sst5e_ppt_ch03.pptx

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STATISTICS

INFORMED DECISIONS USING DATA

Fifth Edition, Global Edition

Chapter 3

Numerically

Summarizing

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3.1 Measures of Central Tendency

Learning Objectives

1. Determine the arithmetic mean of a variable from raw data

2. Determine the median of a variable from raw data

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3.1 Measures of Central Tendency

3.1.1 Determine the Arithmetic Mean of a Variable from Raw

Data

(1 of 9)

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3.1 Measures of Central Tendency

3.1.1 Determine the Arithmetic Mean of a Variable from Raw

Data

(2 of 9)

The

population arithmetic mean

,

μ

(pronounced “mew”), is

computed using all the individuals in a population.

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3.1 Measures of Central Tendency

3.1.1 Determine the Arithmetic Mean of a Variable from Raw

Data

(3 of 9)

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3.1 Measures of Central Tendency

3.1.1 Determine the Arithmetic Mean of a Variable from Raw

Data

(4 of 9)

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3.1 Measures of Central Tendency

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3.1 Measures of Central Tendency

3.1.1 Determine the Arithmetic Mean of a Variable from Raw

Data

(6 of 9)

EXAMPLE Computing a Population Mean and a Sample Mean

The following data represent the travel times (in minutes) to work for all seven employees of a start-up web development company.

23, 36, 23, 18, 5, 26, 43 (a) Compute the population mean of this data.

(b) Then take a simple random sample of n = 3 employees.

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3.1 Measures of Central Tendency

3.1.1 Determine the Arithmetic Mean of a Variable from Raw

Data

(7 of 9)

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3.1 Measures of Central Tendency

3.1.1 Determine the Arithmetic Mean of a Variable from Raw

Data

(8 of 9)

(b) Obtain a simple random sample of size n = 3 from the population of seven employees. Use this simple random sample to determine a

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3.1 Measures of Central Tendency

3.1.1 Determine the Arithmetic Mean of a Variable from Raw

Data

(9 of 9)

IN CLASS ACTIVITY

Population Mean versus Sample Mean

Treat the students in the class as a population. All the students in the class should determine their pulse rates.

a) Compute the population mean pulse rate.

b) Obtain a simple random sample of n = 4 students and compute the sample mean. Does the sample mean equal the population mean?

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3.1 Measures of Central Tendency

3.1.2

Determine the Median of a Variable from Raw Data

(1 of 5)

The

median

of a variable is the value that lies in the middle

of the data when arranged in ascending order.

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3.1 Measures of Central Tendency

3.1.2

Determine the Median of a Variable from Raw Data

(2 of 5)

Steps in Finding the Median of a Data Set

Step 1 Arrange the data in ascending order.

Step 2 Determine the number of observations, n.

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3.1 Measures of Central Tendency

3.1.2

Determine the Median of a Variable from Raw Data

(3 of 5)

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3.1 Measures of Central Tendency

3.1.2

Determine the Median of a Variable from Raw Data

(4 of 5)

EXAMPLE Computing a Median of a Data Set with an Odd Number of Observations

23, 36, 23, 18, 5, 26, 43 Determine the median of this data.

Step 1: 5, 18, 23, 23, 26, 36, 43

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3.1 Measures of Central Tendency

3.1.2

Determine the Median of a Variable from Raw Data

(5 of 5)

EXAMPLE Computing a Median of a Data Set with an Even Number of Observations

Suppose the start-up company hires a new employee. The travel time of the new employee is 70 minutes. Determine the median of the “new” data set.

23, 36, 23, 18, 5, 26, 43, 70

Step 1: 5, 18, 23, 23, 26, 36, 43, 70

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3.1 Measures of Central Tendency

3.1.3

Explain What It Means for a Statistic to Be Resistant

(1 of 6)

EXAMPLE Computing a Median of a Data Set with an Even Number of Observations

23, 36, 23, 18, 5, 26, 43

Suppose a new employee is hired who has a 130 minute commute. How does this impact the value of the mean and median?

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3.1 Measures of Central Tendency

3.1.3

Explain What It Means for a Statistic to Be Resistant

(2 of 6)

A numerical summary of data is said to be

resistant

if

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3.1 Measures of Central Tendency

3.1.3

Explain What It Means for a Statistic to Be Resistant

(3 of 6)

Relation Between the Mean, Median, and Distribution Shape Distribution Shape Mean versus Median

Skewed left Mean substantially smaller than

median

Symmetric Mean roughly equal to median

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3.1 Measures of Central Tendency

3.1.3

Explain What It Means for a Statistic to Be Resistant

(4 of 6)

EXAMPLE Describing the Shape of the Distribution

The following data represent the asking price of homes for sale in Lincoln, NE.

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3.1 Measures of Central Tendency

3.1.3

Explain What It Means for a Statistic to Be Resistant

(5 of 6)

Find the mean and median. Use the mean and median to

identify the shape of the distribution. Verify your result by

drawing a histogram of the data.

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3.1 Measures of Central Tendency

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3.1 Measures of Central Tendency

3.1.4

Determine the Mode of a Variable from Raw Data

(1 of 7)

The

mode

of a variable is the most frequent observation of

the variable that occurs in the data set.

A set of data can have no mode, one mode, or more than

one mode.

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3.1 Measures of Central Tendency

3.1.4

Determine the Mode of a Variable from Raw Data

(2 of 7)

EXAMPLE Finding the Mode of a Data Set

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3.1 Measures of Central Tendency

3.1.4

Determine the Mode of a Variable from Raw Data

(3 of 7)

Vice President

State of Birth Vice President

State of Birth

John Adams Massachusetts Schuyler Colfax

New York Henry Wallace Iowa Thomas

Jefferson Virginia Henry Wilson New Hampshire Harry Truman Missouri Aaron Burr New Jersey William

Wheeler

New York Alben Barkley Kentucky George Clinton New York Chester Arthur Vermont Richard Nixon California Elbridge Gerry Massachusetts Thomas

Hendricks Ohio Lyndon Johnson Texas Daniel

Tompkins New York Levi Morton Vermont Hubert Humphrey South Dakota John Calhoun South Carolina Adlai

Stevenson

Kentucky Spiro Agnew Maryland Martin Van

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3.1 Measures of Central Tendency

3.1.4

Determine the Mode of a Variable from Raw Data

(4 of 7)

Vice President

State of Birth Vice President

State of Birth

Richard Johnson

Kentucky Theodore Roosevelt

New York Nelson Rockefeller

Maine John Tyler Virginia Charles

Fairbanks Ohio Walter Mondale Minnesota George Dallas Pennsylvania James

Sherman

New York George Bush Massachusetts Millard

Fillmore New York Thomas Marshall Indiana Dan Quayle Indiana William King North Carolina Calvin

Coolidge

Vermont Al Gore Washington D.C.

John

Breckinridge Kentucky Charles Dawes Ohio Richard Cheney Nebraska Hannibal

Hamlin

Maine Charles Curtis Kansas Joe Biden Pennsylvania Andrew

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3.1 Measures of Central Tendency

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3.1 Measures of Central Tendency

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3.1 Measures of Central Tendency

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3.2 Measures of Dispersion

Learning Objectives

1. Determine the range of a variable from raw data

2. Determine the standard deviation of a variable from raw

data

3. Determine the variance of a variable from raw data

4. Use the Empirical Rule to describe data that are bell

shaped

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3.2 Measures of Dispersion

Example Comparing Two Sets of Data

(1 of 4)

To order food at a McDonald’s restaurant, one must choose

from multiple lines, while at Wendy’s Restaurant, one enters

a single line. The following data represent the wait time (in

minutes) in line for a simple random sample of 30 customers

at each restaurant during the lunch hour. For each sample,

answer the following:

(a) What was the mean wait time?

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3.2 Measures of Dispersion

Example Comparing Two Sets of Data

(2 of 4)

Wait Time at Wendy’s

1.50 0.79 1.01 1.66 0.94 0.67

2.53 1.20 1.46 0.89 0.95 0.90

1.88 2.94 1.40 1.33 1.20 0.84

3.99 1.90 1.00 1.54 0.99 0.35

0.90 1.23 0.92 1.09 1.72 2.00

Wait Time at McDonald’s

3.50 0.00 0.38 0.43 1.82 3.04

0.00 0.26 0.14 0.60 2.33 2.54

1.97 0.71 2.22 4.54 0.80 0.50

0.00 0.28 0.44 1.38 0.92 1.17

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3.2 Measures of Dispersion

Example Comparing Two Sets of Data

(3 of 4)

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3.2 Measures of Dispersion

Example Comparing Two Sets of Data

(4 of 4)

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3.2 Measures of Dispersion

3.2.1 Determine the Range of a Variable from Raw Data

(1 of 2)

The

range,

R, of a variable is the difference between the

largest data value and the smallest data values. That is,

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3.2 Measures of Dispersion

3.2.1 Determine the Range of a Variable from Raw Data

(2 of 2)

EXAMPLE Finding the Range of a Set of Data

23, 36, 23, 18, 5, 26, 43

Find the range.

Range = 43 − 5

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3.2 Measures of Dispersion

3.2.2 Determine the Standard Deviation of a Variable from Raw Data (1 of 16)

The

population standard deviation

of a variable is the

square root of the sum of squared deviations about the

population mean divided by the number of observations in

the population,

N

. That is, it is the square root of the mean of

the squared deviations about the population mean.

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3.2 Measures of Dispersion

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3.2 Measures of Dispersion

A formula that is equivalent to the one on the previous slide,

called the computational formula, for determining the

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3.2 Measures of Dispersion

EXAMPLE Computing a Population Standard Deviation

23, 36, 23, 18, 5, 26, 43

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3.2 Measures of Dispersion

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3.2 Measures of Dispersion

Using the computational formula, yields the same result.

x_i (x_i)2

23 529 36 1296 23 529 18 324 5 25 26 676 43 1849

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3.2 Measures of Dispersion

The

sample standard deviation

,

s

, of a variable is the

square root of the sum of squared deviations about the

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3.2 Measures of Dispersion

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3.2 Measures of Dispersion

We call

n

− 1 the

degrees of freedom

because the first

n

− 1

observations have freedom to be whatever value they wish,

but the

n

th

value has no freedom. It must be whatever value

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3.2 Measures of Dispersion

EXAMPLE Computing a Sample Standard Deviation

Here are the results of a random sample taken from the travel times (in minutes) to work for all seven employees of a start-up web development company:

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3.2 Measures of Dispersion

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3.2 Measures of Dispersion

Using the computational formula, yields the same result.

x_i (x_i)2

5 25

26 676

36 1296

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3.2 Measures of Dispersion

IN CLASS ACTIVITY

The Sample Standard Deviation

Using the pulse data from Section 3.1, page 000, do the following:

a) Obtain a simple random sample of n = 4 students and compute the sample standard deviation.

b) Obtain a second simple random sample of n = 4 students and compute the sample standard deviation.

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3.2 Measures of Dispersion

EXAMPLE Comparing Standard Deviations

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3.2 Measures of Dispersion

Wait Time at Wendy’s

1.50 0.79 1.01 1.66 0.94 0.67

2.53 1.20 1.46 0.89 0.95 0.90

1.88 2.94 1.40 1.33 1.20 0.84

3.99 1.90 1.00 1.54 0.99 0.35

0.90 1.23 0.92 1.09 1.72 2.00

Wait Time at McDonald’s

3.50 0.00 0.38 0.43 1.82 3.04

0.00 0.26 0.14 0.60 2.33 2.54

1.97 0.71 2.22 4.54 0.80 0.50

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3.2 Measures of Dispersion

EXAMPLE Comparing Standard Deviations

Sample standard deviation for Wendy’s: 0.738 minutes Sample standard deviation for McDonald’s:

1.265 minutes

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3.2 Measures of Dispersion

3.2.3 Determine the Variance of a Variable from Raw Data (1 of 3)

The

variance

of a variable is the square of the standard

deviation. The

population variance

is

σ

2

and the

sample

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3.2 Measures of Dispersion

EXAMPLE Computing a Population Variance

23, 36, 23, 18, 5, 26, 43

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3.2 Measures of Dispersion

EXAMPLE Computing a Population Variance

Recall that the population standard deviation (from slide #49) is

σ = 11.36 so the population variance is σ2_{= 129.05 minutes}

and that the sample standard deviation (from slide #55) is

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3.2 Measures of Dispersion

3.2.4 Use the Empirical Rule to Describe Data that are Bell

Shaped

(1 of 7)

The Empirical Rule

If a distribution is roughly bell shaped, then

• Approximately 68% of the data will lie within 1 standard

deviation of the mean. That is, approximately 68% of the data lie between μ − 1σ and μ + 1σ.

• Approximately 95% of the data will lie within 2 standard

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3.2 Measures of Dispersion

3.2.4 Use the Empirical Rule to Describe Data that are Bell

Shaped

(2 of 7)

The Empirical Rule

If a distribution is roughly bell shaped, then

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3.2 Measures of Dispersion

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3.2 Measures of Dispersion

3.2.4 Use the Empirical Rule to Describe Data that are Bell

Shaped

(4 of 7)

EXAMPLE Using the Empirical Rule

The following data represent the serum HDL cholesterol of the 54 female patients of a family doctor.

41 48 43 38 35 37 44 44 44

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3.2 Measures of Dispersion

3.2.4 Use the Empirical Rule to Describe Data that are Bell

Shaped

(5 of 7)

a) Compute the population mean and standard deviation. b) Draw a histogram to verify the data is bell-shaped.

c) Determine the percentage of all patients that have serum HDL within 3 standard deviations of the mean according to the

Empirical Rule.

d) Determine the percentage of all patients that have serum HDL between 34 and 69.1 according to the Empirical Rule.

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3.2 Measures of Dispersion

3.2.4 Use the Empirical Rule to Describe Data that are Bell

Shaped

(6 of 7)

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3.2 Measures of Dispersion

3.2.4 Use the Empirical Rule to Describe Data that are Bell

Shaped

(7 of 7)

(c) According to the Empirical Rule, 99.7% of the all patients that have serum HDL within 3 standard deviations of the mean.

(d) 13.5% + 34% + 34% = 81.5% of all patients will have a serum HDL between 34.0 and 69.1 according to the Empirical Rule.

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3.2 Measures of Dispersion

3.2.5 Use Chebyshev’s Inequality to Describe Any Set of Data (1 of 2)

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3.2 Measures of Dispersion

3.2.5 Use Chebyshev’s Inequality to Describe Any Set of Data (2 of 2)

EXAMPLE Using Chebyshev’s Theorem

Using the data from the previous example, use Chebyshev’s Theorem to

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3.3 Measures of Central Tendency and Dispersion from

Grouped Data

Learning Objectives

1. Approximate the mean of a variable from grouped data

2. Compute the weighted mean

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3.3 Measures of Central Tendency and Dispersion from

Grouped Data

3.3.1 Approximate the Mean of a Variable from Grouped Data (1 of 4)

We have discussed how to compute descriptive statistics

from raw data, but often the only available data have already

been summarized in frequency distributions (

grouped data

).

Although we cannot find exact values of the mean or

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3.3 Measures of Central Tendency and Dispersion from

Grouped Data

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3.3 Measures of Central Tendency and Dispersion from

Grouped Data

EXAMPLE Approximating the Mean from a Relative Frequency Distribution

The National Survey of Student Engagement is a survey that (among other things) asked first year students at liberal arts colleges how much time they spend preparing for class each week. The results from the 2007 survey are summarized below. Approximate the mean number of hours spent preparing for class each week.

Hours 0 1−5 6−1

0 11−15 16−20 21−25 26−30 31−35

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3.3 Measures of Central Tendency and Dispersion from

Grouped Data

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3.3 Measures of Central Tendency and Dispersion from

Grouped Data

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3.3 Measures of Central Tendency and Dispersion from

Grouped Data

3.3.2 Compute the Weighted Mean

(2 of 2)

EXAMPLE Computed a Weighted Mean

Bob goes to the “Buy the Weigh” Nut store and creates his own

bridge mix. He combines 1 pound of raisins, 2 pounds of chocolate covered peanuts, and 1.5 pounds of cashews. The raisins cost

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3.3 Measures of Central Tendency and Dispersion from

Grouped Data

3.3.3 Approximate the Standard Deviation of a Variable from Grouped Data (1 of 4)

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3.3 Measures of Central Tendency and Dispersion from

Grouped Data

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3.3 Measures of Central Tendency and Dispersion from

Grouped Data

EXAMPLE Approximating the Standard Deviation from a Relative Frequency Distribution

The National Survey of Student Engagement is a survey that (among other things) asked first year students at liberal arts colleges how much time they spend preparing for class each week. The results from the 2007 survey are summarized below. Approximate the standard deviation number of hours spent

preparing for class each week.

Hours 0 1−5 6−10 11−15 16−20 21−25 26−30 31−35

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3.3 Measures of Central Tendency and Dispersion from

Grouped Data

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3.4 Measures of Position and Outliers

Learning Objectives

1. Determine and interpret

z

-scores

2. Interpret percentiles

3. Determine and interpret quartiles

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3.4 Measures of Position and Outliers

3.4.1 Determine and Interpret

z

-Scores

(1 of 3)

The

z

-score represents the distance that a data value is from

the mean in terms of the number of standard deviations. We

find it by subtracting the mean from the data value and

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3.4 Measures of Position and Outliers

3.4.1 Determine and Interpret

z

-Scores

(2 of 3)

EXAMPLE Using Z-Scores

The mean height of males 20 years or older is 69.1 inches with a standard deviation of 2.8 inches. The mean height of females 20 years or older is 63.7 inches with a standard deviation of 2.7

inches. Data is based on information obtained from National Health and Examination Survey. Who is relatively taller?

Kevin Garnett whose height is 83 inches or

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3.4 Measures of Position and Outliers

3.4.1 Determine and Interpret

z

-Scores

(3 of 3)

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3.4 Measures of Position and Outliers

3.4.2 Interpret Percentiles

(1 of 3)

The k

th percentile

, denoted,

P

_k

, of a set of data is a value

such that

k

percent of the observations are less than or

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3.4 Measures of Position and Outliers

3.4.2 Interpret Percentiles

(2 of 3)

EXAMPLE Interpret a Percentile

The Graduate Record Examination (GRE) is a test required for admission to many U.S. graduate schools. The University of

Pittsburgh Graduate School of Public Health requires a GRE score no less than the 70th percentile for admission into their Human

Genetics MPH or MS program.

(Source: http://www.publichealth.pitt.edu/interior.php?pageID=101.)

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3.4 Measures of Position and Outliers

3.4.2 Interpret Percentiles

(3 of 3)

EXAMPLE Interpret a Percentile

In general, the 70th percentile is the score such that 70% of the

individuals who took the exam scored worse, and 30% of the

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3.4 Measures of Position and Outliers

3.4.3 Determine and Interpret Quartiles

(1 of 5)

Quartiles divide data sets into fourths, or four equal parts.

• The 1st quartile, denoted Q

1, divides the bottom 25% the data from the top 75%. Therefore, the 1st quartile is equivalent to the 25th percentile.

• The 2nd quartile divides the bottom 50% of the data from the top 50% of the data, so that the 2nd quartile is equivalent to the 50th percentile, which is equivalent to the median.

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3.4 Measures of Position and Outliers

3.4.3 Determine and Interpret Quartiles

(2 of 5)

Finding Quartiles

Step 1: Arrange the data in ascending order.

Step 2: Determine the median, M, or second quartile, Q₂ .

Step 3: Divide the data set into halves: the observations below (to the left of) M and the observations above M. The first quartile, Q₁, is the median of the bottom half, and the

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3.4 Measures of Position and Outliers

3.4.3 Determine and Interpret Quartiles

(3 of 5)

EXAMPLE Finding and Interpreting Quartiles

A group of Brigham Young University—Idaho students (Matthew Herring, Nathan Spencer, Mark Walker, and Mark Steiner)

collected data on the speed of vehicles traveling through a

construction zone on a state highway, where the posted speed was 25 mph. The recorded speed of 14 randomly selected

vehicles is given below:

20, 24, 27, 28, 29, 30, 32, 33, 34, 36, 38, 39, 40, 40

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3.4 Measures of Position and Outliers

3.4.3 Determine and Interpret Quartiles

(4 of 5)

EXAMPLE Finding and Interpreting Quartiles

Step 1: The data is already in ascending order.

Step 2: There are n = 14 observations, so the median, or second quartile, Q₂, is the mean of the 7th and 8th observations.

Therefore, M = 32.5.

Step 3: The median of the bottom half of the data is the first quartile, Q₁.

20, 24, 27, 28, 29, 30, 32

The median of these seven observations is 28. Therefore, Q₁ = 28. The median of the top half of the data is the third quartile, Q₃.

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3.4 Measures of Position and Outliers

3.4.3 Determine and Interpret Quartiles

(5 of 5)

Interpretation:

• 25% of the speeds are less than or equal to the first quartile, 28 miles per hour, and 75% of the speeds are greater than 28

miles per hour.

• 50% of the speeds are less than or equal to the second quartile, 32.5 miles per hour, and 50% of the speeds are greater than 32.5 miles per hour.

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3.4 Measures of Position and Outliers

3.4.4 Determine and Interpret the Interquartile Range

(1 of 3)

The

interquartile range

,

IQR

, is the range of the middle

50% of the observations in a data set. That is, the IQR is the

difference between the third and first quartiles and is found

using the formula

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3.4 Measures of Position and Outliers

3.4.4 Determine and Interpret the Interquartile Range

(2 of 3)

EXAMPLE Determining and Interpreting the Interquartile

Range

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3.4 Measures of Position and Outliers

3.4.4 Determine and Interpret the Interquartile Range

(3 of 3)

Suppose a 15th car travels through the construction zone at 100 miles per hour. How does this value impact the mean, median, standard deviation, and interquartile range?

blank Without 15th car With 15th car

Mean 32.1 mph 36.7 mph

Median 32.5 mph 33 mph

Standard deviation 6.2 mph 18.5 mph

IQR 10 mph 11 mph

Summary: Which Measures to Report

Shape of Distribution Measures of Central

Tendency Measures of Dispersion

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3.4 Measures of Position and Outliers

3.4.5 Check a Set of Data for Outliers

(1 of 2)

Checking for Outliers by Using Quartiles

Step 1 Determine the first and third quartiles of the data.

Step 2 Compute the interquartile range.

Step 3 Determine the fences. Fences serve as cutoff points for determining outliers.

Lower Fence = Q₁ − 1.5(IQR)

Upper Fence = Q₃ + 1.5(IQR)

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3.4 Measures of Position and Outliers

3.4.5 Check a Set of Data for Outliers

(2 of 2)

EXAMPLE Determining and Interpreting the Interquartile Range

Check the speed data for outliers.

Step 1: The first and third quartiles are Q₁ = 28 mph and Q₃ = 38 mph.

Step 2: The interquartile range is 10 mph.

Step 3: The fences are

Lower Fence = Q₁ − 1.5(IQR) = 28 − 1.5(10) = 13 mph

Upper Fence = Q₃ + 1.5(IQR) = 38 + 1.5(10) = 53 mph

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3.5 The Five-Number Summary and Boxplots

Learning Objectives

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3.5 The Five-Number Summary and Boxplots

3.5.1 Compute the Five-Number Summary

(1 of 4)

The five-number summary of a set of data consists of the

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3.5 The Five-Number Summary and Boxplots

3.5.1 Compute the Five-Number Summary

(2 of 4)

EXAMPLE Obtaining the Five-Number Summary

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3.5 The Five-Number Summary and Boxplots

3.5.1 Compute the Five-Number Summary

(3 of 4)

EXAMPLE Obtaining the Five-Number Summary

Institution Rate

Pulaski Bank and Trust Company 6.5% Rainier Pacific Savings Bank 12.0% Wells Fargo Bank NA 14.4% Firstbank of Colorado 14.4% Lafayette Ambassador Bank 14.3%

Infibank 13.0%

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3.5 The Five-Number Summary and Boxplots

3.5.1 Compute the Five-Number Summary

(4 of 4)

EXAMPLE Obtaining the Five-Number Summary

First, we write the data in ascending order:

6.5%, 9.9%, 12.0%, 13.0%, 13.3%, 13.9%, 14.3%, 14.4%, 14.4%, 14.5%

The smallest number is 6.5%. The largest number is 14.5%. The first quartile is 12.0%. The second quartile is 13.6%. The third quartile is 14.4%.

Five-number Summary:

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3.5 The Five-Number Summary and Boxplots

3.5.2 Draw and Interpret Boxplots

(1 of 6)

Drawing a Boxplot

Step 1 Determine the lower and upper fences. Lower Fence = Q₁ − 1.5(IQR)

Upper Fence = Q₃ + 1.5(IQR)

where IQR = Q₃ − Q₁

Step 2 Draw a number line long enough to include the maximum and minimum values. Insert vertical lines at Q₁, M, and Q₃. Enclose these vertical lines in a box.

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3.5 The Five-Number Summary and Boxplots

3.5.2 Draw and Interpret Boxplots

(2 of 6)

Drawing a Boxplot

Step 4 Draw a line from Q₁ to the smallest data value that is larger than the lower fence. Draw a line from Q₃ to the largest

data value that is smaller than the upper fence. These lines are called whiskers.

Step 5 Any data values less than the lower fence or greater than the upper fence are outliers and are marked with an

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3.5 The Five-Number Summary and Boxplots

3.5.2 Draw and Interpret Boxplots

(3 of 6)

EXAMPLE Constructing a Boxplot

Every six months, the United States Federal Reserve Board conducts a survey of credit card plans in the U.S. The following data are the interest rates charged by 10 credit card issuers

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3.5 The Five-Number Summary and Boxplots

3.5.2 Draw and Interpret Boxplots

(4 of 6)

EXAMPLE Constructing a Boxplot

Institution Rate

Pulaski Bank and Trust Company 6.5% Rainier Pacific Savings Bank 12.0% Wells Fargo Bank NA 14.4% Firstbank of Colorado 14.4% Lafayette Ambassador Bank 14.3%

Infibank 13.0%

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3.5 The Five-Number Summary and Boxplots

3.5.2 Draw and Interpret Boxplots

(5 of 6)

Step 1: The interquartile range (IQR) is 14.4% − 12% = 2.4%. The lower and upper fences are:

Lower Fence = Q₁ − 1.5(IQR) = 12 − 1.5(2.4) = 8.4%

Upper Fence = Q₃ + 1.5(IQR) = 14.4 + 1.5(2.4) = 18.0%

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3.5 The Five-Number Summary and Boxplots

3.5.2 Draw and Interpret Boxplots

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