)!,! !(!5 ! - 3 !.!5 !)!,! !(!2 ! - 3 !.!5 !)!}!.!! =! !{! !0 !.!5 !,! !2 !.!5 !,! !-!2 !.!5 !,! !-!0 !.!5 !,! !1 !.!5 !,! !-!1 !.!5 !}!.

!T!h !e ! !S!D! !i !s! !t!h !e ! !RMS !o !f! !t!h !i !s! !l !i !s!t! !o !f! !d !e !v!i !a !t!i !o !n !s! !f!r!o !m! !t!h !e ! !m!e !a !n !:!!! S!D ! !=! !(! !(!0 !.!5 !2! !+! !2 !.!5!2! !+! !(!-!2 !.!5 !)!2! !+! !(!-!0 !.!5 !)!2! !+! !1 !.!5!2! !+! !(!-!1 !.!5 !)!2!)!/!6 !)!1/2 !

! =! !(!1 !7 !.!5 !/!6 !)!1/2

! =! !1 !.!7 !1 !.

T!h !e ! !u !n !i !t!s! !o !f! !t!h !e ! !S!D ! are !t!h !e ! same as !t!h !e original u !n !i !t!s! !o !f meas!u !r!e !m!e !n !t!.! !F!o !r! !e !x!a !m!p !l !e !,! !i !f!!t!h !e ! !l !i !s!t! !i !s! !c!o !m!p !r!i !s!e !d ! !o !f! !m!e !a !s!u !r!e !m!e !n !t!s! !o !f! !h !e !i !g !h !t!s! !i !n ! !i !n !c!h !e !s!,! !t!h !e ! !S!D ! !h !a !s! !u !n !i !t!s! !o !f! !i !n !c!h !e !s!.!!R !e !c!a !l !l ! !t!h !a !t! !t!h !e ! !RMS! !o !f! !a ! !l !i !s!t! !i !s! !z!e !r!o ! !i !f! !a !n !d ! !o !n !l !y! !i !f! !a !l !l ! !t!h !e ! !e !l !e !m!e !n !t!s! !i !n ! !t!h !e ! !l !i !s!t! !a !r!e ! !z!e !r!o !.!!T!h !u !s! !t!h !e ! !S!D ! !o !f! !a ! !l !i !s!t! !i !s! !z!e !r!o ! !i !f! !a !n !d ! !o !n !l !y! !i !f! !a !l !l ! !t!h !e ! !d !e !v!i !a !t!i !o !n !s! !f!r!o !m! !t!h !e ! !m!e !a !n ! !a !r!e ! !z!e !r!o !,! !t!h !a !t!!i !s!,! !i !f! !a !n !d ! !o !n !l !y! !i !f! !a !l !l ! !t!h !e ! !e !l !e !m!e !n !t!s! !a !r!e ! !e !q !u !a !l ! !t!o ! !e !a !c!h ! !o !t!h !e !r! !(!a !n !d ! !h !e !n !c!e ! !e !q !u !a !l ! !t!o ! !t!h !e !i !r!!m!e !a !n !)!.! !S!i !m!i !l !a !r!l !y!,! !t!h !e ! !r!a !n !g !e ! !o !f! !a ! !l !i !s!t! !i !s! !z!e !r!o ! !i !f! !a !n !d ! !o !n !l !y! !i !f! !a !l !l ! !t!h !e ! !e !l !e !m!e !n !t!s! !a !r!e ! !e !q !u !a !l !.! !I!n !!c!o !n !t!r!a !s!t!,! !t!h !e ! !I!Q!R ! !o !f! !a ! !l !i !s!t! !c!a !n ! !b !e ! !z!e !r!o ! !e !v!e !n ! !i !f! !n !o !t! !a !l !l ! !t!h !e ! !e !l !e !m!e !n !t!s! !a !r!e ! !t!h !e ! !s!a !m!e—only!!t!h !e ! !m!i !d !d !l !e ! !5 !0 !% ! !o !f! !t!h !e ! !o !b !s!e !r!v!a !t!i !o !n !s! !n !e !e !d ! !t!o ! !b !e ! !e !q !u !a !l ! !f!o !r! !t!h !e ! !I!Q!R ! !t!o ! !b !e ! !z!e !r!o !.!

!S!o !m!e ! !c!a !l !c!u !l !a !t!o !r!s! !h !a !v!e ! !a ! !b !u !t!t!o !n ! !l !a !b !e !l !e !d! !s !,! !w !h !i !c!h ! !c!o !m!p !u !t!e !s! !s!o !m!e !t!h !i !n !g ! !r!e !l !a !t!e !d ! !t!o ! !t!h !e ! !S!D !!a !s! !w !e ! !h !a !v!e ! !d !e !f!i !n !e !d ! !i !t!.! !I!n ! !t!h !e ! !u !s!u !a !l ! !d !e !f!i !n !i !t!i !o !n ! !o !f! !s!,! !t!h !e ! !s!u !m! !o !f! !s!q !u !a !r!e !s! !o !f! !r!e !s!i !d !u !a !l !s! !f!r!o !m!!t!h !e ! !m!e !a !n ! !i !s! !d !i !v!i !d !e !d ! !b !y! !(!n !u !m!b !e !r! !o !f! !d !a !t!a ! -1)! !r!a !t!h !e !r! !t!h !a !n ! !b !y! !(!n !u !m!b !e !r! !o !f! !d !a !t!a !)! !b !e !f!o !r!e ! !t!a !k!i !n !g !!t!h !e ! !s!q !u !a !r!e !-!r!o !o !t!.! !T!h !i !s! !i !s! called the ! sample standard d!e !v !i!a !t!i!o!n!.! !W!h !e !n ! !t!h !e ! !n !u !m!b !e !r! !o !f!!d !a !t!a ! !i !s! !l !a !r!g !e !,! !t!h !e !r!e ! !i !s! !n !o !t! !m!u !c!h ! !d !i !f!f!e !r!e !n !c!e ! !b !e !t!w !e !e !n ! !t!h !e ! !s!t!a !n !d !a !r!d ! !d !e !v!i !a !t!i !o !n ! !a !n !d ! !t!h !e !



!s!a !m!p !l !e ! !s!t!a !n !d !a !r!d ! !d !e !v!i !a !t!i !o !n !,! !b !u !t! !w !h !e !n ! !t!h !e ! !n !u !m!b !e !r! !o !f! !d !a !t!a ! !i !s! !s!m!a !l !l !,! !t!h !e ! !d !i !f!f!e !r!e !n !c!e ! !c!a !n ! !b !e !!b !i !g !.!

!!T!h !e ! !f!o !l !l !o !w !i !n !g ! !e !x!e !r!c!i !s!e !s! !c!h !e !c!k! !t!h !a !t! !y!o !u ! !c!a !n ! !c!a !l !c!u !l !a !t!e ! !m!e !a !s!u !r!e !s! !o !f! !s!p !r!e !a !d !,! !a !n !d ! !t!h !a !t! !y!o !u !!u !n !d !e !r!s!t!a !n !d ! !w !h !a !t! !t!h !e !y! !m!e !a !n !.

Exercise 4-6. Refer to Table 3-4, sorted gravity data, in Chapter 3 (picturedbelow).

Table 3-4: Sorted gravity data.

-152 -132 -132 -128 -122 -121 -120 -113 -112 -108-107 -107 -106 -106 -106 -105 -101 -101 -99 -89-87 -86 -83 -83 -80 -80 -79 -74 -74 -74-71 -71 -69 -67 -67 -65 -62 -61 -60 -60-59 -55 -54 -54 -52 -50 -49 -48 -48 -47-44 -43 -38 -37 -35 -34 -34 -29 -27 -27-26 -24 -24 -19 -19 -19 -19 -18 -16 -16-16 -15 -14 -14 -12 -12 -12 -4 -1 00 1 2 7 14 14 14 14 18 1819 24 29 29 41 45 51 72 150 155

1. The range of the gravity data (the tabulated numbers, which are 108 times thedeviations from the reference value) is ____?

2. The IQR of the gravity data is____?

SOLUTIONS: 1. The smallest datum is –152 and the largest is 155, so the rangeis: 155 – (-152) = 307 2. The lower quartile of the gravity data is –80 and theupper quartile is –12, so the interquartile range is –12-(-80) = 68.

Exercise 4.7. TRUE or FALSE: Two students have taken all the same courses,and have the same grade point average (GPA), 3.5. Their grades might not havebeen the same in each class, but overall they must have the same number of Agrades as each other.



SOLUTION: False. Two lists can have the same mean without having the sameentries. For example, the grade lists {3, 3, 4, 4} and {2, 4, 4, 4} both correspondto a GPA of 3.5.

Exercise 4-8. Here is a table of fabricated data.

data -8 7 -10 10 3

1. What is the mean of the data?2. What is the RMS of the data?3. What is the SD of the data?

SOLUTIONS: 1. 0.39 to 0.41 2. 8.015 to 8.035 3. 8.005 to 8.025

Measures of Spread Review

o Measures of spread summarize how much members of a list of numbers differfrom each other.

o The three most common measures of spread are the range, the inter-quartilerange, and the standard deviation.

o The range is the largest element of the list, minus the smallest element of the list:the maximum difference between elements of the list. It is sensitive only to themost extreme values in the list. The range of a list is zero if and only if all theelements of the list are equal.

o The inter-quartile range (IQR) is the upper quartile of the list (75th percentile)minus the lower quartile of the list (25th percentile). It measures the width of theinterval that contains the middle 50% of the data. It is not sensitive to the extremevalues of the list. The IQR of a list is zero if (at least) the middle 50% of the valueare equal.

o The standard deviation (SD) is the average distance from the data to their mean(the rms of the deviations of the data from their mean). It depends on the valuesof all the data. The SD of a list is zero if and only if all the elements in the list areequal (to each other, and hence to their mean).



Affine Transformations

Some variables have simple relationships to other variables, for example,measurements of elevation above sea level in feet, and measurements of elevationabove sea level in meters: Each elevation in meters above sea level is 0.3048 times thecorresponding elevation in feet above sea level. When the relationship betweenvariables is simple, so is the relationship between their measures of location andspread. An affine transformation or change of variables is particularly simple. Affinetransformations have the equation of a line:

!(!t!r!a !n !s!f!o !r!m!e !d ! !v!a !l !u !e ! !o !f! !x!)! !=! !a ! !x! !(!o !r!i !g !i !n !a !l ! !v!a !l !u !e ! !o !f! !x!)! !+! !b !,

where a and b are constants. (Some books call this a linear transformation, because ithas the equation of a straight line.) For example, height in inches is related to height infeet by an affine transformation, with a = 12 and b = 0:

! (!h !e !i !g !h !t! !i !n ! !i !n !c!h !e !s!)! !=! !1 !2 ! !x !(!h !e !i !g !h !t! !i !n ! !f!e !e !t!)! !+! !0 !.

Similarly, temperature in degrees Fahrenheit is related to temperature in degreesCentigrade by an affine transformation with a = 9/5 and b = 32:

! (!t!e !m!p ! !i !n ! !°!F!)! !=! !9 !/!5 ! !x !(!t!e !m!p ! !i !n ! !°!C !)! !+! !3 !2 !.

Currencies are related to each other by affine transformations as well, with a =(exchange rate) and b = 0.

The measures of location and spread introduced in this chapter behave quite regularlywhen a list is transformed by an affine transformation.

How Measures of Location and Spread behave under Affine Transformations

I!f! !a ! !l !i !s!t! !i !s! !t!r!a !n !s!f!o !r!m!e !d ! !s!o ! !t!h !a !t!

! (!t!r!a !n !s!f!o !r!m!e !d ! !v!a !l !u !e !)! !=! !a ! !x !(!o !r!i !g !i !n !a !l ! !v!a !l !u !e !)! !+! !b !,!

!t!h !e !n !! (!M!o !d !e ! !o !f! !t!r!a !n !s!f!o !r!m!e !d ! !l !i !s!t!)! ! !=! a ! !x !(!M!o !d !e ! !o !f! !o !r!i !g !i !n !a !l ! !l !i !s!t!)! !+! !b !! (!M!e !d !i !a !n ! !o !f! !t!r!a !n !s!f!o !r!m!e !d ! !l !i !s!t!)!!=! !a ! !x! !(!M!e !d !i !a !n ! !o !f! !o !r!i !g !i !n !a !l ! !l !i !s!t!)! !+! !b !,! !i !f! !a ! !i !s! !p !o !s!i !t!i !v!e !



! (!M!e !a !n ! !o !f! !t!r!a !n !s!f!o !r!m!e !d ! !l !i !s!t!)! ! !=! !a ! !x! !(!M!e !a !n ! !o !f! !o !r!i !g !i !n !a !l ! !l !i !s!t!)! !+! !b !! (!R !a !n !g !e ! !o !f! !t!r!a !n !s!f!o !r!m!e !d ! !l !i !s!t!)! ! !=! !|!a !|! !x! !(!R !a !n !g !e ! !o !f! !o !r!i !g !i !n !a !l ! !l !i !s!t!)!! (!S!D ! !o !f! !t!r!a !n !s!f!o !r!m!e !d ! !l !i !s!t!)! ! !=! !|!a !|! !x! !(!S!D ! !o !f! !o !r!i !g !i !n !a !l ! !l !i !s!t!)!! (!I!Q!R ! !o !f! !t!r!a !n !s!f!o !r!m!e !d ! !l !i !s!t!)! ! !=! !a ! !x !(!I!Q!R ! !o !f! !o !r!i !g !i !n !a !l ! !l !i !s!t!)!,! !i !f! !a ! !i !s! !p !o !s!i !t!i !v!e !.!

T!h !e ! !m!e !d !i !a !n ! !o !f! !t!h !e ! !t!r!a !n !s!f!o !r!m!e !d ! !l !i !s!t! !c!a !n ! !d !i !f!f!e !r! !s!l !i !g !h !t!l !y! !f!r!o !m! !a ! !x! !(!m!e !d !i !a !n ! !o !f! !o !r!i !g !i !n !a !l ! !l !i !s!t!)! !+! !b !!w !h !e !n ! !a ! !i !s! !n !e !g !a !t!i !v!e !;! !s!i !m!i !l !a !r!l !y!,! !t!h !e ! !I!Q!R ! !o !f! !t!h !e ! !t!r!a !n !s!f!o !r!m!e !d ! !l !i !s!t! !c!a !n ! !d !i !f!f!e !r! !s!l !i !g !h !t!l !y! !f!r!o !m!!|!a !|!x!(!I!Q!R ! !o !f! !o !r!i !g !i !n !a !l ! !l !i !s!t!)! !i !f! !a ! !i !s! !n !e !g !a !t!i !v!e !,! !b !e !c!a !u !s!e ! !o !f! !t!h !e ! !d !e !f!i !n !i !t!i !o !n ! !o !f! !p !e !r!c!e !n !t!i !l !e !s! !a !p !p !l !i !e !d !!t!o ! !a ! !l !i !s!t! !w !i !t!h ! !i !t!s! !s!i !g !n !s! !r!e !v!e !r!s!e !d !.! !S!o !m!e ! !o !f! !t!h !e !s!e ! !r!e !l !a !t!i !o !n !s! are ! !d !e !r!i !v!e !d ! !i !n a f!o !o !t!n !o !t!e !.! !!U !s!i !n !g ! !t!h !e !s!e ! !r!e !l !a !t!i !o !n !s! !c!a !n ! !s!i !m!p !l !i !f!y! !c!a !l !c!u !l !a !t!i !n !g ! !m!e !a !s!u !r!e !s! !o !f! !l !o !c!a !t!i !o !n ! !o !r! !s!p !r!e !a !d ! !w !h !e !n ! !t!h !e !!u !n !i !t!s! !o !f! !m!e !a !s!u !r!e !m!e !n !t! !a !r!e ! !c!h !a !n !g !e !d !.! !T!h !e ! !f!o !l !l !o !w !i !n !g ! !e !x!e !r!c!i !s!e ! !c!h !e !c!ks!! !y!o !u !r! !a !b !i !l !i !t!y! !t!o ! !u !s!e !!t!h !e !s!e ! !r!u !l !e !s!.

Exercise 4-9.

1. The mean of a list is 6. Consider multiplying each element of the list by 8 thenadding 10 to get a new list. What is the mean of the new list?

2. The mode of a particular list is unique and equal to 7. Consider multiplyingeach element of the list by 8 then adding 10 to get a new list. What is themode of the new list?

3. The median of a particular list is 16. Consider multiplying each element of thelist by 8 then adding 10 to get a new list. What is the median of the new list?

4. The SD of a list is 22. Consider multiplying each element of the list by 8 thenadding 10 to get a new list. What is the SD of the new list?

5. The IQR of a list is 24. Consider multiplying each element of the list by 8 thenadding 10 to get a new list. What is the IQR of the new list?

6. The range of a list is 26. Consider multiplying each element of the list by 8then adding 10 to get a new list. What is the range of the new list?

SOLUTIONS: 1. 58 2. 66 3. 138 4. 176 5. 192 6. 208

Markov's Inequality and Chebychev's Inequality



Measures of location and spread can tell us a great deal about lists of numbers. Forexample, for any list, at least half the numbers in the list are no larger than the median,and at least half the numbers in the list are at least as large as the median (this is oneway of defining the median). The mean and SD also can tell us about the fractions ofvalues in a list in various ranges.

Suppose that a list of numbers contains no negative number, and that 10% of thevalues in the list are greater than or equal to 50. What is the smallest the mean of thelist could be? The mean would be smallest if all the values in the list were as small asthey could be, subject to the constraints that the values were not negative, and 10%equal or exceed 50. If 90% of the values were equal to zero, and the rest were equal to50, that would give the smallest mean:

!0 ! !x! !0 !.!9 ! !+! !5 !0 ! !x! !0 !.!1 ! !=! !5 !.

That is, if a list contains no negative number, and 10% of the numbers in the list are 50or larger, then the mean of the list must be at least 5. More generally, if any particularfraction of values in a list exceeds a given threshold, and none of the values in the list isnegative, then the mean of the list cannot be arbitrarily small. Markov's inequality turnsthis idea upside down to limit the fraction of numbers in a list that can exceed any giventhreshold, provided the list contains no negative number. The limit depends on themean of the list, and the threshold.

Markov's Inequality (for lists)

If the mean of a list of numbers is M, and the list contains no negative numberthen:

[fraction of numbers in the list that are greater than or equal to x] ≤ M/x.

!N !o!t!e ! !4 !-!7 !:! !Heuristic d !e !r!i !v!a !t!i !o !n ! !o !f! !M!a !r!k!o !v!'!s! !i !n !e !q !u !a !l !i !t!y!.! !T!h !e ! !b !a !s!i !c! !i !d !e !a ! !i !s! !t!h !a !t! !o !f! !a !!s!e !e !-!s!a !w ! !o !r! !t!e !e !t!e !r!-!t!o !t!t!e !r!:! !t!o ! !b !a !l !a !n !c!e ! !a ! !l !a !r!g !e ! !w !e !i !g !h !t!,! !t!h !e ! !o !t!h !e !r! !w !e !i !g !h !t! !s!h !o !u !l !d ! !b !e ! !a !s!!f!a !r! !a !s! !p !o !s!s!i !b !l !e ! !f!r!o !m! !t!h !e ! !f!u !l !c!r!u !m!.! !T!h !e ! !c!o !n !s!t!r!a !i !n !t! !t!h !a !t! !n !o !n !e ! !o !f! !t!h !e ! !e !l !e !m!e !n !t!s! !i !n ! !t!h !e !!l !i !s!t! !i !s! !b !e !l !o !w ! !z!e !r!o ! !l !i !m!i !t!s! !h !o !w ! !f!a !r! !t!o ! !t!h !e ! !l !e !f!t! !w !e ! !c!a !n ! !p !u !t! !t!h !e ! !b !a !l !a !n !c!i !n !g ! !w !e !i !g !h !t!s!.!!S!u !p !p !o !s!e ! !t!h !e ! !a !v!e !r!a !g !e ! !o !f! !a ! !l !i !s!t! !o !f! !n !o !n !n !e !g !a !t!i !v!e ! !n !u !m!b !e !r!s! !i !s! !M!,! !a !n !d ! !t!h !a !t! !a ! !f!r!a !c!t!i !o !n ! !w !!o !f! !t!h !e ! !e !l !e !m!e !n !t!s! !o !f! !t!h !e ! !l !i !s!t! !a !r!e ! !a !t! !o !r! !a !b !o !v!e ! !s!o !m!e ! !v!a !l !u !e ! !x!.! !W!e ! !w !a !n !t! !t!o ! !m!a !k!e ! !t!h !a !t!!f!r!a !c!t!i !o !n ! !a !s! !l !a !r!g !e ! !a !s! !p !o !s!s!i !b !l !e !,! !w !h !i !l !e ! !k!e !e !p !i !n !g ! !t!h !e ! !a !v!e !r!a !g !e ! !e !q !u !a !l ! !t!o ! !M!.! !T!o ! !m!a !k!e ! !t!h !e !!f!r!a !c!t!i !o !n ! !w ! !a !s! !l !a !r!g !e ! !a !s! !p !o !s!s!i !b !l !e !,! !w !e ! !s!h !o !u !l !d ! !(!a !)! !p !u !t! !t!h !e ! !r!e !s!t! !o !f! !t!h !e ! !l !i !s!t! !a !t! !z!e !r!o !,! !a !n !d !



!(!b !)! !p !u !t! !t!h !e ! !e !n !t!i !r!e ! !f!r!a !c!t!i !o !n ! !w ! !a !t! !t!h !e ! !p !o !i !n !t! !x! !(!a !n !d ! !n !o !t! !p !a !r!t!l !y! !a !b !o !v!e ! !x!)!.! !T!h !e ! !a !v!e !r!a !g !e ! !i !s!!t!h !e !n !!! M! !=! !w ! x !x! !+! !(!1 !0 !0 !% - !w !)!x !0 ! !=! !w !x!x!.!!

! S!o !l !v!i !n !g ! !f!o !r! !w !,! !w !e ! !g !e !t! !w ! !=! !m!/!x! !a !s! !t!h !e ! !l !a !r!g !e !s!t! !p !o !s!s!i !b !l !e ! !f!r!a !c!t!i !o !n ! !o !f! !n !u !m!b !e !r!s! !i !n ! !t!h !e ! !l !i !s!t! !t!h !a !t! !c!a !n ! !b !e ! !e !q !u !a !l ! !t!o ! !o !r! !g !r!e !a !t!e !r! !t!h !a !n ! !x!.! !T!h !i !s! !i !s! Markov’s inequality!.

Example 4-4: Applying Markov's inequality.There are 200 students in a class. The average amount of money in their pocketsis $15. How many could have $75 or more in their pockets?

SOLUTION: No student can have a negative amount of money in his or herpocket, so Markov's inequality applies. Markov's inequality guarantees that

[fraction of students with at least $75 in their pockets] ≤ $15/$75= 0.2 = 20%.

Thus at most 20% of the students (40 students) could have $75 or more in theirpockets.

If we know the mean of a list and its SD, we know something about how many of thenumbers in the list must be in various ranges. Suppose that 25% of the numbers in a listdiffer from the mean by 30 or more. How small could the SD of the list be? To make theSD smallest, all the numbers should be as close as possible to the mean, subject to theconstraint that at least 25% of them differ from the mean by 30 or more. This isachieved by making 75% of the numbers equal to the mean, 12.5% equal to the meanminus 30, and 12.5% equal to the mean plus 30. Thus the SD of the list must be at least

!(! !0 !.!1 !2 !5 ! !x! !3 !0 !2! !+! !0 !.!7 !5 ! !x !0 !2! !+! !0 !.1 !2 !5 ! !x !3 !0 !2! !)!1/2 ! !=! !1 !5 !.

More generally, if a particular fraction of the values differ from the mean of the list by atleast a given threshold, then the SD of the list cannot be too small. Chebychev'sinequality turns this around to find a bound on the fraction of numbers in the list thatdiffer from the mean by more than any given threshold. The bound depends on the SDof the list and the threshold.



C !h !e !b !y!c!h !e !v!'!s! !i !n !e !q !u !a !l !i !t!y! !(!f!o !r! !l !i !s!t!s!)!!!I!f! !t!h !e ! !m!e !a !n ! !o !f! !a ! !l !i !s!t! !o !f! !n !u !m!b !e !r!s! !i !s! !M! !a !n !d ! !t!h !e ! !s!t!a !n !d !a !r!d ! !d !e !v!i !a !t!i !o !n ! !o !f! !t!h !e ! !l !i !s!t! !i !s! !S!D !,!!t!h !e !n ! !f!o !r! !e !v!e !r!y! !p !o !s!i !t!i !v!e ! !n !u !m!b !e !r! !k!,!![!t!h !e ! !f!r!a !c!t!i !o !n ! !o !f! !n !u !m!b !e !r!s! !i !n ! !t!h !e ! !l !i !s!t! !t!h !a !t! !a !r!e ! !k!x!S!D ! !o !r! !f!u !r!t!h !e !r! !f!r!o !m! !M!]! !≤ 1 !/!k!2!.

!N !o!t!e ! !4 !-!8 !:! Heuristic d !e !r!i !v!a !t!i !o !n ! !o !f! !C !h !e !b !y!c!h !e !v!'!s! inequality.! !C !h !e !b !y!c!h !e !v!'!s! !i !n !e !q !u !a !l !i !t!y!!c!a !n ! !b !e ! !d !e !r!i !v!e !d ! !f!r!o !m! !M!a !r!k!o !v!'!s! !i !n !e !q !u !a !l !i !t!y!,! !b !y! !c!o !n !s!i !d !e !r!i !n !g ! !t!h !e ! !l !i !s!t! !o !f! !s!q !u !a !r!e !d !!d !e !v!i !a !t!i !o !n !s! !f!r!o !m! !t!h !e ! !m!e !a !n !.! !T!h !e ! !l !i !s!t! !o !f! !s!q !u !a !r!e !d ! !d !e !v!i !a !t!i !o !n !s! !f!r!o !m! !t!h !e ! !m!e !a !n ! !c!a !n !n !o !t!!h !a !v!e ! !n !e !g !a !t!i !v!e ! !e !n !t!r!i !e !s!,! !s!o !!! a !v!e !r!a !g !e ! !o !f! !s!q !u !a !r!e !d ! !d !e !v!i !a !t!i !o !n !s! !f!r!o !m! !t!h !e ! !m!e !a !n !!f!r!a !c!t!i !o !n ! !o !f! !s!q !u !a !r!e !d ! !d !e !v!i !a !t!i !o !n !s! !t!h !a !t! !a !r!e ! !x! !o !r! !l !a !r!g !e !r ≤ !-!-!-!-!-!-!-!-!-!-!-!-!-!-!-!-!-!-!-!-!-!-!-!-!-!-!-!-!-!-!-!-!-!-!-!-!-!-!-!-!-!-!-!-!-!-!-!-!-!-!-!-!-!-!-!-!-!-!-!-! !.!! x!

!T!h !e ! !f!r!a !c!t!i !o !n ! !o !f! !s!q !u !a !r!e !d ! !d !e !v!i !a !t!i !o !n !s! !f!r!o !m! !t!h !e ! !m!e !a !n ! !t!h !a !t! !a !r!e ! !x! !o !r! !l !a !r!g !e !r! !i !s! !t!h !e ! !s!a !m!e !!a !s! !t!h !e ! !f!r!a !c!t!i !o !n ! !o !f! !d !a !t!a ! !t!h !a !t! !a !r!e ! !x!1/2 !o !r! !m!o !r!e ! !f!r!o !m! !t!h !e ! !m!e !a !n !.! !N !o !w ! !s!u !b !s!t!i !t!u !t!e ! !x!1/2 ! !=!!k!x!S!D !.! !T!h !a !t! !g !i !v!e !s!! !! a !v!e !r!a !g !e ! !o !f! !s!q !u !a !r!e !d ! !d !e !v!i !a !t!i !o !n !s! !f!r!o !m! !t!h !e ! !m!e !a !n !!f!r!a !c!t!i !o !n ! !o !f! !d !a !t!a !! !t!h !a !t! !a !r!e ! !k!xS!D ! !o !r! !f!u !r!t!h !e !r! !f!r!o !m! !!m!e !a !n ≤ !!-!-!-!-!-!-!-!-!-!-!-!-!-!-!-!-!-!-!-!-!-!-!-!-!-!-!-!-!-!-!-!-!-!-!-!-!-!-!-!-!-!-!-!-!-!-!-!-!-!-!-------!-!.! (!k!xS!D !)!2!!R !e !c!a !l !l ! !t!h !a !t! !t!h !e ! !S!D ! !i !s! !t!h !e ! !s!q !u !a !r!e !-!r!o !o !t! !o !f! !t!h !e ! !a !v!e !r!a !g !e ! !o !f! !t!h !e ! !s!q !u !a !r!e !d ! !d !e !v!i !a !t!i !o !n !s!!f!r!o !m! !t!h !e ! !m!e !a !n !,! !s!o ! !t!h !e ! !n !u !m!e !r!a !t!o !r! !o !n ! !t!h !e ! !r!i !g !h !t! !h !a !n !d ! !s!i !d !e ! !i !s! !S!D !2!.! !S!u !b !s!t!i !t!u !t!i !n !g ! !i !n !t!o !!t!h !e ! !n !u !m!e !r!a !t!o !r!,! !a !n !d ! !c!a !n !c!e !l !i !n !g ! !t!h !e ! !f!a !c!t!o !r! !o !f! !S!D !2! !i !n ! !t!h !e ! !n !u !m!e !r!a !t!o !r! !w !i !t!h ! !t!h !a !t! !i !n ! !t!h !e !!d !e !n !o !m!i !n !a !t!o !r!,! !g !i !v!e !s!! !! 1 !!f!r!a !c!t!i !o !n ! !o !f! !d !a !t!a ! ! !t!h !a !t! !a !r!e ! !k!xS!D ! !o !r! !f!u !r!t!h !e !r! !f!r!o !m! !t!h !e ! !m!e !a !n !! ≤ !-!-!-!-!-!-! !,!! k !2!

!w !h !i !c!h ! !i !s! !C !h !e !b !y!c!h !e !v!'!s! !i !n !e !q !u !a !l !i !t!y!.

Chebychev's inequality says that not too many of the numbers in a list can be far fromthe mean, where far is measured in standard deviations. Conversely, if a large fractionof the values are far from the mean, the SD of the list must be large. Table 4-5 listssome specific bounds implied by Chebychev's inequality:



Table 4-5: Bounds implied by Chebychev's inequality.

Number of standard deviations Largest possible fraction of valuesthis far or further from the mean

1 100%2 25%3 11.11%4 6.25%5 4%6 2.78%

Example 4-5 illustrates applying Chebychev's inequality to find bounds on the fraction ofweights in a given range from the mean and SD of a list of weights.

!E!x !a !m !p!l!e ! !4 !-!5 !:! !A!p !p !l !y!i !n !g ! !C !h !e !b !y!c!h !e !v!'!s! !i !n !e !q !u !a !l !i !t!y!.!!!T!h !e ! !m!e !a !n ! !w !e !i !g !h !t! !o !f! !s!t!u !d !e !n !t!s! !i !n ! !a ! !c!e !r!t!a !i !n ! !c!l !a !s!s! !o !f! !s!t!u !d !e !n !t!s! !i !s! !1 !4 !0 ! !l !b !s!,! !a !n !d ! !t!h !e ! !S!D !!o !f! !t!h !e !i !r! !w !e !i !g !h !t!s! !i !s! !3 !0 ! !l !b !s!.! !W!h !a !t! !f!r!a !c!t!i !o !n ! !w !e !i !g !h !s! !b !e !t!w !e !e !n ! !9 !0 ! !l !b !s!! an !d ! !1 !9 !0 ! !l !b !s!!? !!!SOLUTION: !W!e ! !c!a !n !n !o !t! !g !e !t! !a !n ! !e !x!a !c!t! !a !n !s!w !e !r!,! !b !u !t! !w !e ! !c!a !n ! !g !e !t! !a ! !l !o !w !e !r! !b !o !u !n !d ! !u !s!i !n !g !!C !h !e !b !y!c!h !e !v!'!s! !i !n !e !q !u !a !l !i !t!y!.! !T!h !e ! !r!a !n !g !e ! !f!r!o !m! !9 !0 ! !l !b !s!! !t!o ! !1 !9 !0 ! !l !b !s!! !i !s! !t!h !e ! !m!e !a !n !,! !p !l !u !s! !o !r!!m!i !n !u !s! !5 !0 ! !l !b !s!.! !5 !0 ! !l !b !s!! !i !s! !1 ! !2 !/!3 ! !t!i !m!e !s! !t!h !e ! !S!D ! !o !f! !t!h !e ! !w !e !i !g !h !t!s!,! !s!o ! !a !c!c!o !r!d !i !n !g ! !t!o !!C !h !e !b !y!c!h !e !v!'!s! !i !n !e !q !u !a !l !i !t!y!,! !t!h !e ! !f!r!a !c!t!i !o !n ! !o !f! !s!t!u !d !e !n !t!s! !w !h !o ! !w !e !i !g !h ! !l !e !s!s! !t!h !a !n ! !9 !0 ! !l !b !s! !o !r!!m!o !r!e ! !t!h !a !n ! !1 !9 !0 ! !l !bs is! !a !t! !m!o !s!t!!! 1 !/!(!1 ! !2 !/!3 !)!2! !=! !1 !/!(!1 !.!6 !6 !6 !7 !)!2! !=! !0 !.!3 !6 ! !=! !3 !6 !% !.!!

! T!h !u !s! !t!h !e ! !f!r!a !c!t!i !o !n ! !w !h !o ! !w !e !i !g !h ! !b !e !t!w !e !e !n ! !9 !0 ! !l !b !s! a !n !d ! !1 !9 !0 ! !l !b !s! !i !s! !a !t! !l !e !a !s!t! !1 !0 !0 !% ! - 3 !6 !% ! !=! !6 !4 !% !.

In some problems, it is possible to apply both Markov's inequality and Chebychev'sinequality. When that happens, use whichever inequality gives the more preciseanswer—that is, the inequality that limits the fraction most stringently. Example 4-6illustrates this idea.

!E!x !a !m !p!l!e ! !4 !-!6 !:! !S!o !m!e !t!i !m!e !s! !M!a !r!k!o !v!'!s! !i !n !e !q !u !a !l !i !t!y! !a !n !d ! !C !h !e !b !y!c!h !e !v!'!s! !i !n !e !q !u !a !l !i !t!y! !b !o !t!h !!a !p !p !l !y!.!



!O!n ! !t!h !e ! !a !v!e !r!a !g !e !,! !i !t! !t!a !k!e !s! !4 !5 ! !m!i !n !u !t!e !s! !t!o ! !c!r!o !s!s! !t!h !e ! !S!a !n ! !F!r!a !n !c!i !s!c!o ! !B!a !y! !B!r!i !d !g !e !!d !u !r!i !n !g ! !r!u !s!h ! !h !o !u !r!.! !T!h !e ! !S!D ! !o !f! !t!h !e ! !t!i !m!e ! !i !t! !t!a !k!e !s! !t!o ! !c!r!o !s!s! !t!h !e ! !b !r!i !d !g !e ! !i !s! !1 !5 ! !m!i !n !u !t!e !s!.!!W!h !a !t!'!s! !t!h !e ! !l !a !r!g !e !s!t! !f!r!a !c!t!i !o !n ! !o !f! !t!h !e ! !t!i !m!e ! !i !t! !c!o !u !l !d ! !t!a !k!e ! !m!o !r!e ! !t!h !a !n ! !2 ! !h !o !u !r!s! !t!o ! !c!r!o !s!s! !t!h !e !!b !r!i !d !g !e !? !!!SOLUTION: T!r!a !v!e !l ! !t!i !m!e ! !i !s! !p !o !s!i !t!i !v!e !,! !s!o ! !w !e ! can !u !s!e Markov’s inequality.! !B!y!!M!a !r!k!o !v!'!s! !i !n !e !q !u !a !l !i !t!y!,!!! [!f!r!a !c!t!i !o !n ! !o !f! !t!h !e ! !t!i !m!e ! !i !t! !t!a !k!e !s! !m!o !r!e ! !t!h !a !n ! !2 ! !h !o !u !r!s!]! ≤ !(!4 !5 ! !m!i !n !u !t!e !s!)!/!(!2 ! !h !o !u !r!s!)!! =! !(!4 !5 ! !m!i !n !u !t!e !s!)!/!(!1 !2 !0 ! !m!i !n !u !t!e !s!)!! =! !0 !.!3 !7 !5 ! !=! !3 !7 !.!5 !% !.!!!O!n ! !t!h !e ! !o !t!h !e !r hand, !w !e ! can also ! !a !p !p !l !y! !C !h !e !b !y!c!h !e !v!'!s! inequality!,! as f!o !l !l !o !w !s!.!!! 2 ! !h !o !u !r!s! !=! !1 !2 !0 ! !m!i !n !u !t!e !s! !=! !4 !5 ! !m!i !n !u !t!e !s! !+! !7 !5 ! !m!i !n !u !t!e !s!

= mean !t!i !m!e ! !+! !7 !5 ! !m!i !n !u !t!e !s! !=! mean !t!i !m!e ! !+! !5 !S!D !!!T!h !a !t! !i !s!,! !t!w !o ! !h !o !u !r!s! !i !s! !5 !S!D ! !a !b !o !v!e ! !t!h !e ! mean.! !O!n ! !t!h !e ! !o !t!h !e !r! !h !a !n !d !,! !5 !S!D ! !b !e !l !o !w ! !t!h !e !mean is!!! 4 !5 ! !m!i !n !u !t!e !s! - ! !5 ! x (!1 !5 ! !m!i !n !u !t!e !s!)! !=! - 3 !0 ! !m!i !n !u !t!e !s!.!!!T!h !i !s! !i !s! !n !o !t! !a ! !p !o !s!s!i !b !l !e ! !t!r!a !v!e !l ! !t!i !m!e ! !(!i !t! !a !l !w !a !y!s! !t!a !k!e !s! !a ! !p !o !s!i !t!i !v!e ! !a !m!o !u !n !t! !o !f! !t!i !m!e ! !t!o !!c!r!o !s!s! !t!h !e ! !b !r!i !d !g !e !)!.! !T!h !u !s! !t!h !e ! !f!r!a !c!t!i !o !n ! !o !f! !t!h !e ! !t!i !m!e ! !i !t! !t!a !k!e !s! !m!o !r!e ! !t!h !a !n ! !2 ! !h !o !u !r!s! !o !r! !l !e !s!s!!t!h !a !n ! -!3 !0 ! !m!i !n !u !t!e !s! !t!o ! !c!r!o !s!s! !t!h !e ! !b !r!i !d !g !e ! !m!u !s!t! !e !q !u !a !l ! !t!h !e ! !t!i !m!e ! !i !t! !t!a !k!e !s! !m!o !r!e ! !t!h !a !n ! !2 !!h !o !u !r!s! !t!o ! !c!r!o !s!s! !t!h !e ! !b !r!i !d !g !e !.! !B!y! !C !h !e !b !y!c!h !e !v!'!s! !i !n !e !q !u !a !l !i !t!y!,!!![!f!r!a !c!t!i !o !n ! !o !f! !t!h !e ! !t!i !m!e ! !i !t! !t!a !k!e !s! !l !e !s!s! !t!h !a !n ! -!3 !0 ! !m!i !n !u !t!e !s! !o !r! !m!o !r!e ! !t!h !a !n ! !2 ! !h !o !u !r!s!]! ≤! !1 !/!5 !2!!=! !1 !/!2 !5 ! !=! !4 !% !.!!!B!e !c!a !u !s!e ! !t!h !e ! !f!r!a !c!t!i !o !n ! !o !f! !t!h !e ! !t!i !m!e ! !i !t! !t!a !k!e !s! !m!o !r!e ! !t!h !a !n ! !2 ! !h !o !u !r!s! !o !r! !l !e !s!s! !t!h !a !n ! -!3 !0 !!m!i !n !u !t!e !s! !t!o ! !c!r!o !s!s! !t!h !e ! !b !r!i !d !g !e ! !i !s! !t!h !e ! !s!a !m!e ! !a !s! !t!h !e ! !t!i !m!e ! !i !t! !t!a !k!e !s! !m!o !r!e ! !t!h !a !n ! !2 ! !h !o !u !r!s!,!!w !e ! !h !a !v!e !!! [!f!r!a !c!t!i !o !n ! !o !f! !t!h !e ! !t!i !m!e ! !i !t! !t!a !k!e !s! !m!o !r!e ! !t!h !a !n ! !2 ! !h !o !u !r!s!]! ≤ ! !4 !% !.!

! T!h !i !s! !i !s! !a ! !m!o !r!e ! !r!e !s!t!r!i !c!t!i !v!e ! !b !o !u !n !d ! !t!h !a !n ! !t!h !e ! !o !n !e ! !M!a !r!k!o !v!'!s! !i !n !e !q !u !a !l !i !t!y! !g !i !v!e !s! !i !n ! !t!h !i !s! !p !r!o !b !l !e !m! !(!M!a !r!k!o !v!'!s! !i !n !e !q !u !a !l !i !t!y! !g !a !v!e ! !3 !7 !.!5 !% !)! !s!o ! !w !e ! !s!h !o !u !l !d ! !u !s!e ! !i !t! !i !n !s!t!e !a !d !.! !(!L !a !r!g !e !r! !l !o !w !e !r! !b !o !u !n !d !s! !a !r!e ! !b !e !t!t!e !r!;! !s!m!a !l !l !e !r! upper b !o !u !n !d !s! !a !r!e ! !b !e !t!t!e !r!.!)



The following exercises check your ability to apply Markov's inequality and Chebychev'sinequality.

Exercise 4-10. According to Chebychev's inequality, at least what decimalfraction of a list of numbers must be within 5.6 SD of the mean?

SOLUTION: !B!y! !C !h !e !b !y!c!h !e !v!'!s! !i !n !e !q !u !a !l !i !t!y! !f!o !r! !l !i !s!t!s!,! !t!h !e ! !f!r!a !c!t!i !o !n ! !o !f! !o !b !s!e !r!v!a !t!i !o !n !s!!b !e !y!o !n !d ! !5 !.!6 ! !S!D !s! !o !f! !t!h !e ! !m!e !a !n ! !i !s! !a !t! !m!o !s!t! !1 !/!(!5 !.!6 !)!2! !=! !0 !.!0 !3 !2 ! !s!o ! !t!h !e ! !f!r!a !c!t!i !o !n ! !w !i !t!h !i !n ! !5 !.!6 !!S!D !s! !o !f! !t!h !e ! mean !i !s! !a !t! !l !e !a !s!t! !1 ! -! !0 !.!0 !3 !2 ! !=! !0 !.!9 !6 !8 !.

Exercise 4-11. A student has a GPA (grade point average) of 3.5. In eachcourse she takes, she gets a grade between 0 (failing) and 4.0 (A+). What is thelargest decimal fraction of her grades that could be 4 or higher?

SOLUTION: By Markov's inequality, the largest fraction of grades greater than orequal to 4 is at most (3.5)/(4) = 0.88.

Exercise 4-12. A certain type of light bulb has an average lifetime of 10,000hours. The SD of bulb lifetimes is 490 hours. What decimal fraction of bulbscould last more than 12,303 hours?

SOLUTION: 0.0443 to 0.0463.

Summary

This chapter introduced several ways to summarize lists of numbers, quantitative data.Some summaries, measures of location, seek to be as close as possible to everyelement of the list—to typify the elements. The mean, median, and mode areexamples: They represent typical values of the list. The mean, median, and mode eachare "as close as possible" to all the elements in the list, for different definitions of theproximity of two numbers: for the mean, the distance between two numbers is thesquare of their difference; for the median, the distance between two numbers is theabsolute value of their difference; and for the mode, the distance between two numbersis 1 if the numbers differ, 0 if they are equal. The mean is the sum of the elements,divided by the number of elements. The median is the smallest element that is at leastas large as at least half the elements. The mode is the most common value in the list.The mode makes sense for qualitative and categorical data as well as quantitativedata, but the mean and median make sense only for quantitative data. The mean,



median, and mode differ in their sensitivity to changes to the data, or resistance. Astatistic that can be changed arbitrarily by altering a single datum is not resistant. Themedian is resistant. The mean is not resistant. The resistance of the mode depends onthe distribution of values in the list.

The RMS (root mean square) measures the average size of the elements of a list,without regard to their signs. The RMS is not resistant. Other summaries, measures ofspread, reflect how the values of the list differ from each other. Examples include therange, the SD (standard deviation), and the IQR (inter-quartile range). The range of alist of numbers is the largest number minus the smallest number. The range is zero ifand only if all the numbers in the list are equal. The range is not resistant. The SDmeasures the average size of the differences between the mean and the elements ofthe list: It is the RMS of the list of deviations from the mean. The SD of a list is zero ifand only if all the numbers in the list are equal. The SD is not resistant. The IQR is theupper quartile minus the lower quartile. It is the width of an interval that contains themiddle half of the data—25% below the median and 25% above the median. The IQRcan be zero even if not all the numbers are equal, but the middle 50% must be equal.The IQR is resistant. If the units of measurement change by an affine transformation,measures of location and spread in the new units of measurement have simplerelationships to their values in the old units.

Measures of location and spread contain a surprising amount of information about listsof numbers: Markov's inequality limits the fraction of elements of the list that exceedany given threshold, in terms of the mean of the list and the threshold, provided the listcontains no negative number. Chebychev's inequality limits the fraction of elementswhose difference from the mean of the list exceeds any given threshold, in terms of theSD of the list and the threshold.

Key Termso affine transformationo arithmetic meano averageo categoricalo Chebychev's inequalityo class intervalo deviationo discreteo histogramo independent



o interquartile range (IRQ)o lower quartileo Markov's inequalityo meano measures of locationo mediano modeo monotonic functiono percentileo qualitativeo quartileo rangeo resistanto RMSo skewedo spreado standard deviation (SD)o statisticso symmetrico upper quartileo variability

Documents

SticiGui “Chapter 4: Measures of Location and Spread ... · SticiGui “Chapter 4: Measures of Location and Spread” Philip Stark (2013) Summarizing data can help us understand