Slide 3-1 Copyright 2008 Pearson Education, Inc. Chapter 3 Descriptive Measures Slide 3-2 Breaking News Kobe Sucks. Copyright 2008 Pearson Education, Inc. Slide 3-3 MEASURES OF CENTER MEAN, MEDIAN, MODE Copyright 2008 Pearson Education, Inc. Slide 3-4 Copyright 2008 Pearson Education, Inc. Definition 3.1 Mean of a Data Set The mean of a data set is the sum of the observations divided by the number of observations. Slide 3-5 Mean (Average): Example NBA Scoring Leaders ( ) 1. Dwayne Wade: Lebron James: Kobe Bryant: Dirk Nowitzki: Danny Granger:25.8 Copyright 2008 Pearson Education, Inc. Slide 3-6 Mean (Average): Example Continued Mean is calculated as follows: ( )/5 = Copyright 2008 Pearson Education, Inc. Slide 3-7 Copyright 2008 Pearson Education, Inc. Definition 3.2 Median of a Data Set Arrange the data in increasing order. If the number of observations is odd, then the median is the observation exactly in the middle of the ordered list. If the number of observations is even, then the median is the mean of the two middle observations in the ordered list. In both cases, if we let n denote the number of observations, then the median is at position (n + 1) / 2 in the ordered list. Slide 3-8 Median: Example Yearly Salaries (Thousands) Median: ( )/2 = 87.5 Copyright 2008 Pearson Education, Inc. Slide 3-9 Copyright 2008 Pearson Education, Inc. Definition 3.3 Mode of a Data Set Find the frequency of each value in the data set. If no value occurs more than once, then the data set has no mode. Otherwise, any value that occurs with the greatest frequency is a mode of the data set. Slide 3-10 Copyright 2008 Pearson Education, Inc. Table 3.1 Table 3.2 Data Set I Data Set II Example 3.1 Ala spent one summer cutting grass for his mommy-duck. Alas mom also hired some of his friends to help out. The tables list typical weekly earnings for the two halves of the summer. 13 worked during the first half, while 10 worked during the second half. Slide 3-11 Copyright 2008 Pearson Education, Inc. Table 3.4 Solution Example 3.1 Interpretation: The employees who worked in the first half of the summer earned more, on average (a mean salary of $483.85), than those who worked in the second half (a mean salary of $474.00). Slide 3-12 Copyright 2008 Pearson Education, Inc. Figure 3.1 This figure shows the relative positions of the mean and median for right-skewed, symmetric, and left-skewed distributions. Note that the mean is pulled in the direction of skewness, that is, in the direction of the extreme observations. For a right-skewed distribution, the mean is greater than the median; for a symmetric distribution, the mean and the median are equal; and, for a left-skewed distribution, the mean is less than the median. Slide 3-13 Examples: Measures of Center Lizzie takes 4 exams in complex analysis. Her grades are 80, 85, 90, 95. Ala takes 4 exams in abstract algebra. His grades are 0, 90, 92, math Ph.D students took oral exams in measure theory (Pass/Fail). Copyright 2008 Pearson Education, Inc. Slide 3-14 Remarks: Measures of Center Mean is sensitive to outliers (unusual values). When outliers present, median often better measure of center (Alas abstract algebra). Mode only used for qualitative data. Copyright 2008 Pearson Education, Inc. Slide 3-15 Summation Notation: Example Alas abstract algebra scores: 0, 90, 92, 94. Copyright 2008 Pearson Education, Inc. Slide 3-16 Summation Notation: Example (cont.) Copyright 2008 Pearson Education, Inc. Slide 3-17 Summation Notation: Example (cont.) To calculate the mean, let n = 4 (sample size): Copyright 2008 Pearson Education, Inc. Slide 3-18 Copyright 2008 Pearson Education, Inc. Definition 3.4 Slide 3-19 MEASURES OF VARIATION Is there a quantitative measure for determining the spread in our data? Copyright 2008 Pearson Education, Inc. Slide 3-20 Copyright 2008 Pearson Education, Inc. Figure 3.3 The data sets have the same Mean, Median, and Mode yet clearly differ! Measures of Variation or Measures of Spread Slide 3-21 Copyright 2008 Pearson Education, Inc. Definition 3.5 Range of a Data Set The range of a data set is given by the formula Range = Max Min, where Max and Min denote the maximum and minimum observations, respectively. Slide 3-22 Copyright 2008 Pearson Education, Inc. Figure 3.4 Measures of Variation or Measures of Spread: The Range Team I has range 6 inches, Team II has range 17 inches. Slide 3-23 Range: Example Scores: 0, 90, 92, 94 Range: 94 0 = 94 Remark: Range does not take into account the intermediate values. Copyright 2008 Pearson Education, Inc. Slide 3-24 Deviations: Example Weight Deviation 160 = 160 = 160 = 10 The sum of the deviations is always zero: Weight Deviation 160 = 160 = 160 = 10 The sum of the deviations is always zero: Copyright 2008 Pearson Education, Inc. Slide 3-25 So, whatcha gonna do? Copyright 2008 Pearson Education, Inc. Slide 3-26 Squared Deviation WeightDeviation Squared Deviation 160 = 160 = 160 = = 200 (sum of squares) Copyright 2008 Pearson Education, Inc. Slide 3-27 Sample Variance Copyright 2008 Pearson Education, Inc. Slide 3-28 Sample Variance (Example) Copyright 2008 Pearson Education, Inc. Slide 3-29 But arent the units squared? How can we recover the original units? Copyright 2008 Pearson Education, Inc. Slide 3-30 Sample Standard Deviation (Example) Copyright 2008 Pearson Education, Inc. Slide 3-31 Sample Standard Deviation This implies that any given data point has an average spread of 10 units from the mean. Note that the standard deviation is expressed in the same units as the original data. Copyright 2008 Pearson Education, Inc. Slide 3-32 Copyright 2008 Pearson Education, Inc. Definition 3.6 Slide 3-33 Copyright 2008 Pearson Education, Inc. Formula 3.1 Slide 3-34 Copyright 2008 Pearson Education, Inc. Table 3.10 Table 3.11 Standard Deviation: the more variation, the larger the standard deviation. Data set II has greater variation. Slide 3-35 Copyright 2008 Pearson Education, Inc. Figure 3.6 Data set II has greater variation and the visual clearly shows that it is more spread out. Data Set I Figure 3.7 Data Set II Slide 3-36 Exercise 1. Which data set has the largest standard deviation: A={1, 3, 5} or B={2, 3, 4}? 2. By hand, calculate the sample standard deviation for {1, 3, 5}. Copyright 2008 Pearson Education, Inc. Slide 3-37 Answers 1. Set A since the data points have more variation Copyright 2008 Pearson Education, Inc. Slide NBA Finals Hakeem Olajuwon Point Totals: 33, 21, 23, 20, 32, 19 Bird Point Totals: 20, 32, 25, 21, 17, 29 Olajuwon (Mean): 24.7 Bird (Mean): 24.0 Olajuwon (Standard Deviation): 5.67 Bird (Standard Deviation): 5.23 How can we interpret the means and standard deviations? Copyright 2008 Pearson Education, Inc. Slide 3-39 Solutions Olajuwon had a slightly higher points per game average (M = 24.7) than Larry Bird (M = 24.0). Larry Bird was slightly more consistent in game-to-game scoring (Standard Deviation = 5.23) than Hakeem Olajuwon (Standard Deviation = 5.67). Copyright 2008 Pearson Education, Inc. Slide 3-40 Chebychevs Theorem For ANY distribution: At least 75% of the data lie within 2 standard deviations of the mean. At least 89% of the data lie within 3 standard deviations of the mean. Copyright 2008 Pearson Education, Inc. Slide 3-41 Chebychev: Formula In general: At least lie within k standard deviations of the mean. At least lie within k = 2 standard deviations of the mean. At least lie within k = 3 standard deviations of the mean. Copyright 2008 Pearson Education, Inc. Slide 3-42 Chebychev: Example Suppose we collect data from 1,000 individuals and record the number of (long) Costco hotdogs they consume yearly. Also, suppose the mean obtained is 30, with a sample standard deviation of 4 hotdogs. What can you conclude from Chebychev? Copyright 2008 Pearson Education, Inc. Slide 3-43 Copyright 2008 Pearson Education, Inc. Slide 3-44 Chebychev: Solution We may conclude: 1. At least 75% of the participants (or 750) consume between 22 and 38 hotdogs yearly. 2. At least 89% of the participants (or 890) consume between 18 and 42 hotdogs yearly. Copyright 2008 Pearson Education, Inc. Slide 3-45 Copyright 2008 Pearson Education, Inc. Definition 3.7 Quartiles Arrange the data in increasing order and determine the median. The first quartile is the median of the part of the entire data set that lies at or below the median of the entire data set. The second quartile is the median of the entire data set. The third quartile is the median of the part of the entire data set that lies at or above the median of the entire data set. Slide 3-46 Five Number Summary: Example A={1, 2, 3, 4, 5, 6, 7} Minimum: 1 First Quartile: 2 Second Quartile: 4 Third Quartile: 6 Maximum: 7 Copyright 2008 Pearson Education, Inc. Slide 3-47 Copyright 2008 Pearson Education, Inc. Definition 3.9 Five-Number Summary The five-number summary of a data set is Min, Q 1, Q 2, Q 3, Max. Slide 3-48 Copyright 2008 Pearson Education, Inc. Definition 3.8 Interquartile Range The interquartile range, or IQR, is the difference between the first and third quartiles; that is, IQR = Q 3 Q 1. Slide 3-49 Copyright 2008 Pearson Education, Inc. Definition 3.10 Slide 3-50 Copyright 2008 Pearson Education, Inc. Procedure 3.1 Slide 3-51 Copyright 2008 Pearson Education, Inc. Definition 3.11 Slide 3-52 Copyright 2008 Pearson Education, Inc. Definition 3.12 Slide 3-53 Copyright 2008 Pearson Education, Inc. Figure 3.14 Definition 3.13 Slide 3-54 Parameter/Statistic P arameter with P opulation S tatistic with S ample Copyright 2008 Pearson Education, Inc. Slide 3-55 Parameter or Statistic? 1. 1,000 students from Virginia Tech had an average GPA of 1.3. What does 1.3 represent? 2. The average American eats 80 cheeseburgers a year. What does 80 represent? Copyright 2008 Pearson Education, Inc. Slide 3-56 Answers 1. Statistic---refers to measurement of a sample. 2. Parameter---refers to measurement of population. Copyright 2008 Pearson Education, Inc. Slide 3-57 Copyright 2008 Pearson Education, Inc. Definition 3.14 Definition 3.15 Slide 3-58 Z-Scores The standard score, or z-score, represents the number of standard deviations a given value x falls from the population mean. To find the z- score, we use the following formula: Copyright 2008 Pearson Education, Inc. Slide 3-59 Z Score (Remark) Gives us a standard way of comparing values in a given data set: (Mean = 0; standard deviation = 1) Copyright 2008 Pearson Education, Inc. Slide 3-60 Z-Score (Example 1) The mean speed of vehicles along a highway is 56 mph with standard deviation 4 mph. The speed of three cars is 62 mph, 47 mph, 56 mph. Find the z-score that corresponds to each speed. What can you conclude? Copyright 2008 Pearson Education, Inc. Slide 3-61 Z-Score (Example 1) X = 62 mph: z = (62 56)/4 = 1.5 X = 47 mph: z = (47 56)/4 = X = 56 mph: z = (56 56)/4 = 0 Copyright 2008 Pearson Education, Inc. Slide 3-62 Z-Score (Interpretation) From the z-scores, we can conclude that a speed of 62 mph is 1.5 standard deviations above the mean, a speed of 47 mph is 2.25 standard deviations below the mean, and a speed of 56 mph is equal to the mean. Copyright 2008 Pearson Education, Inc. Slide 3-63 Z-Score (Example 2) The midterms are graded for Math 241 for all 55 students. The average score is 85 with a standard deviation of 5. Suppose Lucy receives a score of 75 and Bobby receives a score of 90. What are their corresponding z-scores? Interpret the results. Copyright 2008 Pearson Education, Inc. Slide 3-64 Z-Score (Example 2) Lucy: z = (75 85)/5 = -2 Bobby: z = (90 85)/5 = 1 Thus, Lucys score lies 2 standard deviations below the mean, while Bobbys score is 1 standard deviation above the mean. Copyright 2008 Pearson Education, Inc.