copyright © 2014, 2013, 2010 and 2007 pearson education, inc. chapter numerically summarizing data...

Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc.

Chapter

Numerically Summarizing Data

3


Section

Measures of Central Tendency

3.1

Copyright © 2014, 2013, 2010 and 2007 Pearson Education, Inc.3-3

Objectives

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

2. Determine the median of a variable from raw data

3. Explain what it means for a statistic to be resistant

4. Determine the mode of a variable from raw data


Objective 1

• Determine the Arithmetic Mean of a Variable from Raw Data


The arithmetic mean of a variable is computed by adding all the values of the variable in the data set and dividing by the number of observations.


The population arithmetic mean, μ (pronounced “mew”), is computed using all the individuals in a population.

The population mean is a parameter.


The sample arithmetic mean, (pronounced “x-bar”), is computed using sample data.

The sample mean is a statistic.

x


If x1, x2, …, xN are the N observations of a variable from a population, then the population mean, µ, is

x1 x2 L xN

N

xiN


If x1, x2, …, xn are the n observations of a variable from a sample, then the sample mean, , is

x

x1 x2 L xn

n

xin

x


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. Compute the sample mean. Obtain a second simple random sample of n = 3 employees. Again compute the sample mean.



1 2 7...

7

ix

Nx x x

23 36 23 18 5 26 43

7

174

7

24.9 minutes

(a)



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

1 2 3 4 5 6 7

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

5 36 26

322.3

x

36 23 26

328.3

x


Objective 2

• Determine the Median of a Variable from Raw Data


The median of a variable is the value that lies in the middle of the data when arranged in ascending order.

We use M to represent the median.


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.

Step 3 Determine the observation in the middle of the data set.


Steps in Finding the Median of a Data Set

•If the number of observations is odd, then the median is the data value exactly in the middle of the data set. That is, the median is the observation that lies in then (n + 1)/2 position.

•If the number of observations is even, then the median is the mean of the two middle observations in the data set. That is, the median is the mean of the observations that lie in the n/2 position and the n/2 + 1 position.


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

Step 2: There are n = 7 observations.1 7 1

42 2

n Step 3: M = 23

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


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

Step 2: There are n = 8 observations.1 8 1

4.52 2

n Step 3:

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

23 2624.5 minutes

2M

24.5M


Objective 3

• Explain What it Means for a Statistic to Be Resistant


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?

Mean before new hire: 24.9 minutesMedian before new hire: 23 minutes

Mean after new hire: 38 minutesMedian after new hire: 24.5 minutes


A numerical summary of data is said to be resistant if extreme values (very large or small) relative to the data do not affect its value substantially.


EXAMPLE Describing the Shape of the Distribution

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

Source: http://www.homeseekers.com

79,995 128,950 149,900 189,900

99,899 130,950 151,350 203,950

105,200 131,800 154,900 217,500

111,000 132,300 159,900 260,000

120,000 134,950 163,300 284,900

121,700 135,500 165,000 299,900

125,950 138,500 174,850 309,900

126,900 147,500 180,000 349,900


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.


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.

The mean asking price is $168,320 and the median asking price is $148,700. Therefore, we would conjecture that the distribution is skewed right.


350000300000250000200000150000100000

12

10

8

6

4

2

0

Asking Price

Frequency

Asking Price of Homes in Lincoln, NE


Objective 4

• Determine the Mode of a Variable from Raw Data


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.

If no observation occurs more than once, we say the data have no mode.


EXAMPLE Finding the Mode of a Data Set

The data on the next slide represent the Vice Presidents of the United States and their state of birth. Find the mode.


Joe Biden Pennsylvania


The mode is New York.


Tally data to determine most frequent observation


Section

Measures of Dispersion

3.2


Objectives

1. Determine the range of a variable from raw data2. Determine the standard deviation of a variable from

raw data3. Determine the variance of a variable from raw data 4. Use the Empirical Rule to describe data that are bell

shaped5. Use Chebyshev’s Inequality to describe any data set


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?

(b) Draw a histogram of each restaurant’s wait time.

(c ) Which restaurant’s wait time appears more dispersed? Which line would you prefer to wait in? Why?


1.50 0.79 1.01 1.66 0.94 0.672.53 1.20 1.46 0.89 0.95 0.901.88 2.94 1.40 1.33 1.20 0.843.99 1.90 1.00 1.54 0.99 0.350.90 1.23 0.92 1.09 1.72 2.00

3.50 0.00 0.38 0.43 1.82 3.040.00 0.26 0.14 0.60 2.33 2.541.97 0.71 2.22 4.54 0.80 0.500.00 0.28 0.44 1.38 0.92 1.173.08 2.75 0.36 3.10 2.19 0.23

Wait Time at Wendy’s

Wait Time at McDonald’s


(a) The mean wait time in each line is 1.39 minutes.


(b)


Objective 1

• Determine the Range of a Variable from Raw Data


The range, R, of a variable is the difference between the largest data value and the smallest data values. That is,

Range = R = Largest Data Value – Smallest Data Value


EXAMPLE Finding the Range of a Set of Data


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

Find the range.

Range = 43 – 5

= 38 minutes


Objective 2

• Determine the Standard Deviation of a Variable from Raw Data


The population standard deviation is the measure of variation of values about the mean.

The population standard deviation is symbolically represented by σ (lowercase Greek sigma).

The mean of the waiting time at Wendy’s and McDonald’s are the same but the samples are different in the amount of waiting times. We want to develop the ability to measure and understand such variation.


where x1, x2, . . . , xN are the N observations in the population and μ is the population mean.

x1 2

x2 2L xN 2

N

xi 2N


A formula that is equivalent to the one on the previous slide, called the computational formula, for determining the population standard deviation is

xi

2 xi 2

NN


EXAMPLE Computing a Population Standard Deviation


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

Compute the population standard deviation of this data.


xi μ xi – μ (xi – μ)2

23 24.85714 -1.85714 3.44898

36 24.85714 11.14286 124.1633

23 24.85714 -1.85714 3.44898

18 24.85714 -6.85714 47.02041

5 24.85714 -19.8571 394.3061

26 24.85714 1.142857 1.306122

43 24.85714 18.14286 329.1633

902.8571 2

ix

xi 2N

902.8571

711.36 minutes


xi (xi )2

23 529

36 1296

23 529

18 324

5 25

26 676

43 1849

Σ xi = 174 Σ (xi)2 = 5228

xi

2 xi 2

NN

5228

174 2

77

11.36 minutes

Using the computational formula, yields the same result.


The sample standard deviation, s, of a variable is the square root of the sum of squared deviations about the sample mean divided by n – 1, where n is the sample size.

where x1, x2, . . . , xn are the n observations in the sample and is the sample mean.

s xi x 2n 1

x1 x 2

x2 x 2L xn x 2

n 1

x


A formula that is equivalent to the one on the previous slide, called the computational formula, for determining the sample standard deviation is

s xi

2 xi 2

nn 1


We call n - 1 the degrees of freedom because the first n - 1 observations have freedom to be whatever value they wish, but the nth value has no freedom. It must be whatever value forces the sum of the deviations about the mean to equal zero.


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:

5, 26, 36

Find the sample standard deviation.


xi

5 22.33333 -17.333 300.432889

26 22.33333 3.667 13.446889

36 22.33333 13.667 186.786889

500.66667

xi x 2

s xi x 2n 1

500.66667

215.82 minutes

x xi x xi x 2


xi (xi )2

5 25

26 676

36 1296

Σ xi = 67 Σ (xi)2 = 1997

xi

2 xi 2

nn 1

1997

67 2

32

15.82 minutes

Using the computational formula, yields the same result.


EXAMPLE Comparing Standard Deviations

Determine the standard deviation waiting time for Wendy’s and McDonald’s. Which is larger? Why?


1.50 0.79 1.01 1.66 0.94 0.672.53 1.20 1.46 0.89 0.95 0.901.88 2.94 1.40 1.33 1.20 0.843.99 1.90 1.00 1.54 0.99 0.350.90 1.23 0.92 1.09 1.72 2.00

3.50 0.00 0.38 0.43 1.82 3.040.00 0.26 0.14 0.60 2.33 2.541.97 0.71 2.22 4.54 0.80 0.500.00 0.28 0.44 1.38 0.92 1.173.08 2.75 0.36 3.10 2.19 0.23

Wait Time at Wendy’s

Wait Time at McDonald’s


EXAMPLE Comparing Standard Deviations

Sample standard deviation for Wendy’s:

0.738 minutes

Sample standard deviation for McDonald’s:

1.265 minutes

Recall from earlier that the data is more dispersed for McDonald’s resulting in a larger standard deviation.


Objective 3

• Determine the Variance of a Variable from Raw Data


The variance of a variable is the square of the standard deviation. The population variance is σ2 and the sample variance is s2.


EXAMPLE Computing a Population Variance


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

Compute the population and sample variance of this data.


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 s = 15.82, so the sample variance iss2 = 250.27 minutes


Objective 4

• Use the Empirical Rule to Describe Data That Are Bell Shaped


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 betweenμ – 1σ and μ + 1σ.

• Approximately 95% of the data will lie within 2 standard deviations of the mean. That is, approximately 95% of the data lie betweenμ – 2σ and μ + 2σ.


The Empirical Rule

If a distribution is roughly bell shaped, then

• Approximately 99.7% of the data will lie within 3 standard deviations of the mean. That is, approximately 99.7% of the data lie between μ – 3σ and μ + 3σ.

Note: We can also use the Empirical Rule based on sample data with used in place of μ and s used in place of σ.

x


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 4462 75 77 58 82 39 85 55 5467 69 69 70 65 72 74 74 7460 60 60 61 62 63 64 64 6454 54 55 56 56 56 57 58 5945 47 47 48 48 50 52 52 53


(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.

(e) Determine the actual percentage of patients that have serum HDL between 34 and 69.1.


(a) Using a TI-83 plus graphing calculator, we find

(b)

7.11 and 4.57


22.3 34.0 45.7 57.4 69.1 80.8 92.5

(e) 45 out of the 54 or 83.3% of the patients have a serum HDL between 34.0 and 69.1.

(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.


Objective 5

• Use Chebyshev’s Inequality to Describe Any Set of Data


Chebyshev’s Inequality

For any data set or distribution, at least

1 1

k2

100% of the observations lie within k

Note: We can also use Chebyshev’s Inequality based on sample data.

standard deviations of the mean, where k is any number greater than 1. That is, at least

of the data lie between μ – kσ

and μ + kσ for k > 1.

1 1

k2

100%


EXAMPLE Using Chebyshev’s Theorem

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

(a) determine the percentage of patients that have serum HDL within 3 standard deviations of the mean.

k = 3

(b) determine the actual percentage of patients that have serum HDL between 34 and 80.8 (within 3 SD of mean).

52/54 ≈ 0.96 ≈ 96%

1 1

32

100% 88.9%


Section

Measures of Position and Outliers

3.4


Objectives

1. Determine and interpret z-scores

2. Interpret percentiles

3. Determine and interpret quartiles

4. Determine and interpret the interquartile range

5. Check a set of data for outliers


Objective 1Consider a comparison of two individual data values:

a)A man’s height is 76.2 in, the heights from a data set have a mean of 68.34 in and a standard deviation 3.02 in

b)A man’s weight of 237.2 lb, the weights from a data set have a mean of 172.55 lb and a standard deviation of 26.33 lb

We cannot compare heights and weights directly(apples and oranges), we need to determine the relative standings


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 dividing this result by the standard deviation. There is both a population z-score and a sample z-score:

z x

Sample z-scorePopulation z-score

The z-score is unitless. It has mean 0 and standard deviation 1.

z x x

s


If we calculate the z score for the height and weight problem, we get:

Height: z = (76.2 in -68.34 in)/3.02 in =2.60 2.60 SD above the meanWeight: z = (237.1-172.55)/26.33=2.45 2.45 SD above the meanHeight is more extreme value.

The height of 76.2 in is more extreme that the weight of 237.1


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

Candace Parker whose height is 76 inches


Kevin Garnett’s height is 4.96 standard deviations above the mean. Candace Parker’s height is 4.56 standard deviations above the mean. Kevin Garnett is relatively taller.

zkg 83 69.1

2.84.96

zcp 76 63.7

2.74.56


Objective 2

• Interpret Percentiles

Percentiles divide the data set into 100 equal groups.


The kth percentile, denoted, Pk , of a set of data is a value such that k percent of the observations are less than or equal to the value.


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.)

Interpret this admissions requirement.


EXAMPLES 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 individuals scores better. In order to be admitted to this program, an applicant must score as high or higher than 70% of the people who take the GRE. Put another way, the individual’s score must be in the top 30%.

If a student’s test score of 74 corresponds to the 64th percentile, then the student did better than 64% of the students in his class


Objective 3

• Determine and Interpret Quartiles


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

• The 1st quartile, denoted Q1, divides the bottom 25% of 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.

• The 3rd quartile divides the bottom 75% of the data from the top 25% of the data, so that the 3rd quartile is equivalent to the 75th percentile.


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 third quartile, Q3 , is the median of the top half.


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

Find and interpret the quartiles for speed in the construction zone.

EXAMPLE Finding and Interpreting Quartiles


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. Therefore, Q3 = 38.


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.

• 75% of the speeds are less than or equal to the third quartile, 38 miles per hour, and 25% of the speeds are greater than 38 miles per hour.


Objective 4

• Determine and Interpret the Interquartile Range


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

IQR = Q3 – Q1


EXAMPLE Determining and Interpreting the Interquartile Range

Determine and interpret the interquartile range of the speed data.

Q1 = 28 Q3 = 38

The range of the middle 50% of the speed of cars traveling through the construction zone is 10 miles per hour.

IQR Q3 Q

1

38 28

10


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?

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


Objective 5

• Check a Set of Data for Outliers

An Outlier is an extremely high or an extremely low data value when compared with the rest of the data values


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)

Step 4 If a data value is less than the lower fence or greater than the upper fence, it is considered an outlier.


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

Step 4: There are no values less than 13 mph or greater than 53 mph. Therefore, there are no outliers.


Section

The Five-Number Summary and Boxplots

3.5


Objectives

1. Compute the five-number summary

2. Draw and interpret boxplots


Objective 1

• Compute the Five-Number Summary


The five-number summary of a set of data consists of the smallest data value, Q1, the median, Q3, and the largest data value. We organize the five-number summary as follows:


EXAMPLE Obtaining the Five-Number Summary

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 randomly selected for the July 2005 survey. Determine the five-number summary of the data.



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%

United Bank, Inc. 13.3%

First National Bank of The Mid-Cities 13.9%

Bank of Louisiana 9.9%

Bar Harbor Bank and Trust Company 14.5%Source: http://www.federalreserve.gov/pubs/SHOP/survey.htm



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:

6.5% 12.0% 13.6% 14.4% 14.5%


Objective 2

• Draw and Interpret Boxplots


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.

Step 3 Label the lower and upper fences.


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 asterisk (*).



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 randomly selected for the July 2005 survey. Construct a boxplot of the data.



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%

United Bank, Inc. 13.3%

First National Bank of The Mid-Cities 13.9%

Bank of Louisiana 9.9%

Bar Harbor Bank and Trust Company 14.5%Source: http://www.federalreserve.gov/pubs/SHOP/survey.htm


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%

Step 2:

[ ]*


The interest rate boxplot indicates that the distribution is skewed left.

Use a boxplot and quartiles to describe the shape of a distribution.

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