Numerically Summarizing Data Chapter 3 Numerically Summarizing Data 3.1 3.2 3.3 3.4 3.5

Measures of Central Tendency Section 3.1 Measures of Central Tendency 3.1 3.2 3.3 3.4 3.5

3.1 Objectives Determine the arithmetic mean of a variable from raw data Determine the median of a variable from raw data Explain what it means for a statistic to be resistant Determine the mode of a variable from raw data 3

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. 4

The population mean is a parameter. 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. 5

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 Compute the population mean of this data. 6

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. 7

Determine the median of this data. 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. Step 1: 5, 18, 23, 23, 26, 36, 43 Step 2: There are n = 7 observations. 5, 18, 23, 23, 26, 36, 43 M = 23 8

There are 8 observations. 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 There are 8 observations. 5, 18, 23, 23, 26, 36, 43, 70 9

Objective 3 Explain What it Means for a Statistic to Be Resistant 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. 10

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Mean before new hire: 24.9 minutes Median before new hire: 23 minutes EXAMPLE Computing a Median of a Data Set with an Even 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 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 minutes Median before new hire: 23 minutes Mean after new hire: 38 minutes Median after new hire: 24.5 minutes After the new hire, the data will be right skewed. 12

EXAMPLE Describing the Shape of the Distribution The following data represent the asking price of homes for sale in Lincoln, NE. 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 Source: http://www.homeseekers.com 13

Therefore, we would conjecture that the distribution is skewed right. 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. 14

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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. 16

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. 17

Joe Biden Pennsylvania The mode is New York. 18

Tally data to determine most frequent observation 19

Summary 3.1 Objectives Determine the arithmetic mean of a variable from raw data Determine the median of a variable from raw data Explain what it means for a statistic to be resistant Determine the mode of a variable from raw data 20

Measures of Dispersion Section 3.2 Measures of Dispersion

3.2 Objectives Determine the range of a variable from raw data Determine the standard deviation and variance of a variable from raw data Use the Empirical Rule to describe data that are bell shaped Use Chebyshev’s Inequality to describe any data set 22

(a) What was the mean wait time? 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? 23

Which restaurant’s wait time appears more dispersed? Wait Time at Wendy’s Wait Time at McDonald’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 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 3.08 2.75 0.36 3.10 2.19 0.23 (a) The mean wait time in each line is 1.39 minutes. Which restaurant’s wait time appears more dispersed? 24

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 25

EXAMPLE Finding the Range of a Set of Data 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 Find the range. Range = 43 – 5 = 38 minutes 26

Objective 2 Determine the Standard Deviation and Variance of a Variable from Raw Data The standard deviation of a variable is the square root of the mean of the squared deviations about the population mean. The population standard deviation is represented by σ (lowercase Greek sigma). The sample standard deviation is represented by s. The variance of a variable is the square of the standard deviation. The population variance is σ2 and the sample variance is s2. 27

Standard Deviation Formulas Sample Population population variance = σ2 sample variance = s2

Compute the population standard deviation of this data. EXAMPLE Computing a Population Standard Deviation 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 standard deviation of this data. 29

xi μ xi – μ (xi – μ)2 23 24.85714 -1.85714 3.44898 36 11.14286 124.1633 18 -6.85714 47.02041 5 -19.8571 394.3061 26 1.142857 1.306122 43 18.14286 329.1633 902.8571 Variance = 128.98 30

Using the computational formula, yields the same standard deviation. xi (xi )2 23 529 36 1296 18 324 5 25 26 676 43 1849 Σ xi = 174 Σ (xi)2 = 5228 Variance = 128.98 31

Find the sample standard deviation. 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 three employees of a start-up web development company: 5, 26, 36 Find the sample standard deviation. 32

xi 5 22.33333 -17.333 300.432889 26 3.667 13.446889 36 13.667 186.786889 500.66667 Variance = 250.33 33

xi (xi )2 Using the computational formula, yields the same result. 5 25 26 676 36 1296 Σ xi = 67 Σ (xi)2 = 1997 Variance = 250.33 34

Reminder of Wendy’s and McDonald’s Histogram Which do we suspect will have a larger standard deviation? Wait Time at Wendy’s Wait Time at McDonald’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 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 3.08 2.75 0.36 3.10 2.19 0.23 35

EXAMPLE Comparing Standard Deviations Determine the standard deviation waiting time for Wendy’s and McDonald’s. Which is larger? Why? Don’t actually calculate the standard deviation Wait Time at Wendy’s Wait Time at McDonald’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 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 3.08 2.75 0.36 3.10 2.19 0.23 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. 36

Objective 3 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. • Approximately 95% of the data will lie within 2 standard deviations of the mean. Approximately 99.7% of the data will lie within 3 standard deviations of the mean. 37

34% 34% 0.15% 2.35% 2.35% 0.15% 13.5% 13.5% 38

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 62 75 77 58 82 39 85 55 54 67 69 69 70 65 72 74 74 74 60 60 60 61 62 63 64 64 64 54 54 55 56 56 56 57 58 59 45 47 47 48 48 50 52 52 53 39

22.3 34.0 45.7 57.4 69.1 80.8 92.5 Determine the percentage of all patients that have serum HDL within 3 standard deviations of the mean according to the Empirical Rule. (b) Determine the percentage of all patients that have serum HDL between 34 and 69.1 according to the Empirical Rule. (c) Determine the actual percentage of patients that have serum HDL between 34 and 69.1. According to the Empirical Rule, 99.7% 13.5% + 34% + 34% = 81.5% 45 out of the 54 or 83.3% For part c) go back a slide and count how many out of 45 40

Objective 4 Use Chebyshev’s Inequality to Describe Any Set of Data For any data set or distribution, at least of the observations lie within k standard deviations of the mean, where k is any number greater than 1. Note: We can also use Chebyshev’s Inequality based on sample data. 41

Chebychev’s Theorem If k = 2, what would Chebychev’s Theorem imply? And, k = 3? k = 3 k = 2: In any data set, at least 75% of the data lie within 2 standard deviations of the mean. In any data set, at least 88.9% of the data lie within 3 standard deviations of the mean. Larson/Farber 4th ed.

Example 1: 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? k = 2 = 0.75 -10.4 14.4 39.2 64 88.8 At least 75% of the population of Florida is between 0 and 88.8 years old. Larson/Farber 4th ed.

Example2 : Using Chebychev’s Theorem The mean time in a women’s 400 meter dash is 57.07 seconds, with a standard deviation of 1.05. Apply Chebychev’s Theorem to estimate how many out of 52 runners will have a time between 52.87 and 61.27 seconds. Finally: Ask, how many standard deviations from the mean? k = 4 52.87 53.92 54.97 56.02 57.07 58.12 59.17 60.22 61.27 How many women of the 52 runners have a time between 52.87 and 61.27 sec? 15/16 of 52 = 48.75 ≈ 48 runners (they will tell you to round down) Larson/Farber 4th ed.

Summary 3.2 Objectives Determine the range of a variable from raw data Determine the standard deviation and variance of a variable from raw data Use the Empirical Rule to describe data that are bell shaped 68% - 95% - 99.7% Use Chebyshev’s Inequality to describe any data set 45

Measures of Central Tendency and Dispersion from Grouped Data Section 3.3 Measures of Central Tendency and Dispersion from Grouped Data

3.3 Objectives Approximate the mean of a variable from grouped data Compute the weighted mean of a variable from grouped data Approximate the standard deviation of a variable from grouped data 47

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 standard deviation without raw data, we can approximate these measures using the techniques discussed in this section. 48

Objective 1 Approximate the Mean of a Variable from Grouped Data Approximate the Mean of a Variable from a Frequency Distribution Population Mean Sample Mean where xi is the midpoint or value of the ith class fi is the frequency of the ith class 49

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 1-5 6-10 11-15 16-20 21-25 26-30 31-35 Frequency 130 250 230 180 100 60 50 Source:http://nsse.iub.edu/NSSE_2007_Annual_Report/docs/withhold/NSSE_2007_Annual_Report.pdf 50

Time Frequency xi xi fi 1 - 5 130 3 390 6 - 10 250 8 2000 11 - 15 230 1 - 5 130 3 390 6 - 10 250 8 2000 11 - 15 230 13 2990 16 - 20 180 18 3240 21 - 25 100 23 2300 26 – 30 60 28 1680 31 – 35 50 33 1650 51

Objective 2 Compute the Weighted Mean 52

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 $1.25 per pound, the chocolate covered peanuts cost $3.25 per pound, and the cashews cost $5.40 per pound. What is the cost per pound of this mix. 53

Objective 3 Approximate the Standard Deviation of a Variable from Grouped Data 54

Population Standard Deviation Sample Standard Deviation Approximate the Standard Deviation of a Variable from a Frequency Distribution Population Standard Deviation Sample Standard Deviation where xi is the midpoint or value of the ith class fi is the frequency of the ith class 55

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 standard deviation number of hours spent preparing for class each week. Hours 1-5 6-10 11-15 16-20 21-25 26-30 31-35 Frequency 130 250 230 180 100 60 50 Source:http://nsse.iub.edu/NSSE_2007_Annual_Report/docs/withhold/NSSE_2007_Annual_Report.pdf 56

L2 L1 L3 Time Freq xi 1 - 5 130 3 –11.25 16,453.125 6 - 10 250 8 –6.25 9765.625 11 - 15 230 13 –1.25 359.375 16 - 20 180 18 3.75 2531.25 21 - 25 100 23 8.75 7656.25 26 – 30 60 28 13.75 11,343.75 31 – 35 50 33 18.75 17,578.125 57

3.3 Objectives Approximate the mean of a variable from grouped data Compute the weighted mean Approximate the standard deviation of a variable from grouped data 58

Measures of Position and Outliers Section 3.4 Measures of Position and Outliers

3.4 Objectives Determine and interpret z-scores Interpret percentiles Determine and interpret quartiles Determine and interpret the interquartile range Check a set of data for outliers 60

Objective 1 Determine and Interpret z-scores The z-score represents the distance that a data value is from the mean in terms of the number of standard deviations. Population z-score Sample z-score 61

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. 62

Objective 2 Interpret Percentiles 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. 63

Interpret this admissions requirement. 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. In general, the 70th percentile is the score such that 70% of the those 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%. 64

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% the data from the top 75%. Therefore, Q1 = P25. The 2nd quartile divides the bottom 50% of the data from the top 50% of the data. Q2 = P50 = M The 3rd quartile divides the bottom 75% of the data from the top 25% of the data. Q3 = P75 65

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 Find and interpret the quartiles for speed in the construction zone. The data values are already in the correct order Q1 = 28 = P25 Q2 = 32.5 = M = P50 Q3 = 38 = P75 66

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. IQR = Q3 – Q1 67

Determine and interpret the interquartile range of the speed data. EXAMPLE Determining and Interpreting the Interquartile Range Determine and interpret the interquartile range of the speed data. 20, 24, 27, 28, 29, 30, 32, 33, 34, 36, 38, 39, 40, 40 The range of the middle 50% of the speed of cars traveling through the construction zone is 10 miles per hour. 68

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 69

Objective 5 Check a Set of Data for Outliers Checking for Outliers by Using Quartiles Step 1 Determine Q1, Q3, and IQR. Step 2 Determine the fences. Fences serve as cutoff points for determining outliers. Lower Fence = Q1 – 1.5(IQR) Upper Fence = Q3 + 1.5(IQR) Step 3 If a data value is less than the lower fence or greater than the upper fence, it is considered an outlier. 70

Check the speed data for outliers. EXAMPLE Determining and Interpreting the Interquartile Range Check the speed data for outliers. 20, 24, 27, 28, 29, 30, 32, 33, 34, 36, 38, 39, 40, 40 Step 1: Q1 = 28 mph, Q3 = 38 mph, and IQR = 10. Step 2: 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 3: There are no values less than 13 mph or greater than 53 mph. Therefore, there are no outliers. 71

3.4 Objectives Determine and interpret z-scores Interpret percentiles Determine and interpret quartiles Determine and interpret the interquartile range Check a set of data for outliers 72

The Five-Number Summary and Boxplots Section 3.5 The Five-Number Summary and Boxplots

3.5 Objectives Compute the five-number summary Draw and interpret boxplots 74

Objective 1 Compute the Five-Number Summary The five-number summary of a set of data consists of the minimum value, Q1, the median, Q3, and the maximum data value. 75

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. 76

Pulaski Bank and Trust Company 6.5% Rainier Pacific Savings Bank 12.0% 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 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% Notice: the data is NOT in order Source: http://www.federalreserve.gov/pubs/SHOP/survey.htm 77

First, we write the data in ascending order: 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% Five-number Summary: 6.5% 12.0% 13.6% 14.4% 14.5% 78

Objective 2 Draw and Interpret Boxplots 79

Step 1 Determine the lower and upper fences. Drawing a Boxplot Step 1 Determine the lower and upper fences. 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. 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 (*). 80

* Construct a boxplot of the data from the last example. EXAMPLE Obtaining the Five-Number Summary Construct a boxplot of the data from the last example. 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% Five-number Summary: 6.5% 12.0% 13.6% 14.4% 14.5% 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% Skewed: Left Discuss percentiles from the graph Outliers are 6.5% * –––––––– – 81

Use a boxplot and quartiles to describe the shape of a distribution. 82

3.5 Objectives Compute the five-number summary Draw and interpret boxplots 83