Chapter 2

Chapter 2

Descriptive Statistics

1Larson/Farber 4th ed.

Chapter Outline

• 2.1 Frequency Distributions and Their Graphs

• 2.2 More Graphs and Displays

• 2.3 Measures of Central Tendency

• 2.4 Measures of Variation

• 2.5 Measures of Position


Section 2.1

Frequency Distributions and Their Graphs


Section 2.1 Objectives

• Construct frequency distributions• Construct frequency histograms, frequency polygons,

relative frequency histograms, and ogives

Larson/Farber 4th ed. 4

Frequency Distribution

Frequency Distribution• A table that shows

classes or intervals of data with a count of the number of entries in each class.

• The frequency, f, of a class is the number of data entries in the class.


Class Frequency, f

1 – 5 5

6 – 10 8

11 – 15 6

16 – 20 8

21 – 25 5

26 – 30 4

Lower classlimits

Upper classlimits

Class width 6 – 1 = 5

Constructing a Frequency Distribution


1. Decide on the number of classes. Usually between 5 and 20; otherwise, it may be

difficult to detect any patterns. 2. Find the class width.

Determine the range of the data. Divide the range by the number of classes. Round up to the next convenient number.


3. Find the class limits. You can use the minimum data entry as the lower

limit of the first class. Find the remaining lower limits (add the class

width to the lower limit of the preceding class). Find the upper limit of the first class. Remember

that classes cannot overlap. Find the remaining upper class limits.



4. Make a tally mark for each data entry in the row of the appropriate class.

5. Count the tally marks to find the total frequency f for each class.


Example: Constructing a Frequency Distribution

The following sample data set lists the number of minutes 50 Internet subscribers spent on the Internet during their most recent session. Construct a frequency distribution that has seven classes.50 40 41 17 11 7 22 44 28 21 19 23 37 51 54 42 8641 78 56 72 56 17 7 69 30 80 56 29 33 46 31 39 2018 29 34 59 73 77 36 39 30 62 54 67 39 31 53 44


Solution: Constructing a Frequency Distribution

1. Number of classes = 7 (given)2. Find the class width


max min 86 7 11.29#classes 7

Round up to 12

50 40 41 17 11 7 22 44 28 21 19 23 37 51 54 42 8641 78 56 72 56 17 7 69 30 80 56 29 33 46 31 39 2018 29 34 59 73 77 36 39 30 62 54 67 39 31 53 44



Lower limit

Upper limit

7Class width = 12

3. Use 7 (minimum value) as first lower limit. Add the class width of 12 to get the lower limit of the next class.

7 + 12 = 19Find the remaining lower limits.

193143556779


The upper limit of the first class is 18 (one less than the lower limit of the second class). Add the class width of 12 to get the upper limit of the next class.

18 + 12 = 30Find the remaining upper limits.


Lower limit

Upper limit

7193143556779

Class width = 1230

4254667890

18


4. Make a tally mark for each data entry in the row of the appropriate class.

5. Count the tally marks to find the total frequency f for each class.


Class Tally Frequency, f

7 – 18 IIII I 6

19 – 30 IIII IIII 10

31 – 42 IIII IIII III 13

43 – 54 IIII III 8

55 – 66 IIII 5

67 – 78 IIII I 6

79 – 90 II 2Σf = 50

Determining the Midpoint

Midpoint of a class


(Lower class limit) (Upper class limit)2

Class Midpoint Frequency, f

7 – 18 6

19 – 30 10

31 – 42 13

7 18 12.52

19 30 24.52

31 42 36.52

Class width = 12

Determining the Relative FrequencyRelative Frequency of a class • Portion or percentage of the data that falls in a

particular class.


nf

sizeSample

frequencyclassfrequencyrelative

Class Frequency, f Relative Frequency

7 – 18 6

19 – 30 10

31 – 42 13

6 0.1250

10 0.2050

13 0.2650

•

Determining the Cumulative Frequency

Cumulative frequency of a class• The sum of the frequency for that class and all

previous classes.


Class Frequency, f Cumulative frequency

7 – 18 6

19 – 30 10

31 – 42 13

+

+

6

16

29

Expanded Frequency Distribution


Class Frequency, f MidpointRelative

frequencyCumulative frequency

7 – 18 6 12.5 0.12 6

19 – 30 10 24.5 0.20 16

31 – 42 13 36.5 0.26 29

43 – 54 8 48.5 0.16 37

55 – 66 5 60.5 0.10 42

67 – 78 6 72.5 0.12 48

79 – 90 2 84.5 0.04 50

Σf = 50 1nf

Graphs of Frequency Distributions

Frequency Histogram• A bar graph that represents the frequency distribution.• The horizontal scale is quantitative and measures the

data values.• The vertical scale measures the frequencies of the

classes.• Consecutive bars must touch.


data valuesfr

eque

ncy

Class BoundariesClass boundaries• The numbers that separate classes without forming

gaps between them.


ClassClass

BoundariesFrequency,

f 7 – 18 619 – 30 1031 – 42 13

• The distance from the upper limit of the first class to the lower limit of the second class is 19 – 18 = 1.

• Half this distance is 0.5.

• First class lower boundary = 7 – 0.5 = 6.5• First class upper boundary = 18 + 0.5 = 18.5

6.5 – 18.5

Class Boundaries


ClassClass

boundariesFrequency,

f 7 – 18 6.5 – 18.5 619 – 30 18.5 – 30.5 1031 – 42 30.5 – 42.5 1343 – 54 42.5 – 54.5 855 – 66 54.5 – 66.5 567 – 78 66.5 – 78.5 679 – 90 78.5 – 90.5 2

Example: Frequency Histogram

Construct a frequency histogram for the Internet usage frequency distribution.


ClassClass

boundaries MidpointFrequency,

f 7 – 18 6.5 – 18.5 12.5 6

19 – 30 18.5 – 30.5 24.5 10

31 – 42 30.5 – 42.5 36.5 13

43 – 54 42.5 – 54.5 48.5 8

55 – 66 54.5 – 66.5 60.5 5

67 – 78 66.5 – 78.5 72.5 6

79 – 90 78.5 – 90.5 84.5 2

Solution: Frequency Histogram (using Midpoints)


Solution: Frequency Histogram (using class boundaries)


6.5 18.5 30.5 42.5 54.5 66.5 78.5 90.5

You can see that more than half of the subscribers spent between 19 and 54 minutes on the Internet during their most recent session.


Frequency Polygon• A line graph that emphasizes the continuous change

in frequencies.


data values

freq

uenc

y

Example: Frequency Polygon

Construct a frequency polygon for the Internet usage frequency distribution.


Class Midpoint Frequency, f

7 – 18 12.5 6

19 – 30 24.5 10

31 – 42 36.5 13

43 – 54 48.5 8

55 – 66 60.5 5

67 – 78 72.5 6

79 – 90 84.5 2

Solution: Frequency Polygon

02468101214

0.5 12.5 24.5 36.5 48.5 60.5 72.5 84.5 96.5

Freq

uenc

y

Time online (in minutes)

Internet Usage


You can see that the frequency of subscribers increases up to 36.5 minutes and then decreases.

The graph should begin and end on the horizontal axis, so extend the left side to one class width before the first class midpoint and extend the right side to one class width after the last class midpoint.


Relative Frequency Histogram• Has the same shape and the same horizontal scale as

the corresponding frequency histogram.• The vertical scale measures the relative frequencies,

not frequencies.


data valuesre

lativ

e fr

eque

ncy

Example: Relative Frequency Histogram

Construct a relative frequency histogram for the Internet usage frequency distribution.


ClassClass


fRelative

frequency 7 – 18 6.5 – 18.5 6 0.12

19 – 30 18.5 – 30.5 10 0.20

31 – 42 30.5 – 42.5 13 0.26

43 – 54 42.5 – 54.5 8 0.16

55 – 66 54.5 – 66.5 5 0.10

67 – 78 66.5 – 78.5 6 0.12

79 – 90 78.5 – 90.5 2 0.04

Solution: Relative Frequency Histogram


6.5 18.5 30.5 42.5 54.5 66.5 78.5 90.5

From this graph you can see that 20% of Internet subscribers spent between 18.5 minutes and 30.5 minutes online.


Cumulative Frequency Graph or Ogive• A line graph that displays the cumulative frequency

of each class at its upper class boundary.• The upper boundaries are marked on the horizontal

axis.• The cumulative frequencies are marked on the

vertical axis.


data valuescu

mul

ativ

e fr

eque

ncy

Constructing an Ogive

1. Construct a frequency distribution that includes cumulative frequencies as one of the columns.

2. Specify the horizontal and vertical scales. The horizontal scale consists of the upper class

boundaries. The vertical scale measures cumulative

frequencies.3. Plot points that represent the upper class boundaries

and their corresponding cumulative frequencies.


Constructing an Ogive

4. Connect the points in order from left to right.5. The graph should start at the lower boundary of the

first class (cumulative frequency is zero) and should end at the upper boundary of the last class (cumulative frequency is equal to the sample size).


Example: Ogive

Construct an ogive for the Internet usage frequency distribution.


ClassClass


fCumulative frequency

7 – 18 6.5 – 18.5 6 6

19 – 30 18.5 – 30.5 10 16

31 – 42 30.5 – 42.5 13 29

43 – 54 42.5 – 54.5 8 37

55 – 66 54.5 – 66.5 5 42

67 – 78 66.5 – 78.5 6 48

79 – 90 78.5 – 90.5 2 50

Solution: Ogive

0

10

20

30

40

50

60

Cum

ulati

ve fr

eque

ncy

Time online (in minutes)

Internet Usage


6.5 18.5 30.5 42.5 54.5 66.5 78.5 90.5

From the ogive, you can see that about 40 subscribers spent 60 minutes or less online during their last session. The greatest increase in usage occurs between 30.5 minutes and 42.5 minutes.

Section 2.1 Summary

• Constructed frequency distributions• Constructed frequency histograms, frequency

polygons, relative frequency histograms and ogives


Section 2.2

More Graphs and Displays



• Graph quantitative data using stem-and-leaf plots and dot plots

• Graph qualitative data using pie charts and Pareto charts

• Graph paired data sets using scatter plots and time series charts


Graphing Quantitative Data Sets

Stem-and-leaf plot• Each number is separated into a stem and a leaf.• Similar to a histogram.• Still contains original data values.


Data: 21, 25, 25, 26, 27, 28, 30, 36, 36, 45

26

2 1 5 5 6 7 83 0 6 6 4 5

Example: Constructing a Stem-and-Leaf Plot

The following are the numbers of text messages sent last month by the cellular phone users on one floor of a college dormitory. Display the data in a stem-and-leaf plot.


155 159 144 129 105 145 126 116 130 114 122 112 112 142 126156 118 108 122 121 109 140 126 119 113 117 118 109 109 119139 139 122 78 133 126 123 145 121 134 124 119 132 133 124129 112 126 148 147

Solution: Constructing a Stem-and-Leaf Plot


• The data entries go from a low of 78 to a high of 159.• Use the rightmost digit as the leaf.

For instance, 78 = 7 | 8 and 159 = 15 | 9

• List the stems, 7 to 15, to the left of a vertical line.• For each data entry, list a leaf to the right of its stem.

155 159 144 129 105 145 126 116 130 114 122 112 112 142 126156 118 108 122 121 109 140 126 119 113 117 118 109 109 119139 139 122 78 133 126 123 145 121 134 124 119 132 133 124129 112 126 148 147

Solution: Constructing a Stem-and-Leaf Plot


Include a key to identify the values of the data.

From the display, you can conclude that more than 50% of the cellular phone users sent between 110 and 130 text messages.

Graphing Quantitative Data Sets

Dot plot• Each data entry is plotted, using a point, above a

horizontal axis


Data: 21, 25, 25, 26, 27, 28, 30, 36, 36, 45

26

20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45

Example: Constructing a Dot Plot

Use a dot plot organize the text messaging data.


• So that each data entry is included in the dot plot, the horizontal axis should include numbers between 70 and 160.

• To represent a data entry, plot a point above the entry's position on the axis.

• If an entry is repeated, plot another point above the previous point.

155 159 144 129 105 145 126 116 130 114 122 112 112 142 126156 118 108 122 121 109 140 126 119 113 117 118 109 109 119139 139 122 78 133 126 123 145 121 134 124 119 132 133 124129 112 126 148 147

Solution: Constructing a Dot Plot


From the dot plot, you can see that most values cluster between 105 and 148 and the value that occurs the most is 126. You can also see that 78 is an unusual data value.

155 159 144 129 105 145 126 116 130 114 122 112 112 142 126156 118 108 122 121 109 140 126 119 113 117 118 109 109 119139 139 122 78 133 126 123 145 121 134 124 119 132 133 124129 112 126 148 147

Graphing Qualitative Data Sets

Pie Chart• A circle is divided into sectors that represent

categories.• The area of each sector is proportional to the

frequency of each category.


Example: Constructing a Pie Chart

The numbers of motor vehicle occupants killed in crashes in 2005 are shown in the table. Use a pie chart to organize the data. (Source: U.S. Department of Transportation, National Highway Traffic Safety Administration)


Vehicle type KilledCars 18,440Trucks 13,778Motorcycles 4,553Other 823

Solution: Constructing a Pie Chart

• Find the relative frequency (percent) of each category.


Vehicle type Frequency, f Relative frequency

Cars 18,440

Trucks 13,778

Motorcycles 4,553

Other 823

37,594

18440 0.4937594

13778 0.3737594

4553 0.1237594

823 0.0237594


• Construct the pie chart using the central angle that corresponds to each category. To find the central angle, multiply 360º by the

category's relative frequency. For example, the central angle for cars is

360(0.49) ≈ 176º




Vehicle type Frequency, fRelative

frequency Central angle

Cars 18,440 0.49

Trucks 13,778 0.37

Motorcycles 4,553 0.12

Other 823 0.02

360º(0.49)≈176º

360º(0.37)≈133º

360º(0.12)≈43º

360º(0.02)≈7º



Vehicle typeRelative

frequencyCentral angle

Cars 0.49 176ºTrucks 0.37 133ºMotorcycles 0.12 43º

Other 0.02 7º

From the pie chart, you can see that most fatalities in motor vehicle crashes were those involving the occupants of cars.

Graphing Qualitative Data Sets

Pareto Chart• A vertical bar graph in which the height of each bar

represents frequency or relative frequency.• The bars are positioned in order of decreasing height,

with the tallest bar positioned at the left.


Categories

Freq

uenc

y

Example: Constructing a Pareto Chart

In a recent year, the retail industry lost $41.0 million in inventory shrinkage. Inventory shrinkage is the loss of inventory through breakage, pilferage, shoplifting, and so on. The causes of the inventory shrinkage are administrative error ($7.8 million), employee theft ($15.6 million), shoplifting ($14.7 million), and vendor fraud ($2.9 million). Use a Pareto chart to organize this data. (Source: National Retail Federation and Center for Retailing Education, University of Florida)


Solution: Constructing a Pareto Chart


Cause $ (million)

Admin. error 7.8Employee theft 15.6

Shoplifting 14.7Vendor fraud 2.9

From the graph, it is easy to see that the causes of inventory shrinkage that should be addressed first are employee theft and shoplifting.

Graphing Paired Data Sets

Paired Data Sets• Each entry in one data set corresponds to one entry in

a second data set.• Graph using a scatter plot.

The ordered pairs are graphed aspoints in a coordinate plane.

Used to show the relationship between two quantitative variables.


x

y

Example: Interpreting a Scatter Plot

The British statistician Ronald Fisher introduced a famous data set called Fisher's Iris data set. This data set describes various physical characteristics, such as petal length and petal width (in millimeters), for three species of iris. The petal lengths form the first data set and the petal widths form the second data set. (Source: Fisher, R. A., 1936)


Example: Interpreting a Scatter Plot

As the petal length increases, what tends to happen to the petal width?


Each point in the scatter plot represents thepetal length and petal width of one flower.

Solution: Interpreting a Scatter Plot


Interpretation From the scatter plot, you can see that as the petal length increases, the petal width also tends to increase.

Graphing Paired Data Sets

Time Series• Data set is composed of quantitative entries taken at

regular intervals over a period of time. e.g., The amount of precipitation measured each

day for one month. • Use a time series chart to graph.


timeQ

uant

itativ

e da

ta

Example: Constructing a Time Series Chart

The table lists the number of cellular telephone subscribers (in millions) for the years 1995 through 2005. Construct a time series chart for the number of cellular subscribers. (Source: Cellular Telecommunication & Internet Association)


Solution: Constructing a Time Series Chart

• Let the horizontal axis represent the years.

• Let the vertical axis represent the number of subscribers (in millions).

• Plot the paired data and connect them with line segments.


Solution: Constructing a Time Series Chart


The graph shows that the number of subscribers has been increasing since 1995, with greater increases recently.

Section 2.2 Summary

• Graphed quantitative data using stem-and-leaf plots and dot plots

• Graphed qualitative data using pie charts and Pareto charts

• Graphed paired data sets using scatter plots and time series charts


Section 2.3

Measures of Central Tendency



• Determine the mean, median, and mode of a population and of a sample

• Determine the weighted mean of a data set and the mean of a frequency distribution

• Describe the shape of a distribution as symmetric, uniform, or skewed and compare the mean and median for each


Measures of Central Tendency

Measure of central tendency• A value that represents a typical, or central, entry of a

data set.• Most common measures of central tendency:

Mean Median Mode


Measure of Central Tendency: Mean

Mean (average)• The sum of all the data entries divided by the number

of entries.• Sigma notation: Σx = add all of the data entries (x)

in the data set.• Population mean:

• Sample mean:


xN

xxn

Example: Finding a Sample Mean

The prices (in dollars) for a sample of roundtrip flights from Chicago, Illinois to Cancun, Mexico are listed. What is the mean price of the flights?

872 432 397 427 388 782 397


Solution: Finding a Sample Mean

872 432 397 427 388 782 397


• The sum of the flight prices isΣx = 872 + 432 + 397 + 427 + 388 + 782 + 397 = 3695

• To find the mean price, divide the sum of the prices by the number of prices in the sample

3695 527.97

xxn

The mean price of the flights is about $527.90.

Measure of Central Tendency: Median

Median• The value that lies in the middle of the data when the

data set is ordered.• Measures the center of an ordered data set by

dividing it into two equal parts.• If the data set has an

odd number of entries: median is the middle data entry.

even number of entries: median is the mean of the two middle data entries.


Example: Finding the Median

The prices (in dollars) for a sample of roundtrip flights from Chicago, Illinois to Cancun, Mexico are listed. Find the median of the flight prices.

872 432 397 427 388 782 397


Solution: Finding the Median

872 432 397 427 388 782 397


• First order the data.388 397 397 427 432 782 872

• There are seven entries (an odd number), the median is the middle, or fourth, data entry.

The median price of the flights is $427.

Example: Finding the Median

The flight priced at $432 is no longer available. What is the median price of the remaining flights?

872 397 427 388 782 397


Solution: Finding the Median

872 397 427 388 782 397


• First order the data.388 397 397 427 782 872

• There are six entries (an even number), the median is the mean of the two middle entries.

The median price of the flights is $412.

397 427Median 4122

Measure of Central Tendency: Mode

Mode• The data entry that occurs with the greatest frequency.• If no entry is repeated the data set has no mode.• If two entries occur with the same greatest frequency,

each entry is a mode (bimodal).


Example: Finding the Mode

The prices (in dollars) for a sample of roundtrip flights from Chicago, Illinois to Cancun, Mexico are listed. Find the mode of the flight prices.

872 432 397 427 388 782 397


Solution: Finding the Mode

872 432 397 427 388 782 397


• Ordering the data helps to find the mode.388 397 397 427 432 782 872

• The entry of 397 occurs twice, whereas the other data entries occur only once.

The mode of the flight prices is $397.

Example: Finding the Mode

At a political debate a sample of audience members was asked to name the political party to which they belong. Their responses are shown in the table. What is the mode of the responses?


Political Party Frequency, fDemocrat 34Republican 56Other 21Did not respond 9

Solution: Finding the Mode


Political Party Frequency, fDemocrat 34Republican 56Other 21Did not respond 9

The mode is Republican (the response occurring with the greatest frequency). In this sample there were more Republicans than people of any other single affiliation.

Comparing the Mean, Median, and Mode

• All three measures describe a typical entry of a data set.

• Advantage of using the mean: The mean is a reliable measure because it takes

into account every entry of a data set.• Disadvantage of using the mean:

Greatly affected by outliers (a data entry that is far removed from the other entries in the data set).


Example: Comparing the Mean, Median, and Mode

Find the mean, median, and mode of the sample ages of a class shown. Which measure of central tendency best describes a typical entry of this data set? Are there any outliers?


Ages in a class20 20 20 20 20 20 21

21 21 21 22 22 22 23

23 23 23 24 24 65

Solution: Comparing the Mean, Median, and Mode


Mean: 20 20 ... 24 65 23.8 years20

xxn

Median: 21 22 21.5 years2

20 years (the entry occurring with thegreatest frequency)

Ages in a class20 20 20 20 20 20 21

21 21 21 22 22 22 23

23 23 23 24 24 65

Mode:



Mean ≈ 23.8 years Median = 21.5 years Mode = 20 years

• The mean takes every entry into account, but is influenced by the outlier of 65.

• The median also takes every entry into account, and it is not affected by the outlier.

• In this case the mode exists, but it doesn't appear to represent a typical entry.



Sometimes a graphical comparison can help you decide which measure of central tendency best represents a data set.

In this case, it appears that the median best describes the data set.

Weighted Mean

Weighted Mean• The mean of a data set whose entries have varying

weights.

• where w is the weight of each entry x


( )x wxw

Example: Finding a Weighted Mean

You are taking a class in which your grade is determined from five sources: 50% from your test mean, 15% from your midterm, 20% from your final exam, 10% from your computer lab work, and 5% from your homework. Your scores are 86 (test mean), 96 (midterm), 82 (final exam), 98 (computer lab), and 100 (homework). What is the weighted mean of your scores? If the minimum average for an A is 90, did you get an A?


Solution: Finding a Weighted Mean


Source Score, x Weight, w x∙wTest Mean 86 0.50 86(0.50)= 43.0Midterm 96 0.15 96(0.15) = 14.4Final Exam 82 0.20 82(0.20) = 16.4Computer Lab 98 0.10 98(0.10) = 9.8Homework 100 0.05 100(0.05) = 5.0

Σw = 1 Σ(x∙w) = 88.6

( ) 88.6 88.61

x wxw

Your weighted mean for the course is 88.6. You did not get an A.

Mean of Grouped Data

Mean of a Frequency Distribution• Approximated by

where x and f are the midpoints and frequencies of a class, respectively


( )x fx n fn

Finding the Mean of a Frequency Distribution

In Words In Symbols


( )x fxn

(lower limit)+(upper limit)2

x

( )x f

n f

1. Find the midpoint of each class.

2. Find the sum of the products of the midpoints and the frequencies.

3. Find the sum of the frequencies.

4. Find the mean of the frequency distribution.

Example: Find the Mean of a Frequency Distribution

Use the frequency distribution to approximate the mean number of minutes that a sample of Internet subscribers spent online during their most recent session.


Class Midpoint Frequency, f 7 – 18 12.5 619 – 30 24.5 1031 – 42 36.5 1343 – 54 48.5 855 – 66 60.5 567 – 78 72.5 679 – 90 84.5 2

Solution: Find the Mean of a Frequency Distribution


Class Midpoint, x Frequency, f (x∙f) 7 – 18 12.5 6 12.5∙6 = 75.019 – 30 24.5 10 24.5∙10 = 245.031 – 42 36.5 13 36.5∙13 = 474.543 – 54 48.5 8 48.5∙8 = 388.055 – 66 60.5 5 60.5∙5 = 302.567 – 78 72.5 6 72.5∙6 = 435.079 – 90 84.5 2 84.5∙2 = 169.0

n = 50 Σ(x∙f) = 2089.0

( ) 2089 41.8 minutes50

x fxn

The Shape of Distributions


Symmetric Distribution• A vertical line can be drawn through the middle of

a graph of the distribution and the resulting halves are approximately mirror images.



Uniform Distribution (rectangular)• All entries or classes in the distribution have equal

or approximately equal frequencies.• Symmetric.



Skewed Left Distribution (negatively skewed)• The “tail” of the graph elongates more to the left.• The mean is to the left of the median.



Skewed Right Distribution (positively skewed)• The “tail” of the graph elongates more to the right.• The mean is to the right of the median.

Section 2.3 Summary

• Determined the mean, median, and mode of a population and of a sample

• Determined the weighted mean of a data set and the mean of a frequency distribution

• Described the shape of a distribution as symmetric, uniform, or skewed and compared the mean and median for each


Section 2.4

Measures of Variation



• Determine the range of a data set• Determine the variance and standard deviation of a

population and of a sample• Use the Empirical Rule and Chebychev’s Theorem to

interpret standard deviation• Approximate the sample standard deviation for

grouped data


Range

Range• The difference between the maximum and minimum

data entries in the set.• The data must be quantitative.• Range = (Max. data entry) – (Min. data entry)


Example: Finding the Range

A corporation hired 10 graduates. The starting salaries for each graduate are shown. Find the range of the starting salaries.

Starting salaries (1000s of dollars)41 38 39 45 47 41 44 41 37 42


Solution: Finding the Range

• Ordering the data helps to find the least and greatest salaries.

37 38 39 41 41 41 42 44 45 47

• Range = (Max. salary) – (Min. salary) = 47 – 37 = 10

The range of starting salaries is 10 or $10,000.


minimum maximum

Deviation, Variance, and Standard Deviation

Deviation• The difference between the data entry, x, and the

mean of the data set.• Population data set:

Deviation of x = x – μ• Sample data set:

Deviation of x = x – x


Example: Finding the Deviation

A corporation hired 10 graduates. The starting salaries for each graduate are shown. Find the deviation of the starting salaries.



Solution:• First determine the mean starting salary.

415 41.510

xN

Solution: Finding the Deviation


• Determine the deviation for each data entry.

Salary ($1000s), x Deviation: x – μ41 41 – 41.5 = –0.538 38 – 41.5 = –3.539 39 – 41.5 = –2.545 45 – 41.5 = 3.547 47 – 41.5 = 5.541 41 – 41.5 = –0.544 44 – 41.5 = 2.541 41 – 41.5 = –0.537 37 – 41.5 = –4.542 42 – 41.5 = 0.5

Σx = 415 Σ(x – μ) = 0


Population Variance

•

Population Standard Deviation

•


22 ( )x

N

Sum of squares, SSx

22 ( )x

N

Finding the Population Variance & Standard Deviation

In Words In Symbols


1. Find the mean of the population data set.

2. Find deviation of each entry.

3. Square each deviation.

4. Add to get the sum of squares.

xN

x – μ

(x – μ)2

SSx = Σ(x – μ)2

Finding the Population Variance & Standard Deviation


5. Divide by N to get the population variance.

6. Find the square root to get the population standard deviation.

22 ( )x

N

2( )xN

In Words In Symbols

Example: Finding the Population Standard Deviation

A corporation hired 10 graduates. The starting salaries for each graduate are shown. Find the population variance and standard deviation of the starting salaries.


Recall μ = 41.5.


Solution: Finding the Population Standard Deviation


• Determine SSx

• N = 10

Salary, x Deviation: x – μ Squares: (x – μ)2

41 41 – 41.5 = –0.5 (–0.5)2 = 0.2538 38 – 41.5 = –3.5 (–3.5)2 = 12.2539 39 – 41.5 = –2.5 (–2.5)2 = 6.2545 45 – 41.5 = 3.5 (3.5)2 = 12.2547 47 – 41.5 = 5.5 (5.5)2 = 30.2541 41 – 41.5 = –0.5 (–0.5)2 = 0.2544 44 – 41.5 = 2.5 (2.5)2 = 6.2541 41 – 41.5 = –0.5 (–0.5)2 = 0.2537 37 – 41.5 = –4.5 (–4.5)2 = 20.2542 42 – 41.5 = 0.5 (0.5)2 = 0.25

Σ(x – μ) = 0 SSx = 88.5

Solution: Finding the Population Standard Deviation


Population Variance

•

Population Standard Deviation

•

22 ( ) 88.5 8.9

10x

N

2 8.85 3.0

The population standard deviation is about 3.0, or $3000.


Sample Variance

•

Sample Standard Deviation

•


22 ( )

1x xsn

22 ( )

1x xs sn

Finding the Sample Variance & Standard Deviation

In Words In Symbols


1. Find the mean of the sample data set.

2. Find deviation of each entry.

3. Square each deviation.

4. Add to get the sum of squares.

xxn

2( )xSS x x

2( )x x

x x

Finding the Sample Variance & Standard Deviation


5. Divide by n – 1 to get the sample variance.

6. Find the square root to get the sample standard deviation.

In Words In Symbols2

2 ( )1

x xsn

2( )1

x xsn

Example: Finding the Sample Standard Deviation

The starting salaries are for the Chicago branches of a corporation. The corporation has several other branches, and you plan to use the starting salaries of the Chicago branches to estimate the starting salaries for the larger population. Find the sample standard deviation of the starting salaries.



Solution: Finding the Sample Standard Deviation


• Determine SSx

• n = 10

Salary, x Deviation: x – μ Squares: (x – μ)2

41 41 – 41.5 = –0.5 (–0.5)2 = 0.2538 38 – 41.5 = –3.5 (–3.5)2 = 12.2539 39 – 41.5 = –2.5 (–2.5)2 = 6.2545 45 – 41.5 = 3.5 (3.5)2 = 12.2547 47 – 41.5 = 5.5 (5.5)2 = 30.2541 41 – 41.5 = –0.5 (–0.5)2 = 0.2544 44 – 41.5 = 2.5 (2.5)2 = 6.2541 41 – 41.5 = –0.5 (–0.5)2 = 0.2537 37 – 41.5 = –4.5 (–4.5)2 = 20.2542 42 – 41.5 = 0.5 (0.5)2 = 0.25

Σ(x – μ) = 0 SSx = 88.5

Solution: Finding the Sample Standard Deviation


Sample Variance

•


•

22 ( ) 88.5 9.8

1 10 1x xsn

2 88.5 3.19

s s

The sample standard deviation is about 3.1, or $3100.

Example: Using Technology to Find the Standard Deviation

Sample office rental rates (in dollars per square foot per year) for Miami’s central business district are shown in the table. Use a calculator or a computer to find the mean rental rate and the sample standard deviation. (Adapted from: Cushman & Wakefield Inc.)


Office Rental Rates35.00 33.50 37.0023.75 26.50 31.2536.50 40.00 32.0039.25 37.50 34.7537.75 37.25 36.7527.00 35.75 26.0037.00 29.00 40.5024.50 33.00 38.00

Solution: Using Technology to Find the Standard Deviation


Sample Mean


Interpreting Standard Deviation

• Standard deviation is a measure of the typical amount an entry deviates from the mean.

• The more the entries are spread out, the greater the standard deviation.


Interpreting Standard Deviation: Empirical Rule (68 – 95 – 99.7 Rule)

For data with a (symmetric) bell-shaped distribution, the standard deviation has the following characteristics:


• About 68% of the data lie within one standard deviation of the mean.

• About 95% of the data lie within two standard deviations of the mean.

• About 99.7% of the data lie within three standard deviations of the mean.

Interpreting Standard Deviation: Empirical Rule (68 – 95 – 99.7 Rule)


3x s x s 2x s 3x sx s x2x s

68% within 1 standard deviation

34% 34%

99.7% within 3 standard deviations

2.35% 2.35%

95% within 2 standard deviations

13.5% 13.5%

Example: Using the Empirical Rule

In a survey conducted by the National Center for Health Statistics, the sample mean height of women in the United States (ages 20-29) was 64 inches, with a sample standard deviation of 2.71 inches. Estimate the percent of the women whose heights are between 64 inches and 69.42 inches.


Solution: Using the Empirical Rule


3x s x s 2x s 3x sx s x2x s55.87 58.58 61.29 64 66.71 69.42 72.13

34%

13.5%

• Because the distribution is bell-shaped, you can use the Empirical Rule.

34% + 13.5% = 47.5% of women are between 64 and 69.42 inches tall.

Chebychev’s Theorem

• The portion of any data set lying within k standard deviations (k > 1) of the mean is at least:


2

11k

• k = 2: In any data set, at least 2

1 31 or 75%2 4

of the data lie within 2 standard deviations of the mean.

• k = 3: In any data set, at least 2

1 81 or 88.9%3 9

of the data lie within 3 standard deviations of the mean.

Example: Using Chebychev’s Theorem

The age distribution for Florida is shown in the histogram. Apply Chebychev’s Theorem to the data using k = 2. What can you conclude?


Solution: Using Chebychev’s Theorem

k = 2: μ – 2σ = 39.2 – 2(24.8) = -10.4 (use 0 since age can’t be negative)

μ + 2σ = 39.2 + 2(24.8) = 88.8


At least 75% of the population of Florida is between 0 and 88.8 years old.

Standard Deviation for Grouped Data

Sample standard deviation for a frequency distribution

•

• When a frequency distribution has classes, estimate the sample mean and standard deviation by using the midpoint of each class.


2( )1

x x fsn

where n= Σf (the number of entries in the data set)

Example: Finding the Standard Deviation for Grouped Data


You collect a random sample of the number of children per household in a region. Find the sample mean and the sample standard deviation of the data set.

Number of Children in 50 Households

1 3 1 1 1

1 2 2 1 0

1 1 0 0 0

1 5 0 3 6

3 0 3 1 1

1 1 6 0 1

3 6 6 1 2

2 3 0 1 1

4 1 1 2 2

0 3 0 2 4

x f xf0 10 0(10) = 01 19 1(19) = 192 7 2(7) = 143 7 3(7) =214 2 4(2) = 85 1 5(1) = 56 4 6(4) = 24

Solution: Finding the Standard Deviation for Grouped Data

• First construct a frequency distribution.

• Find the mean of the frequency distribution.


Σf = 50 Σ(xf )= 91

91 1.850

xfxn

The sample mean is about 1.8 children.


• Determine the sum of squares.


x f0 10 0 – 1.8 = –1.8 (–1.8)2 = 3.24 3.24(10) = 32.401 19 1 – 1.8 = –0.8 (–0.8)2 = 0.64 0.64(19) = 12.162 7 2 – 1.8 = 0.2 (0.2)2 = 0.04 0.04(7) = 0.283 7 3 – 1.8 = 1.2 (1.2)2 = 1.44 1.44(7) = 10.084 2 4 – 1.8 = 2.2 (2.2)2 = 4.84 4.84(2) = 9.685 1 5 – 1.8 = 3.2 (3.2)2 = 10.24 10.24(1) = 10.246 4 6 – 1.8 = 4.2 (4.2)2 = 17.64 17.64(4) = 70.56

x x 2( )x x 2( )x x f

2( ) 145.40x x f


• Find the sample standard deviation.


x x 2( )x x 2( )x x f2( ) 145.40 1.71 50 1

x x fsn

The standard deviation is about 1.7 children.

Section 2.4 Summary

• Determined the range of a data set• Determined the variance and standard deviation of a

population and of a sample• Used the Empirical Rule and Chebychev’s Theorem

to interpret standard deviation• Approximated the sample standard deviation for

grouped data


Section 2.5

Measures of Position



• Determine the quartiles of a data set• Determine the interquartile range of a data set• Create a box-and-whisker plot• Interpret other fractiles such as percentiles• Determine and interpret the standard score (z-score)


Quartiles• Fractiles are numbers that partition (divide) an

ordered data set into equal parts.• Quartiles approximately divide an ordered data set

into four equal parts. First quartile, Q1: About one quarter of the data

fall on or below Q1. Second quartile, Q2: About one half of the data

fall on or below Q2 (median). Third quartile, Q3: About three quarters of the

data fall on or below Q3.Larson/Farber 4th ed. 134

Example: Finding Quartiles

The test scores of 15 employees enrolled in a CPR training course are listed. Find the first, second, and third quartiles of the test scores.13 9 18 15 14 21 7 10 11 20 5 18 37 16 17


Solution:• Q2 divides the data set into two halves.

5 7 9 10 11 13 14 15 16 17 18 18 20 21 37

Q2

Lower half Upper half

Solution: Finding Quartiles• The first and third quartiles are the medians of the

lower and upper halves of the data set.

5 7 9 10 11 13 14 15 16 17 18 18 20 21 37


Q2

Lower half Upper half

Q1 Q3

About one fourth of the employees scored 10 or less, about one half scored 15 or less; and about three fourths scored 18 or less.

Interquartile Range

Interquartile Range (IQR)• The difference between the third and first quartiles.• IQR = Q3 – Q1


Example: Finding the Interquartile Range

Find the interquartile range of the test scores.Recall Q1 = 10, Q2 = 15, and Q3 = 18


Solution:• IQR = Q3 – Q1 = 18 – 10 = 8

The test scores in the middle portion of the data set vary by at most 8 points.

Box-and-Whisker Plot

Box-and-whisker plot• Exploratory data analysis tool.• Highlights important features of a data set.• Requires (five-number summary):

Minimum entry First quartile Q1 Median Q2 Third quartile Q3 Maximum entry


Drawing a Box-and-Whisker Plot

1. Find the five-number summary of the data set.2. Construct a horizontal scale that spans the range of

the data.3. Plot the five numbers above the horizontal scale.4. Draw a box above the horizontal scale from Q1 to Q3

and draw a vertical line in the box at Q2.5. Draw whiskers from the box to the minimum and

maximum entries.


Whisker Whisker

Maximum entry

Minimum entry

Box

Median, Q2 Q3Q1

Example: Drawing a Box-and-Whisker Plot

Draw a box-and-whisker plot that represents the 15 test scores. Recall Min = 5 Q1 = 10 Q2 = 15 Q3 = 18 Max = 37


5 10 15 18 37

Solution:

About half the scores are between 10 and 18. By looking at the length of the right whisker, you can conclude 37 is a possible outlier.

Percentiles and Other Fractiles

Fractiles Summary SymbolsQuartiles Divides data into 4 equal

partsQ1, Q2, Q3

Deciles Divides data into 10 equal parts

D1, D2, D3,…, D9

Percentiles Divides data into 100 equal parts

P1, P2, P3,…, P99


Example: Interpreting Percentiles

The ogive represents the cumulative frequency distribution for SAT test scores of college-bound students in a recent year. What test score represents the 72nd percentile? How should you interpret this? (Source: College Board Online)


Solution: Interpreting Percentiles

The 72nd percentile corresponds to a test score of 1700. This means that 72% of the students had an SAT score of 1700 or less.


The Standard Score

Standard Score (z-score)• Represents the number of standard deviations a given

value x falls from the mean μ.

•


value - meanstandard deviation

xz

Example: Comparing z-Scores from Different Data Sets

In 2007, Forest Whitaker won the Best Actor Oscar at age 45 for his role in the movie The Last King of Scotland. Helen Mirren won the Best Actress Oscar at age 61 for her role in The Queen. The mean age of all best actor winners is 43.7, with a standard deviation of 8.8. The mean age of all best actress winners is 36, with a standard deviation of 11.5. Find the z-score that corresponds to the age for each actor or actress. Then compare your results.


Solution: Comparing z-Scores from Different Data Sets


• Forest Whitaker45 43.7 0.15

8.8xz

• Helen Mirren61 36 2.17

11.5xz

0.15 standard deviations above the mean

2.17 standard deviations above the mean

Solution: Comparing z-Scores from Different Data Sets


The z-score corresponding to the age of Helen Mirren is more than two standard deviations from the mean, so it is considered unusual. Compared to other Best Actress winners, she is relatively older, whereas the age of Forest Whitaker is only slightly higher than the average age of other Best Actor winners.

z = 0.15 z = 2.17

Section 2.5 Summary

• Determined the quartiles of a data set• Determined the interquartile range of a data set• Created a box-and-whisker plot• Interpreted other fractiles such as percentiles• Determined and interpreted the standard score

(z-score)


Documents

Chapter 2