14.2 variables 14.3 numerical summaries 14.4 …jga001/math 1332 chapter 14 slides.pdf... 14.1 - 12...

Excursions in Modern Mathematics, 7e: 14.1 - 1 Copyright © 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


A data set is a collection of data values.

Statisticians often refer to the individual

data values in a data set as data points.

For the sake of simplicity, we will work with

data sets in which each data point consists

of a single number, but in more complicated

settings, a single data point can consist of

many numbers.

Data Set


• Use the letter N to represent the size of

the data set. In real- life applications,

data sets can range in size from

reasonably small (a dozen or so data

points) to very large (hundreds of millions

of data points), and the larger the data

set is, the more we need a good way to

describe and summarize it.

Data Set


The day after the midterm exam in his Stat 101 class, Dr.Blackbeard has posted the results online. The data set consists of N = 75 data points (the number of students who took the test). Each data point (listed in the second column) is a score between 0 and 25 (Dr. Blackbeard gives no partial credit). Notice that the numbers listed in the first column are not data points–they are numerical IDs used as substitutes for names to protect the students’ rights of privacy.

Example 14.1 Stat 101 Test Scores


How do we package the results into a compact, organized, and intelligible whole?



The first step in summarizing the information in Table 14-1 is to organize the scores in a frequency table such as Table 14-2. In this table, the number below each score gives the frequency of the score–that is, the number of students getting that particular score.

Example 14.2 Stat 101 Test Scores:

Part 2


We can readily see from Table 14-2 that there was one student with a score of 1, one with a score of 6, two with a score of 7, six with a score of 8, and so on. Notice that the scores with a frequency of zero are not listed in the table.


Part 2


• Figure 14-1 (next slide) shows the same information in a much more visual way called a bar graph, with the test scores listed in increasing order on a horizontal axis and the frequency of each test score displayed by the height of the column above that test score.

• Notice that in the bar graph, even the test scores with a frequency of zero show up–there simply is no column above these scores.


Part 2


Figure 14-1


Part 2


• Bar graphs are easy to read, and they are a nice way to present a good general picture of the data.

• Outliers are extreme data points that do not fit into the overall pattern of the data. In this example there are two obvious outliers–the score of 24 (head and shoulders above the rest of the class) and the score of 1 (lagging way behind the pack).


Part 2


• Sometimes it is more convenient to express the bar graph in terms of relative frequencies –that is, the frequencies given in terms of percentages of the total population.

• Figure 14-2 shows a relative frequency bar graph for the Stat 101 data set. Notice that we indicated on the graph that we are dealing with percentages rather than total counts and that the size of the data set is N = 75.


Part 2


Figure 14-2


Part 2


• This allows anyone who wishes to do so to compute the actual frequencies. For example, Fig. 14-2 indicates that 12% of the 75 students scored a 12 on the exam, so the actual frequency is given by 75 0.12 = 9 students.

• The change from actual frequencies to percentages (or vice versa) does not change the shape of the graph–it is basically a change of scale.


Part 2


• Page 545, problem 6

Examples


• Frequency charts that use icons or

pictures instead of bars to show the

frequencies are commonly referred to as

pictograms.

Bar Graph versus Pictogram


• The point of a pictogram is that a graph

is often used not only to inform but also

to impress and persuade, and, in such

cases, a well-chosen icon or picture can

be a more effective tool than just a bar.

• Here’s a pictogram displaying the same

data as in figure 14-2.



Figure 14-3



This figure is a pictogram showing the growth

in yearly sales of the XYZ Corporation

between 2001 and 2006. It’s a good picture to

Example 14.3 Selling the XYZ

Corporation

show at a

shareholders

meeting, but

the picture is

actually quite

misleading.


This figure shows a pictogram for exactly the

same data with a much more accurate and

sobering picture of how well the XYZ


Corporation

Corporation

had been

doing.


The difference between the two pictograms

can be attributed to a couple of standard

tricks of the trade: (1) stretching the scale of

the vertical axis and (2) “cheating” on the

choice of starting value on the vertical axis.

As an educated consumer, you should always

be on the lookout for these tricks. In graphical

descriptions of data, a fine line separates

objectivity from propaganda.


Corporation




14.2 Variables




• A variable is any characteristic that varies

with the members of a population.

• A variable that represents a measurable

quantity is called a numerical (or

quantitative) variable.

Variable


• Example: the students in Dr. Blackbeard’s

Stat 101 course (the population) did not

all perform equally on the exam (see

chapter 14.1). Thus, the test score is a

variable, which in this particular case is a

whole number between 0 and 25.

Variable


In some instances, such as when the

instructor gives partial credit, a test score

may take on a fractional value, such as 18.5

or 18.25. Even in these cases, however, the

possible increments for the values of the

variable are given by some minimum

amount–a quarter-point, a half-point,

whatever. In these cases, the variable is a

discrete variable.

Variable


• In contrast to a discrete variable, the amount of time each student studied for the exam is a continuous variable. In this case the variable can take on values that differ by any amount: an hour, a minute, a second, a tenth of a second, and so on.

Variable


• When the difference between the values of a numerical variable can be arbitrarily small, we call the variable continuous (person’s height, weight, foot size, time it takes to run one mile);

• when possible values of the numerical variable change by minimum increments, the variable is called discrete (person’s IQ, SAT score, shoe size, score of a basketball game).

Numerical Variable


Variables can also describe characteristics that cannot be measured numerically: nationality, gender, hair color, and so on. Variables of this type are called categorical (or qualitative) variables.

Categorical Variable


• In some ways, categorical variables must be treated differently from numerical variables–they cannot, for example, be added, multiplied, or averaged.

• In other ways, categorical variables can be treated much like discrete numerical variables, particularly when it comes to graphical descriptions, such as bar graphs and pictograms.

Categorical Variable


Table 14-3 shows undergraduate enrollments

in each of the five schools at Tasmania State

Example 14.4 Enrollments at Tasmania

State University

University. A sixth category

(“other”) includes

undeclared students,

interdisciplinary majors,

and so on. The variable

“school” is a categorical

variable.


Vertical and horizontal bar graphs displaying the data for table 14-3.


State University


When the number of categories is small, as is the case here, another common way to describe the relative frequencies of the categories is by using a pie chart. In a pie chart the “pie” represents the entire population (100%), and the “slices” represent the categories (or classes), with the size (angle) of each slice being proportional to the relative frequency of the corresponding category.


State University


This figure shows an accurate pie chart for the school-enrollment data given in Table 14-3.


State University


• Page 546, problems 14(a)

Examples


• When it comes to deciding how best to display graphically the frequencies of a population, a critical issue is the number of categories into which numerical (quantitative) data can fall.

• When the number of categories is too big (say, in the dozens), a bar graph or pictogram can become muddled and ineffective.

How Many Categories


The SAT consists of three sections: a math

section, a writing section, and a critical

reading section, with the scores for each

section ranging from a minimum of 200 to a

maximum of 800 and going up in increments

of 10 points. In 2007, there were 1,494,531

college-bound seniors who took the SAT.

How do we describe the math section results

for this group of students?

Example 14.6 2007 SAT Math Scores


We could set up a frequency table (or a bar

graph) with the number of students scoring

each of the possible scores–

200, 210, 220,… ,790, 800

The problem is that there are 61 different

possible scores between 200 and 800, and

this number is too large for an effective bar

graph.



• In this case the data can be grouped

together, or aggregated, into sets of scores

into categories called class intervals.

• The decision as to how the class intervals

are defined and how many there are will

depend on how much or how little detail is

desired, but as a general rule of thumb, the

number of class intervals should be

somewhere between 5 and 20.



SAT scores are usually

aggregated into 12 class

intervals of essentially the

same size:

200–249,

250–299,

300–349,

700–749,

750–800.



Here is the associated bar graph.



• Page 546,

problem 7

Example



Solution:

Examples


When a numerical variable is continuous, its

possible values can vary by infinitesimally

small increments. As a consequence, there

are no gaps between the class intervals,

and our old way of doing things (using

separated columns or stacks) will no longer

work. In this case we use a variation of a

bar graph called a histogram.

Histogram


Suppose we want to use a graph to display the distribution of starting salaries for last year’s graduating class at Tasmania State University.

The starting salaries of the N = 3258 graduates range from a low of $40,350 to a high of $74,800.

Example 14.8 Starting Salaries of TSU

Graduates


Based on this range and the amount of detail we want to show, we must decide on the length of the class intervals. A reasonable choice would be to use class intervals defined in increments of $5000.


Graduates



Graduates


• The superscript “plus” marks in Table 14-6

indicate how we chose to deal with the

endpoints in Fig. 14-11.

• 45,000+–50,000 indicates numbers greater

than 45,000 but less than or equal to

50,000


Graduates


• A starting salary of exactly $50,000, for

example, would be listed under the

45,000+–50,000 class interval rather than

the 50,000+–55,000 class interval.


Graduates


Here is the histogram

showing the relative

frequency of each

class interval. As we

can see, a histogram

is very similar to a bar

graph.


Graduates


• The differences between a bar graph and a

histogram are:

– Unlike a bar graph, a histogram is used for

continuous variables and there can be no

gaps between the class intervals

– Unlike a bar graph, the bars of a histogram

touch each other


Graduates


When creating histograms, we should try,

as much as possible, to define class

intervals of equal length.

Use Class Intervals of Equal Length



Examples




14.2 Variables




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


• Page 548, problem 24(a)

Examples



• Solution: 0.1625

Examples


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.

Test scores are given in Table 14-1 (next

slide)


Part 4


The addition of the 75 test scores can be

simplified considerably if we use a frequency

table.


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.


Part 4


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



Examples



• Solution: 1.5875

Examples


This example (next slide) shows that the average can be a misleading statistic.

Example 14.10 Starting Salaries of

Philosophy Majors


The average starting salary of philosophy majors who recently graduated from Tasmania State University is $76,400 a year! But, 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.


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.


Philosophy Majors


■ The total of the other 74 salaries (excluding

Hoops’s cool 3.5 mill) is $2,230,000

■ The average of the remaining 74 salaries is

then

$2,230,000/74 ≈ $30,135


Philosophy Majors


Let p be any integer between 0 and 100.

The pth percentile of a data set is a value

for which p percent of the values in the data

set are less than or equal to this value.

Percentile


First, sort the data from small to large.

If you are finding the pth percentile of a data set of size N, calculate p percent of N which is the position locator:

Calculate pth percentile

Np

L100


If L is an integer, the pth percentile is the

average of the data values in positions L

and L+1.

If L is not an integer, round up and use the

value in this position as the pth percentile.



Example of Finding Percentile

• Find the 25th and 75th percentiles

53 37 38 50 65 44 47 39 36 57 44 69



• Sort the data from small to large:

36 37 38 39 44 44 47 50 53 57 65 69

There are 12 data values.



25% of 12

312100

25L



• The data can be grouped as follows:

36 37 38 39 44 44 47 50 53 57 65 69

25% of the data is less than or equal to 38.5

(the average of 38 and 39).

The 25th percentile is 38.5

3rd position



75% of 12

912100

75L



36 37 38 39 44 44 47 50 53 57 65 69

75% of the data is less than or equal to 55


The 75th percentile is 55

9th position



• Find the 25th and 75th percentiles

44 50 39 36 47 38 65



• First sort the data

36 38 39 44 47 50 65




25% of 7

75.17100

25L

1.75 round up to get 2



36 38 39 44 47 50 65

2nd position


Approximately 25% of the data is less than or equal to 38



75% of 7

25.57

100

75L




36 38 39 44 47 50 65

6th position




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


Percentiles


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.


Percentiles


$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 = 3.85 (the average between 3.82 and

3.88). Thus, three students (the ones with

GPAs of 3.88, 3.91 and 4.0) get $5000

scholarships.


Percentiles


$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 by

d9 = 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.


Percentiles


• 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


First, sort the data from small to large.

Median is the 50th percentile so the locator is 50 percent of N:

Calculate Median

250.0

100

50 NNNL


If L is an integer, the median is the average

of the data values in positions L and L+1.

If L is not an integer, round up and use the

value in this position as the median.



Example of Finding Median

• Find the median

53 37 38 50 65 44 47 39 36 57 44 69


• Sort the data from small to large:

36 37 38 39 44 44 47 50 53 57 65 69




50% of 12

612100

50L



• The data can be grouped as follows:

36 37 38 39 44 44 47 50 53 57 65 69

50% of the data is less than or equal to 45.5


The median is 45.5

6th position



• Find the median

44 50 39 36 47 38 65



• First sort the data

36 38 39 44 47 50 65




50% of 7

5.37100

50L




36 38 39 44 47 50 65

2nd position

The median is 44




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


Hills


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, and

8.25 to 9.


Hills


Thus, the median home price is given by

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

the first quartile is given by

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

and the third quartile is given by

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


Hills


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 gives

221, 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 252 and 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 of

234 and 238 (i.e., Q1 = $236,000).

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


Example 14.13 Another Home Sells in

Green Hills


• We can calculate the median using a

frequency table as the following example

shows.

Calculate median and quartiles using

a Frequency Table


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.


Part 5


• 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.


Part 5


• 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.


Part 5


• The locator for the first quartile is L = (0.25)

75 = 18.75 which is rounded up to 19.

• 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.


Part 5


• The tenth test score is 8 (the last of the 8’s)

and the next ten test scores are all 9.

Hence, the first quartile of the Stat 101

midterm scores is the 19th test score which

is Q1 = 9.


Part 5


Similarly, we find the third quartile of the Stat

101 data set is Q3 = 12.


Part 5


• The five-number summary is

(1) smallest value in the data set (called the Min),

(2) first quartile Q1,

(3) median M,

(4) third quartile Q3, and

(5) the largest value in the data set (called the

Max).

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?

–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.


Part 6


• 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 and M = 11 points and no

more than 18 students scored in the 1–8

point range.


Part 6


• 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.

Box Plots


• 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.






14.2 Variables




• The difference between the highest and

lowest values of the data in a data set is.

called the range of the data set denoted

by R. Thus,

R = Max – Min

• The range of a data set is a useful piece

of information when there are no outliers

in the data. In the presence of outliers

the range tells a distorted story.

The Range



The range of the test scores in the Stat 101

exam is 24 – 1 = 23 points


• The range is sensitive to outliers.

• In the previous example, if we discount

the two outliers, the remaining 73 test

scores would have a much smaller range

of 16 – 6 = 10 points.

The Range


• To eliminate the possible distortion

caused by outliers, a common practice

when measuring the spread of a data set

is to use the interquartile range,

denoted by the acronym IQR.

• The interquartile range is the difference

between the third quartile and the first

quartile

IQR = Q3 – Q1

The Interquartile Range


• The IQR tells us how spread out the

middle 50% of the data values are.

• For many types of real-world data, the

interquartile range is a useful measure of

spread.

The Interquartile Range


For the Stat 101 data set, the five-number

summary is

Min = 1, Q1 = 9, M = 11, Q3 = 12, Max = 24

IQR = Q3 – Q1 = 12 – 9 = 3

The middle 50% of the exam scores differ by

at most 3 points.




• Note: use the solutions from exercise 37

which are that the first quartile is 29 and

the third quartile is 32

Examples



• Solution: there are 10 outliers which are 6

ages of 37 and 4 ages of 39.

Examples


• The most important and most commonly used measure of spread for a data set is the standard deviation.

• The key concept for understanding the standard deviation is the concept of deviation from the mean. If A is the average of the data set and x is an arbitrary data value, the difference x – A is the deviation from the mean for the data value x.

Standard Deviation


• The deviations from the mean tell us how “far” the data values are from the average value of the data.

• We will use this information to figure out how spread out the data is.

Standard Deviation


■ Let A denote the mean of the data set. For each number x in the data set, compute its deviation from the mean (x – A) and square each of these numbers. These numbers are called the squared deviations.

■ Find the average of the squared deviations. This number is called the variance V.

THE STANDARD DEVIATION

OF A DATA SET


■ The standard deviation, denoted by the greek letter , is the square

root of the variance

THE STANDARD DEVIATION

OF A DATA SET

V


Over the course of the semester, Angela

turned in all of her homework assignments.

Her grades in the 10 assignments (sorted

from lowest to highest) were

85, 86, 87, 88, 89, 91, 92, 93, 94, 95

Calculate the standard deviation of this data

set.

Example 14.19 Calculation of a SD


First calculate the average of the data set:

85, 86, 87, 88, 89, 91, 92, 93, 94, 95

Adding all of the data values gives 900.

Average = 900/10 =90



Next calculate the deviations of each data

value in the data set. After subtracting 90

from each data value we get:

-5, -4, -3, -2, -1, 1, 2, 3, 4, 5

Next square each of these values to get the

squared deviations:

25, 16, 9, 4, 1, 1, 4, 9, 16, 25



Next we average the squared deviations to

get the variance:

V = 110/10 = 11

Finally we take the square root of the

variance to get the standard deviation:


3.311


Calculating the deviations

and squared deviations are

can be easily summarized in

a table. Then we add the

numbers in the last column

(squared deviations) and

divide by 10 to get the

standard deviation.

Example 14.19


• It is clear from just a casual look at

Angela’s homework scores that she was

pretty consistent in her homework, never

straying too much above or below her

average score of 90 points.

• The standard deviation is, in effect, a way

to measure this degree of consistency (or

lack thereof).

Interpreting the Standard Deviation


• A small standard deviation tells us that the

data are consistent and the spread of the

data is small, as is the case with Angela’s

homework scores.



The ultimate in consistency within a data set

is when all the data values are the same (like

Angela’s friend Chloe, who got a 20 in every

homework assignment). When this happens

the standard deviation is 0.



On the other hand, when there is a lot of

inconsistency within the data set, we are

going to get a large standard deviation. This

is illustrated by Angela’s other friend, Tiki,

whose homework scores were

5, 15, 25, 35, 45, 55, 65, 75, 85, 95

The standard deviation of this data is almost

29 points.



The standard deviation is arguably the most

important and frequently used measure of

data spread. Here are a few facts.

Summary of the Standard Deviation


• The standard deviation of a data set is

measured in the same units as the original

data.

• For example, if the data are points are

dollar amounts then the standard deviation

is given in dollars; if the data have units of

gallons then the standard deviation has

units of gallons.



• If the standard deviation is small, we can

conclude that the data points are all

bunched together–there is very little

spread. A standard deviation of 0 means

that every data value is the same.

• As the standard deviation is large, we can

conclude that the data points are spread

out.


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