Chapter 2 Descriptive Statistics 1 Larson/Farber 4th ed.

Chapter Outline 2.1 Frequency Distributions and Their Graphs 2.2 More Graphs and Displays 2.3 Measures of Central Tendency 2.4 Measures of Variation 2.5 Measures of Position 2 Larson/Farber 4th ed.

Section 2.4 Measures of Variation Larson/Farber 4th ed. 3

Section 2.4 Objectives Determine the range of a data set Determine the variance and standard deviation of a population and of a sample Use the Empirical Rule and Chebychev’s Theorem to interpret standard deviation Approximate the sample standard deviation for grouped data Larson/Farber 4th ed. 4

Range The difference between the maximum and minimum data entries in the set. The data must be quantitative. Range = (Max. data entry) – (Min. data entry) Larson/Farber 4th ed. 5

Example: Finding the Range A corporation hired 10 graduates. The starting salaries for each graduate are shown. Find the range of the starting salaries. Starting salaries (1000s of dollars) Larson/Farber 4th ed. 6

Solution: Finding the Range Ordering the data helps to find the least and greatest salaries Range = (Max. salary) – (Min. salary) = 47 – 37 = 10 The range of starting salaries is 10 or $10,000. Larson/Farber 4th ed. 7 minimum maximum

Deviation, Variance, and Standard Deviation Deviation The difference between the data entry, x, and the mean of the data set. Population data set:  Deviation of x = x – μ Sample data set:  Deviation of x = x – x Larson/Farber 4th ed. 8

Example: Finding the Deviation A corporation hired 10 graduates. The starting salaries for each graduate are shown. Find the deviation of the starting salaries. Starting salaries (1000s of dollars) Larson/Farber 4th ed. 9 Solution: First determine the mean starting salary.

Solution: Finding the Deviation Larson/Farber 4th ed. 10 Determine the deviation for each data entry. Salary ($1000s), xDeviation: x – μ 4141 – 41.5 = – – 41.5 = – – 41.5 = – – 41.5 = – 41.5 = – 41.5 = – – 41.5 = – 41.5 = – – 41.5 = – – 41.5 = 0.5 Σx = 415 Σ(x – μ) = 0

Deviation, Variance, and Standard Deviation Population Variance Population Standard Deviation Larson/Farber 4th ed. 11 Sum of squares, SS x

Finding the Population Variance & Standard Deviation In Words In Symbols Larson/Farber 4th ed Find the mean of the population data set. 2.Find deviation of each entry. 3.Square each deviation. 4.Add to get the sum of squares. x – μ (x – μ) 2 SS x = Σ(x – μ) 2

Finding the Population Variance & Standard Deviation Larson/Farber 4th ed Divide by N to get the population variance. 6.Find the square root to get the population standard deviation. In Words In Symbols

Example: Finding the Population Standard Deviation A corporation hired 10 graduates. The starting salaries for each graduate are shown. Find the population variance and standard deviation of the starting salaries. Starting salaries (1000s of dollars) Recall μ = Larson/Farber 4th ed. 14

Solution: Finding the Population Standard Deviation Larson/Farber 4th ed. 15 Determine SS x N = 10 Salary, xDeviation: x – μSquares: (x – μ) – 41.5 = –0.5(–0.5) 2 = – 41.5 = –3.5(–3.5) 2 = – 41.5 = –2.5(–2.5) 2 = – 41.5 = 3.5(3.5) 2 = – 41.5 = 5.5(5.5) 2 = – 41.5 = –0.5(–0.5) 2 = – 41.5 = 2.5(2.5) 2 = – 41.5 = –0.5(–0.5) 2 = – 41.5 = –4.5(–4.5) 2 = – 41.5 = 0.5(0.5) 2 = 0.25 Σ(x – μ) = 0 SS x = 88.5

Solution: Finding the Population Standard Deviation Larson/Farber 4th ed. 16 Population Variance Population Standard Deviation The population standard deviation is about 3.0, or $3000.

Deviation, Variance, and Standard Deviation Sample Variance Sample Standard Deviation Larson/Farber 4th ed. 17

Finding the Sample Variance & Standard Deviation In Words In Symbols Larson/Farber 4th ed Find the mean of the sample data set. 2.Find deviation of each entry. 3.Square each deviation. 4.Add to get the sum of squares.

Finding the Sample Variance & Standard Deviation Larson/Farber 4th ed Divide by n – 1 to get the sample variance. 6.Find the square root to get the sample standard deviation. In Words In Symbols

Example: Finding the Sample Standard Deviation The starting salaries are for the Chicago branches of a corporation. The corporation has several other branches, and you plan to use the starting salaries of the Chicago branches to estimate the starting salaries for the larger population. Find the sample standard deviation of the starting salaries. Starting salaries (1000s of dollars) Larson/Farber 4th ed. 20

Solution: Finding the Sample Standard Deviation Larson/Farber 4th ed. 21 Determine SS x n = 10 Salary, xDeviation: x – μSquares: (x – μ) – 41.5 = –0.5(–0.5) 2 = – 41.5 = –3.5(–3.5) 2 = – 41.5 = –2.5(–2.5) 2 = – 41.5 = 3.5(3.5) 2 = – 41.5 = 5.5(5.5) 2 = – 41.5 = –0.5(–0.5) 2 = – 41.5 = 2.5(2.5) 2 = – 41.5 = –0.5(–0.5) 2 = – 41.5 = –4.5(–4.5) 2 = – 41.5 = 0.5(0.5) 2 = 0.25 Σ(x – μ) = 0 SS x = 88.5

Solution: Finding the Sample Standard Deviation Larson/Farber 4th ed. 22 Sample Variance Sample Standard Deviation The sample standard deviation is about 3.1, or $3100.

Example: Using Technology to Find the Standard Deviation Sample office rental rates (in dollars per square foot per year) for Miami’s central business district are shown in the table. Use a calculator or a computer to find the mean rental rate and the sample standard deviation. (Adapted from: Cushman & Wakefield Inc.) Larson/Farber 4th ed. 23 Office Rental Rates

Solution: Using Technology to Find the Standard Deviation Larson/Farber 4th ed. 24 Sample Mean Sample Standard Deviation

Interpreting Standard Deviation Standard deviation is a measure of the typical amount an entry deviates from the mean. The more the entries are spread out, the greater the standard deviation. Larson/Farber 4th ed. 25

Interpreting Standard Deviation: Empirical Rule (68 – 95 – 99.7 Rule) For data with a (symmetric) bell-shaped distribution, the standard deviation has the following characteristics: Larson/Farber 4th ed. 26 About 68% of the data lie within one standard deviation of the mean. About 95% of the data lie within two standard deviations of the mean. About 99.7% of the data lie within three standard deviations of the mean.

Interpreting Standard Deviation: Empirical Rule (68 – 95 – 99.7 Rule) Larson/Farber 4th ed % within 1 standard deviation 34% 99.7% within 3 standard deviations 2.35% 95% within 2 standard deviations 13.5%

Example: Using the Empirical Rule In a survey conducted by the National Center for Health Statistics, the sample mean height of women in the United States (ages 20-29) was 64 inches, with a sample standard deviation of 2.71 inches. Estimate the percent of the women whose heights are between 64 inches and inches. Larson/Farber 4th ed. 28

Solution: Using the Empirical Rule Larson/Farber 4th ed % 13.5% Because the distribution is bell-shaped, you can use the Empirical Rule. 34% % = 47.5% of women are between 64 and inches tall.

Chebychev’s Theorem The portion of any data set lying within k standard deviations (k > 1) of the mean is at least: Larson/Farber 4th ed. 30 k = 2: In any data set, at least of the data lie within 2 standard deviations of the mean. k = 3: In any data set, at least of the data lie within 3 standard deviations of the mean.

Example: 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? Larson/Farber 4th ed. 31

Solution: Using Chebychev’s Theorem k = 2: μ – 2σ = 39.2 – 2(24.8) = (use 0 since age can’t be negative) μ + 2σ = (24.8) = 88.8 Larson/Farber 4th ed. 32 At least 75% of the population of Florida is between 0 and 88.8 years old.

Standard Deviation for Grouped Data Sample standard deviation for a frequency distribution When a frequency distribution has classes, estimate the sample mean and standard deviation by using the midpoint of each class. Larson/Farber 4th ed. 33 where n= Σf (the number of entries in the data set)

Example: Finding the Standard Deviation for Grouped Data Larson/Farber 4th ed. 34 You collect a random sample of the number of children per household in a region. Find the sample mean and the sample standard deviation of the data set. Number of Children in 50 Households

xfxf 0100(10) = (19) = (7) = (7) =21 424(2) = 8 515(1) = 5 646(4) = 24 Solution: Finding the Standard Deviation for Grouped Data First construct a frequency distribution. Find the mean of the frequency distribution. Larson/Farber 4th ed. 35 Σf = 50 Σ(xf )= 91 The sample mean is about 1.8 children.

Solution: Finding the Standard Deviation for Grouped Data Determine the sum of squares. Larson/Farber 4th ed. 36 xf 0100 – 1.8 = –1.8(–1.8) 2 = (10) = – 1.8 = –0.8(–0.8) 2 = (19) = – 1.8 = 0.2(0.2) 2 = (7) = – 1.8 = 1.2(1.2) 2 = (7) = – 1.8 = 2.2(2.2) 2 = (2) = – 1.8 = 3.2(3.2) 2 = (1) = – 1.8 = 4.2(4.2) 2 = (4) = 70.56

Solution: Finding the Standard Deviation for Grouped Data Find the sample standard deviation. Larson/Farber 4th ed. 37 The standard deviation is about 1.7 children.

Section 2.4 Summary Determined the range of a data set Determined the variance and standard deviation of a population and of a sample Used the Empirical Rule and Chebychev’s Theorem to interpret standard deviation Approximated the sample standard deviation for grouped data Larson/Farber 4th ed. 38