Excursions in Modern Mathematics, 7e: 14.3 - 2Copyright © 2010 Pearson Education, Inc. 14...

Excursions in Modern Mathematics, 7e: 14.3 - 2Copyright © 2010 Pearson Education, Inc.

14 Descriptive Statistics

14.1 Graphical Descriptions of Data

14.2 Variables

14.3 Numerical Summaries

14.4 Measures of Spread


Measures of Location

Measures of location such as the mean (or average), the median, and the quartiles, are numbers that provide information about the values of the data.

Numerical Summaries of a Data Set


The best known of all numerical summaries of data is the average, also called the mean. There is no universal agreement as to which of these names is a better choice–in some settings mean is a better choice than average, in other settings it’s the other way around. In this chapter we will use whichever seems the better choice at the moment.

Average or Mean


The average (or mean) of a set of N numbers is found by adding the numbers and dividing the total by N. In other words, the average of the numbers

d1, d2, d3,…, dN

is

A = (d1 + d2 + d3 +…+ dN)/N.

Average or Mean


In this example we will find the average test score in the Stat 101 exam first introduced in Example 14.1. To find this average we need to add all the test scores and divide by 75. The addition of the 75 test scores can be simplified considerably if we use a frequency table.

Example 14.9 Stat 101 Test Scores: Part 4


From the frequency table we can find the sum S of all the test scores as follows: Multiply each test score by its corresponding frequency and then add these products. Thus, the sum of all the test scores is

S = (1 1) + (6 1) + (7 2) + (8 6) + …+

(16 1) + (24 1) = 814

If we divide this sum by N = 75, we get the average test score A = 814/75 ≈ 10.85 points.



In general, to find the average A of a data set given by a frequency table such as Table 14-8 we do the following:Step 1.

S = d1•f1 + d2•f2 +… + dk•fk

To Find the Average

Step 2.

N = f1 + f2 +…+ fk Step 3.A = S/N


Imagine that you just read in the paper the following remarkable tidbit: The average starting salary of philosophy majors who recently graduated from Tasmania State University is $76,400 a year! This is quite an impressive number, but before we all rush out to change majors, let’s point out that one of the graduating philosophy majors happens to be basketball star “Hoops” Tallman, who is doing his thing in the NBA for a starting salary of $3.5 million a year.

Example 14.10 Starting Salaries of Philosophy Majors


If we were to take this one outlier out of the population of 75 philosophy majors, we would have a more realistic picture of what philosophy majors are making.Here is how we can do it.


■ The total of all 75 salaries is 75 times the average salary:

75 $76,400 = $5,730,000


■ The total of the other 74 salaries (excluding Hoops’s cool 3.5 mill) is

$5,730,000 – $3,500,000 = $2,230,000

■ The average of the remaining 74 salaries is $2,230,000/74 ≈ $30,135



Table 14-9 shows the monthly balance (monthly income minus monthly spending) in Billy’s budget over the past year. A negative amount indicates that Billy spent more than what he had coming in (adding to his credit card debt).

Example 14.11 Living Beyond Your Means


In spite of his consistent overspending, Billy’s average monthly balance for the year is $26 (check it out!). This average hides the true picture of what is going on. Billy is living well beyond his means but was bailed out by a lucky break and a generous mom.

Example 14.11 Living Beyond Your Means


While a single numerical summary–such as the average–can be useful, it is rarely sufficient to give a meaningful description of a data set. A better picture of the data set can be presented by using a well-organized cadre of numerical summaries. The most common way to do this is by means of percentiles. The pth percentile of a data set is a value such that p percent of the numbers fall at or below this value and the rest fall at or above it. It essentially splits a data set into two parts: the lower p% of the data values and the upper of the data values.

Percentiles


There are several different ways to compute percentiles that will satisfy the definition,and different statistics books describe different methods. We will illustrate one such method.The first step is to sort the numbers from smallest to largest. Let’s denote the sorted data values by d1, d2, d3, … , dN , where d1 represents the smallest number in the data set, d2 the second smallest number, and so on; d3.5 represents the average of the data values d3 and d4, d7.5 represents the average of the data values d7 and d8.

Percentiles


The next, and most important, step is to identify which d represents the pth percentile of the data set. To do this, we compute the pth percent of N, which we will call the locator and denote by the letter L. [In other words, ] If L happens to be a whole number, then the pth percentile will be dL.5 (the average of dL and dL+1). If L is not a whole number, then the pth percentile will be dL+ where L+ represents the value of L rounded up.

Percentiles


■ Step 0.Sort the data set from smallest to largest. Let d1, d2, d3, … , dN represent the sorted data.

■ Step 1.Find the locator L = (p/100)•N.

FINDING THE pTH PERCENTILE OF A DATA SET


■ Step 2.Depending on whether L is a whole number or not, the pth percentile is given by

■ dL.5 if L is a whole number.

■ dL+ if L is not a whole number(L+ is L rounded up).

FINDING THE pTH PERCENTILE OF A DATA SET


To reward good academic performance from its athletes, Tasmania State University has a program in which athletes with GPAs in the top 20th percentile of their team’s GPAs get a $5000 scholarship and athletes with GPAs in the top forty-fifth percentile of their team’s GPAs who did not get the $5000 scholarship get a $2000 scholarship.

Example 14.12 Scholarships by Percentiles


The women’s soccer team has N = 15 players. A list of their GPAs is as follows:

3.42, 3.91, 3.33, 3.65, 3.57, 3.45, 4.0, 3.71, 3.35, 3.82, 3.67, 3.88, 3.76, 3.41, 3.62

When we sort these GPAs we get the list

3.33, 3.35, 3.41, 3.42, 3.45, 3.57, 3.62, 3.65, 3.67, 3.71, 3.76, 3.82, 3.88, 3.91, 4.0



Since this list goes from lowest to highest GPA, we are looking for the 80th percentile and above (top 20th percentile) for the $5000 scholarships and the 55th percentile and above (top 45th percentile) for the $2000 scholarships.



$5000 scholarships: The locator for the 80th percentile is (0.8) 15 = 12. Here the locator is a whole number, so the 80th percentile is given by d12.5 = 3.85 (the average between d12 = 3.82 and d13 = 3.88). Thus, three students (the ones with GPAs of 3.88, 3.91 and 4.0) get $5000 scholarships.



$2000 scholarships: The locator for the 55th percentile is (0.55) 15 = 8.25. Here the locator is not a whole number, so we round it up to 9, and the 55th percentile is given byd9 = 3.67. Thus, the students with GPAs of 3.67, 3.71, 3.76 and 3.82 (all students with GPAs of 3.67 or higher except the ones that already received $5000 scholarships) get $2000 scholarships.



The 50th percentile of a data set is known as the median and denoted by M. The median splits a data set into two halves–half of the data is at or below the median and half of the data is at or above the median.

Median


■ Sort the data set from smallest to largest. Let d1, d2, d3, … , dN represent the sorted data.

■ If N is odd, the median is

■ If N is even, the median is the average of

FINDING THE MEDIANOF A DATA SET

d

N+12

.

d

N

2

and dN

2+1.


After the median, the next most commonly used set of percentiles are the first and third quartiles. The first quartile (denoted by Q1) is the 25th percentile, and the third quartile (denoted by Q3) is the 75th percentile.

Quartiles


During the last year, 11 homes sold in the Green Hills subdivision. The selling prices, in chronological order, were $267,000, $252,000, $228,000, $234,000, $292,000, $263,000, $221,000, $245,000, $270,000, $238,000, and $255,000. We are going to find the median and the quartiles of the N = 11 home prices.

Example 14.13 Home Prices in Green Hills


Sorting the home prices from smallest to largest (and dropping the 000’s) gives the sorted list

221, 228, 234, 238, 245, 252, 255, 263, 267, 270, 292



The locator for the median is (0.5) 11 = 5.5, the locator for the first quartile is(0.25) 11 = 2.75, and the locator for the third quartile is (0.75) 11 = 8.25. Since these locators are not whole numbers, they must be rounded up:5.5 to 6,2.75 to 3, and8.25 to 9.



Thus, the median home price is given by

d6 = 252 (i.e., M = $252,000),

the first quartile is given by

d3 = 234 (i.e., M = $234,000),

and the third quartile is given by

d9 = 267 (i.e., M = $267,000).



Oops! Just this morning a home sold in Green Hills for $264,000. We need to recalculate the median and quartiles for what are now N = 12 home prices. We can use the sorted data set that we already had–all we have to do is insert the new home price (264) in the right spot (remember, we drop the 000’s!). This gives221, 228, 234, 238, 245, 252, 255, 263, 264, 267, 270, 292

Example 14.13 Another Home Sells in Green Hills


Now N = 12 and in this case the median is the average of d6 = 252 and d7 = 255. It follows that

the median home price is M = $253,500. The locator for the first quartile is(0.25) 12 = 3, since the locator is a whole number, the first quartile is the average ofd3 = 234 and d4 = 238 (i.e., Q1 = $236,000).

Similarly, the third quartile is Q3 = $265,500 (the

average of d9 = 264 and d10 = 267).

Example 14.13 Another Home Sells in Green Hills


We will now find the median and quartile scores for the Stat 101 data set (shown again in Table 14-10). Having the frequency table available eliminates the need for sorting the scores–the frequency table has, in fact, done this for us.



Here N = 75 (odd), so the median is the thirty-eighth score (counting from the left) in the frequency table. To find the thirty-eighth number in Table 14-10, we tally frequencies as we move from left to right: 1 + 1= 2; 1 + 1 + 2 = 4; 1 + 1 + 2 + 6 = 10; 1 + 1 + 2 + 6 + 10 = 20; 1 + 1 + 2 + 6 + 10 + 16 = 36. At this point, we know that the 36th test score on the list is a 10 (the last of the 10’s) and the next 13 scores are all 11’s. We can conclude that the 38th test score is 11. Thus, M = 11.



The locator for the first quartile is L = (0.25) 75 = 18.75. Thus, Q1 = d19. To find the

nineteenth score in the frequency table, we tally frequencies from left to right: 1 + 1 = 2; 1 + 1 + 2 = 4; 1 + 1 + 2 + 6 = 10; 1 + 1 + 2 + 6 + 10 = 20. At this point we realize that d10 = 8

(the last of the 8’s) and that d11 through d10 all

equal 9. Hence, the first quartile of the Stat 101 midterm scores is Q1 = d19 = 9.



Since the first and third quartiles are at an equal “distance” from the two ends of the sorted data set, a quick way to locate the third quartile now is to look for the nineteenth score in the frequency table when we count frequencies from right to left. We find the third quartile of the Stat 101 data set is Q3 = 12.



In this example we continue the discussion of the 2007 SAT math scores introduced in Example 14.6. Recall that the number of college-bound high school seniors taking the test was N = 1,494,531. As reported by the College Board, the median score in the test was M = 510, the first quartile score was Q1

=430, and the third quartile was Q3 = 590.

What can we make of this information?

Example 14.15 2007 SAT Math Scores: Part 2


Let’s start with the median. From N = 1,494,531 (an odd number), we can conclude that the median (510 points) is the 747,266th score in the sorted list of test scores. This means that there were at least 747,266 students who scored 510 or less in the math section of the 2007 SAT. Why did we use “at least” in the preceding sentence?



Could there have been more than that number who scored 510 or less? Yes, almost surely. Since the number of students who scored 510 is in the thousands, it is very unlikely that the 747,266th score is the last of the 510s.

In a similar vein, we can conclude that there were at least 373,633 scores of Q1 = 430 or less [the locator for the first quartile is(0.25) 1,494,531 = 373,632.75] and at least 1,120,899 scores of Q3 = 590 or less.



Medians, quartiles, and general percentiles are often computed using statistical calculators or statistical software packages, which is all well and fine since the whole process can be a bit tedious. The problem is that there is no universally agreed upon procedure for computing percentiles, so different types of calculators and different statistical packages may give different answers from each other and from those given in this book for quartiles and other percentiles (everyone agrees on the median).

A Note of Warning


A common way to summarize a large data

set is by means of its five-number summary.

The five-number summary is given by (1)

the smallest value in the data set (called the

Min), (2) the first quartile Q1, (3) the median

M, (4) the third quartile Q3, and (5) the largest

value in the data set (called the Max). These

five numbers together often tell us a great

deal about the data.

The Five-Number Summary


For the Stat 101 data set, the five-number summary is Min = 1, Q1 = 9, M = 11, Q3 = 12, Max = 24. What useful information can we get out of this? Right away we can see that the N = 75 test scores were not evenly spread out over the range of possible scores. For example, from M = 11 and Q3 = 12 we can conclude that at least 25% of the class (that means at least 19 students) scored either 11 or 12 on the test.



At the same time, from Q3 = 12 and Max = 24 we can conclude that less than one-fourth of the class (i.e., at most 18 students) had scores in the 13–24 point range. Using similar arguments, we can conclude that at least 19 students had scores between Q1 = 9 andM = 11 points and no more than 18 students scored in the 1–8 point range.



The “big picture” we get from the five-number summary of the Stat 101 test scores is that there was a lot of bunching up in a narrow band of scores (at least half of the students in the class scored in the range 9–12 points), and the rest of the class was all over the place. In general, this type of “bumpy” distribution of test scores is indicative of a test with an uneven level of difficulty–a bunch of easy questions and a bunch of really hard questions with little in between.



Invented in 1977 by statistician John Tukey, a box plot (also known as a box-and-whisker plot) is a picture of the five-number summary of a data set. The box plot consists of a rectangular box that sits above a scale and extends from the first quartile Q1 to the third quartile Q3 on that scale. A vertical line crosses the box, indicating the position of the median M. On both sides of the box are “whiskers” extending to the smallest value, Min, and largest value, Max, of the data.

Box Plots


This figure shows a generic box plot for a data set.

Box Plots


This figure shows a box plot for the Stat 101 data set. The long whiskers in this box plot are largely due to the outliers 1 and 24.

Box Plots


This figure shows a variation of the same box plot, but with the two outliers, marked with two crosses, segregated from the rest of the data.

Box Plots


This figure shows box plots for the starting salaries of two different populations: first-year agriculture and engineering graduates of Tasmania State University.

Example 14.17 Comparing Agriculture and Engineering Salaries


Superimposing the two box plots on the same scale allows us to make some useful comparisons. It is clear, for instance, that engineering graduates are doing better overall than agriculture graduates, even though at the very top levels agriculture graduates are better paid.



Another interesting point is that the median salary of agriculture graduates ($43,000) is less than the first quartile of the salaries of engineering graduates ($45,000).



The very short whisker on the left side of the agriculture box plot tells us that the bottom 25% of agriculture salaries are concentrated in a very narrow salary range ($32,500–$35,000).



We can also see that agriculture salaries are much more spread out than engineering salaries,even though most of the spread occurs at the higher end of the salary scale.


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Excursions in Modern Mathematics, 7e: 14.3 - 2Copyright © 2010 Pearson Education, Inc. 14...

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