3.1 measures of central tendency. ch. 3 numerically summarizing data the arithmetic mean of a...

3.1

Measures of Central Tendency

Ch. 3 Numerically Summarizing Data

• The arithmetic mean of a variable is computed by determining the sum of all the values of the variable in the data set divided by the number of observations.

• The population arithmetic mean is computed using all the individuals in a population.– The population mean is a parameter.– The population arithmetic mean is denoted by

the symbol μ

Population Mean

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

1 2 Nx x x

N

Sample Mean

• The sample arithmetic mean is computed using sample data.

• The sample mean is a statistic.

• The sample arithmetic mean is denoted by

x

Sample Mean

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

x

1 2 nx x xx

n

Sample Problem 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

Compute the population mean of this data.

Median

• 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.

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

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

Determine the median of this data.

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 mean and median of the “new” data set.

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

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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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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

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Resistance

• 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

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Sample Problem• 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.

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350000300000250000200000150000100000

12

10

8

6

4

2

0

Asking Price

Frequency

Asking Price of Homes in Lincoln, NE

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Mode

• The mode of a variable is the most frequent observation of the variable that occurs in the data set.

• If there is no observation that occurs with the most frequency, we say the data has no mode.– The data on the next slide represent the Vice

Presidents of the United States and their state of birth. Find the mode.

Tally data to determine most frequent observation

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One-Variable Statistics Nspire1. Create a list &

spreadsheets page

2. Title column

3. Enter data into column

4. Create a calculator page

5. Click Menu

6. 6:Statistics– 1:Stat Calculations– 1: One-Variable Stats

•

meanSx = sample S.D.σx = population S.D.Record min, Q1, Median, Q3, Max

Practice Problem

• The following is a list of pulse rates for nine students enrolled in a section of statistics. Determine the mean, median, & mode of the data set using technology.

76

60

60

81

72

80

80

68

73

3.2

Measures of Dispersion

Range

• 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.

Population Variance

• The population variance of a variable is the sum of squared deviations about the population mean divided by the number of observations in the population, N.

• That is it is the mean of the sum of the squared deviations about the population mean. It let’s us know how spread out data is from the average.

The population variance is symbolically represented by σ2 (lower case Greek sigma squared).

Note: When using the above formula, do not round until the last computation. Use as many decimals as allowed by your calculator in order to avoid round off errors.

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EXAMPLE Computing a Population Variance


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

Compute the population variance of this data. Recall that

17424.85714

7

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 2

2 902.8571

7ix

N

129.0 minutes2

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Sample Variance

• The sample variance is computed by determining the sum of squared deviations about the sample mean and then dividing this result by n – 1.

Note: Whenever a statistic consistently overestimates or underestimates a parameter, it is called biased. To obtain an unbiased estimate of the population variance, we divide the sum of the squared deviations about the mean by n - 1.

33

The population standard deviation is denoted by

It is obtained by taking the square root of the population variance, so that

The sample standard deviation is denoted by

s

It is obtained by taking the square root of the sample variance, so that

2s s3-34

EXAMPLE Computing a Population Standard Deviation


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

Compute the population standard deviation and variance of this data using technology.

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One-Variable Statistics Nspire1. Create a list &

spreadsheets page

2. Title column

3. Enter data into column

4. Create a calculator page

5. Click Menu

6. 6:Statistics– 1:Stat Calculations– 1: One-Variable Stats

•

meanSx = sample S.D.σx = population S.D.Record min, Q1, Median, Q3, Max

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

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(a) Compute the population mean and standard deviation.

(b) Draw a histogram to verify the data is bell-shaped.

(c) Determine the percentage of patients that have serum HDL within 3 standard deviations of the mean according to the Empirical Rule.

(d) Determine the percentage of 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.

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(a) Using a TI-nspire graphing calculator, we find

(b)

7.11 and 4.57

3-41© 2010 Pearson Prentice Hall. All rights reserved

(c) Determine the percentage of patients that have serum HDL within 3 standard deviations of the mean according to the Empirical Rule.

(d) Determine the percentage of 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. (look back at original data!)

One measure of intelligence is the Stanford-Binet Intelligence Quotient (IQ). IQ scores have bell-shaped distribution with a

mean of 100 and a standard deviation of 15

A. What percentage of people has an IQ score between 70 and 130?

B. What percentage of people has an IQ score less than 70 or greater than 130?

C. What percentage of people has an IQ score below 85?

3.4

Measures of Position and Outliers

The Z-Score

EXAMPLE Using Z-ScoresThe 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 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

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Sample Problem

• Score on ACT was 26 with a mean of 22 and sd of 3. Score on SAT was 950 with mean of 925 and sd of 25. Which score is "better"?

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

• The 1st quartile, denoted Q1, 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.

• 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.

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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. In addition find the mean, median, and standard deviation. (using technology)

EXAMPLE Finding and Interpreting Quartiles

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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.

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Interquartile Range

EXAMPLE Determining and Interpreting the Interquartile Range

Determine and interpret the interquartile range of the speed data.

Q1 = 28 Q3 = 38

3 1IQR

38 28

10

Q Q

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

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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?

3-54

With Out 15th Car With 15th Car

Mean

Median

Standard Deviation

IQR

Which measures should we report now?

When we add the 15th car which changes less – the mean or median (measures of center)?

When we add the 15th car which changes les – the standard deviation or the IQR

(measures of dispersion)?

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

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Outliers

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) Upper Fence = Q3 + 1.5(IQR)

= 28 – 1.5(10) = 38 + 1.5(10)

= 13 mph = 53 mph

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

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Sample Problem• For the following data

of rainfall for Chicago, IL determine the following:

1. The Quartiles & Median

2. The IQR

3. Determine if there are any outliers.

.97 2.47 3.94 4.11 5.79

1.14 2.78 3.97 4.77 6.14

1.85 3.41 4.00 5.22 6.28

2.34 3.48 4.02 5.50 7.69

3.5

5-Number Summary and BoxPlots

5-Number Summary

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 randomly selected for the July 2005 survey. Draw 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

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

Lower Fence = Q1 – 1.5(IQR) Upper Fence = Q3 + 1.5(IQR)

= 12 – 1.5(2.4) = 14.4 + 1.5(2.4)

= 8.4% = 18.0%

Step 2:

[ ]*

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

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TI-nspire – Creating a BoxPlot• See handout• Use the Nspire calculator to create a

boxplot of the rainfall data from 3.4 data.

.97 2.47 3.94 4.11 5.79

1.14 2.78 3.97 4.77 6.14

1.85 3.41 4.00 5.22 6.28

2.34 3.48 4.02 5.50 7.69

3.1 measures of central tendency. ch. 3 numerically summarizing data the arithmetic mean of a...

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