© 2010 Pearson Prentice Hall. All rights reserved Numerical Descriptions of 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. 3-2© 2010 Pearson Prentice Hall. All rights reserved

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

If x 1, x 2, …, x N are the N observations of a variable from a population, then the population mean, µ, is 3-4© 2010 Pearson Prentice Hall. All rights reserved

The sample arithmetic mean is computed using sample data. The sample mean is a statistic. The sample arithmetic mean is denoted by. 3-5© 2010 Pearson Prentice Hall. All rights reserved

If x 1, x 2, …, x n are the n observations of a variable from a sample, then the sample mean,, is 3-6© 2010 Pearson Prentice Hall. All rights reserved

EXAMPLEComputing the population mean and sample mean of a data set 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 population mean of this data. Step 1: There are N = 7 observations. Step 2: 3-7© 2010 Pearson Prentice Hall. All rights reserved = 174 Step 3: 174 / 7 ≈

EXAMPLEComputing the population mean and sample mean of a data set Now suppose that from the seven times given we take a random sample of 3 observations. Those observations were: 5, 36, 26 Determine the sample mean for these observations Step 1: There are n = 3 observations. Step 2: 3-8© 2010 Pearson Prentice Hall. All rights reserved = 67 Step 3: 67 / 3≈

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. 3-9© 2010 Pearson Prentice Hall. All rights reserved

3-10© 2010 Pearson Prentice Hall. All rights reserved

EXAMPLEComputing 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. Step 3: M = 23 5, 18, 23, 23, 26, 36, © 2010 Pearson Prentice Hall. All rights reserved

EXAMPLEComputing 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. Step 3: 5, 18, 23, 23, 26, 36, 43, © 2010 Pearson Prentice Hall. All rights reserved

EXAMPLEComputing 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 3-13© 2010 Pearson Prentice Hall. All rights reserved

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. 3-14© 2010 Pearson Prentice Hall. All rights reserved

3-15© 2010 Pearson Prentice Hall. All rights reserved

EXAMPLE Describing the Shape of the Distribution The following data represent the asking price of homes for sale in Lincoln, NE. Source: 79,995128,950149,900189,900 99,899130,950151,350203, ,200131,800154,900217, ,000132,300159,900260, ,000134,950163,300284, ,700135,500165,000299, ,950138,500174,850309, ,900147,500180,000349, © 2010 Pearson Prentice Hall. All rights reserved

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. 3-17© 2010 Pearson Prentice Hall. All rights reserved

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. 3-18© 2010 Pearson Prentice Hall. All rights reserved

3-19© 2010 Pearson Prentice Hall. All rights reserved

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. 3-20© 2010 Pearson Prentice Hall. All rights reserved

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. 3-21© 2010 Pearson Prentice Hall. All rights reserved

3-22© 2010 Pearson Prentice Hall. All rights reserved

3-23© 2010 Pearson Prentice Hall. All rights reserved

The mode is New York. 3-24© 2010 Pearson Prentice Hall. All rights reserved

Tally data to determine most frequent observation 3-25© 2010 Pearson Prentice Hall. All rights reserved

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 3-26© 2010 Pearson Prentice Hall. All rights reserved

EXAMPLEFinding 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 3-27© 2010 Pearson Prentice Hall. All rights reserved

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. 3-28© 2010 Pearson Prentice Hall. All rights reserved

The population variance is symbolically represented by σ 2 (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. 3-29© 2010 Pearson Prentice Hall. All rights reserved

EXAMPLE Computing a Population Variance 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 variance of this data. Recall that 3-30© 2010 Pearson Prentice Hall. All rights reserved

xixi μ x i – μ(x i – μ) minutes © 2010 Pearson Prentice Hall. All rights reserved

The Computational Formula 3-32© 2010 Pearson Prentice Hall. All rights reserved

EXAMPLE Computing a Population Variance Using the Computational Formula 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 variance of this data using the computational formula. 3-33© 2010 Pearson Prentice Hall. All rights reserved

23, 36, 23, 18, 5, 26, © 2010 Pearson Prentice Hall. All rights reserved

The sample variance is computed by determining the sum of squared deviations about the sample mean and then dividing this result by n – © 2010 Pearson Prentice Hall. All rights reserved

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 © 2010 Pearson Prentice Hall. All rights reserved

EXAMPLE Computing a Sample Variance Previously, we obtained the following simple random sample for the travel time data: 5, 36, 26. Compute the sample variance travel time. Travel Time, x i Sample Mean,Deviation about the Mean, Squared Deviations about the Mean, – = ( ) 2 = square minutes 3-37© 2010 Pearson Prentice Hall. All rights reserved

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 3-38© 2010 Pearson Prentice Hall. All rights reserved

EXAMPLEComputing a Sample Standard Deviation Recall the sample data 5, 26, 36 results in a sample variance of square minutes Use this result to determine the sample standard deviation. 3-39© 2010 Pearson Prentice Hall. All rights reserved

EXAMPLE Comparing Standard Deviations Determine the standard deviation waiting time for Wendy’s and McDonald’s. Which is larger? Why? 3-40© 2010 Pearson Prentice Hall. All rights reserved

Wait Time at Wendy’s Wait Time at McDonald’s 3-41© 2010 Pearson Prentice Hall. All rights reserved

EXAMPLE Comparing Standard Deviations Determine the standard deviation waiting time for Wendy’s and McDonald’s. Which is larger? Why? Sample standard deviation for Wendy’s: minutes Sample standard deviation for McDonald’s: minutes 3-42© 2010 Pearson Prentice Hall. All rights reserved

Quartiles divide data sets into fourths, or four equal parts. The 1 st quartile, denoted Q 1, divides the bottom 25% the data from the top 75%. Therefore, the 1 st quartile is equivalent to the 25 th percentile. The 2 nd quartile divides the bottom 50% of the data from the top 50% of the data, so that the 2 nd quartile is equivalent to the 50 th percentile, which is equivalent to the median. The 3 rd quartile divides the bottom 75% of the data from the top 25% of the data, so that the 3 rd quartile is equivalent to the 75 th percentile. 3-43© 2010 Pearson Prentice Hall. All rights reserved

3-44© 2010 Pearson Prentice Hall. All rights reserved

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 Step 1: The data is already in ascending order. Step 2: There are n = 14 observations, so the median, or second quartile, Q 2, is the mean of the 7 th and 8 th observations. Therefore, M = Step 3: The median of the bottom half of the data is the first quartile, Q 1. 20, 24, 27, 28, 29, 30, 32 The median of these seven observations is 28. Therefore, Q 1 = 28. The median of the top half of the data is the third quartile, Q 3. Therefore, Q 3 = © 2010 Pearson Prentice Hall. All rights reserved

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. 3-46© 2010 Pearson Prentice Hall. All rights reserved

3-47© 2010 Pearson Prentice Hall. All rights reserved

EXAMPLE Determining and Interpreting the Interquartile Range Determine and interpret the interquartile range of the speed data. Q 1 = 28 Q 3 = 38 The range of the middle 50% of the speed of cars traveling through the construction zone is 10 miles per hour. 3-48© 2010 Pearson Prentice Hall. All rights reserved

Suppose a 15 th 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 15 th carWith 15 th car Mean32.1 mph36.7 mph Median32.5 mph33 mph Standard deviation6.2 mph18.5 mph IQR10 mph11 mph 3-49© 2010 Pearson Prentice Hall. All rights reserved

3-50© 2010 Pearson Prentice Hall. All rights reserved

EXAMPLE Determining and Interpreting the Interquartile Range Check the speed data for outliers. Step 1: The first and third quartiles are Q 1 = 28 mph and Q 3 = 38 mph. Step 2: The interquartile range is 10 mph. Step 3: The fences are Lower Fence = Q 1 – 1.5(IQR) Upper Fence = Q (IQR) = 28 – 1.5(10) = (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. 3-51© 2010 Pearson Prentice Hall. All rights reserved

3-52© 2010 Pearson Prentice Hall. All rights reserved

EXAMPLEObtaining 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. InstitutionRate Pulaski Bank and Trust Company6.5% Rainier Pacific Savings Bank12.0% Wells Fargo Bank NA14.4% Firstbank of Colorado14.4% Lafayette Ambassador Bank14.3% Infibank13.0% United Bank, Inc.13.3% First National Bank of The Mid-Cities13.9% Bank of Louisiana9.9% Bar Harbor Bank and Trust Company14.5% Source: First, we write the data is 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% 3-53© 2010 Pearson Prentice Hall. All rights reserved

3-54© 2010 Pearson Prentice Hall. All rights reserved

EXAMPLEConstructing 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. InstitutionRate Pulaski Bank and Trust Company6.5% Rainier Pacific Savings Bank12.0% Wells Fargo Bank NA14.4% Firstbank of Colorado14.4% Lafayette Ambassador Bank14.3% Infibank13.0% United Bank, Inc.13.3% First National Bank of The Mid-Cities13.9% Bank of Louisiana9.9% Bar Harbor Bank and Trust Company14.5% Source: © 2010 Pearson Prentice Hall. All rights reserved

Step 1: The interquartile range (IQR) is 14.4% - 12% = 2.4%. The lower and upper fences are: Lower Fence = Q 1 – 1.5(IQR) Upper Fence = Q (IQR) = 12 – 1.5(2.4) = (2.4) = 8.4% = 18.0% Step 2: [ ] * 3-56© 2010 Pearson Prentice Hall. All rights reserved

The interest rate boxplot indicates that the distribution is skewed left. 3-57© 2010 Pearson Prentice Hall. All rights reserved

3-58© 2010 Pearson Prentice Hall. All rights reserved

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. (b) determine the actual percentage of patients that have serum HDL between 34 and © 2010 Pearson Prentice Hall. All rights reserved

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3-61© 2010 Pearson Prentice Hall. All rights reserved

EXAMPLE Using the Empirical Rule The following data represent the serum HDL cholesterol of the 54 female patients of a family doctor © 2010 Pearson Prentice Hall. All rights reserved

(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 © 2010 Pearson Prentice Hall. All rights reserved

3-64© 2010 Pearson Prentice Hall. All rights reserved

EXAMPLE Using the Empirical Rule The following data represent the serum HDL cholesterol of the 54 female patients of a family doctor © 2010 Pearson Prentice Hall. All rights reserved

(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 © 2010 Pearson Prentice Hall. All rights reserved

(a) Using the formulas for mean and standard deviation (b) 3-67© 2010 Pearson Prentice Hall. All rights reserved

(e) 45 out of the 54 or 83.3% of the patients have a serum HDL between 34.0 and (c) According to the Empirical Rule, 99.7% of the patients that have serum HDL within 3 standard deviations of the mean. (d) 13.5% + 34% + 34% = 81.5% of patients will have a serum HDL between 34.0 and 69.1 according to the Empirical Rule. 3-68© 2010 Pearson Prentice Hall. All rights reserved

3-69© 2010 Pearson Prentice Hall. All rights reserved

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 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 3-70© 2010 Pearson Prentice Hall. All rights reserved

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. 3-71© 2010 Pearson Prentice Hall. All rights reserved

The kth percentile, denoted, P k, of a set of data is a value such that k percent of the observations are less than or equal to the value. 3-72© 2010 Pearson Prentice Hall. All rights reserved