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Chapter 2 Descriptive Statistics © 2012 Pearson Education, Inc. All rights reserved. Edited by Tonya Jagoe 1 of 149 The Shape of Distributions Symmetric Distribution • A vertical line can be drawn through the middle of a graph of the distribution and the resulting halves are approximately mirror images. © 2012 Pearson Education, Inc. All rights reserved. 2 of 149 The Shape of Distributions Uniform Distribution (rectangular) • All entries or classes in the distribution have equal or approximately equal frequencies. • Symmetric. © 2012 Pearson Education, Inc. All rights reserved. 3 of 149 The Shape of Distributions Skewed Left Distribution (negatively skewed) • The “tail” of the graph elongates more to the left. • The mean is to the left of the median. © 2012 Pearson Education, Inc. All rights reserved. 4 of 149 The Shape of Distributions Skewed Right Distribution (positively skewed) • The “tail” of the graph elongates more to the right. • The mean is to the right of the median. © 2012 Pearson Education, Inc. All rights reserved. 5 of 149 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. © 2012 Pearson Education, Inc. All rights reserved. 6 of 149 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: • 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. © 2012 Pearson Education, Inc. All rights reserved. 7 of 149 Interpreting Standard Deviation: Empirical Rule (68 – 95 – 99.7 Rule) 99.7% within 3 standard deviations 95% within 2 standard deviations 68% within 1 standard deviation 34% 34% 2.35% 2.35% 13.5% 13.5% x s x xs x 3s x 2s x 2s x 3s Notice: this just involves adding/subtracting the standard deviation to/from the mean, just like we did for the standard deviation distribution last week. © 2012 Pearson Education, Inc. All rights reserved. 8 of 149 Interpreting Standard Deviation: Empirical Rule (68 – 95 – 99.7 Rule) If 68% of data lies within 1 standard deviation of the mean… what % lies outside that interval? 100% 68% 68% within 1 standard deviation 34% x 3s x 2s x s 32% 34% x xs x 2s x 3s Interpreting Standard Deviation: Empirical Rule (68 – 95 – 99.7 Rule) If 32% of data lies outside 1 standard deviation of the mean… what % lies above +1 std. dev.’s? Below -1 std. dev.’s? 32% 16% 2 68% within 1 standard deviation 16% x 3s x 2s 34% x s 16% 34% x xs x 2s x 3s Interpreting Standard Deviation: Empirical Rule (68 – 95 – 99.7 Rule) If 95% of data lies within 2 standard deviations of the mean… what % lies outside that interval? 95% within 2 standard deviations 100% 95% 5% x 3s x 2s x s x xs x 2s x 3s 11 of 149 Interpreting Standard Deviation: Empirical Rule (68 – 95 – 99.7 Rule) If 5% of data lies outside the 2 standard deviation interval of the mean… what % lies above +2 std. dev.’s? Below -2 std. dev.’s? 5% 2.5% 2 95% within 2 standard deviations 2.5% x 3s x 2s 2.5% x s x xs x 2s x 3s 12 of 149 Interpreting Standard Deviation: Empirical Rule (68 – 95 – 99.7 Rule) If 99.7% of data lies within 3 standard deviations of the mean… what % lies outside that interval? 99.7% within 3 standard deviations 100% 99.7% 0. 3% x 3s x 2s x s x xs x 2s x 3s 13 of 149 Interpreting Standard Deviation: Empirical Rule (68 – 95 – 99.7 Rule) If 0.3% of data lies outside the 3 standard deviation interval of the mean… what % lies above +3 std. dev.’s? Below -3 std. dev’s? 0.3% 0.15% 2 99.7% within 3 standard deviations 0.15% 0.15% x 3s x 2s x s x xs x 2s x 3s 14 of 149 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 64.3 inches, with a sample standard deviation of 2.62 inches. Estimate the percent of the women whose heights are between 59.06 inches and 64.3 inches. © 2012 Pearson Education, Inc. All rights reserved. 15 of 149 Solution: Using the Empirical Rule • Because the distribution is bell-shaped (NORMAL), you can use the Empirical Rule. Step 2: Reread problem to identify the specific values you are seeking – circle or highlight those. Step 1: Add and subtract std. dev. to find +/-1, +/-2, & +/-3 std. dev. values. Write all values on diagram. Step 3: Shade the appropriate region. +2.62 72.16 +2.62 69.54 +2.62 66.92 -2.62 64.3 -2.62 61.68 56.44 -2.62 59.06 Step 4: Add the %’s in the shaded region. 13.5% 34% 47.5% Solution: Using the Empirical Rule 64.3 59.06 • Finally, be sure to interpret your solution in the context of the problem. 47.5% of women are between 59.06 and 64.3 inches tall. © 2012 Pearson Education, Inc. All rights reserved. 17 of 149 Solution: Using the Empirical Rule What % of data falls in each interval? 1. From 59.06 to 61.68? 13.5% 2. From 56.44 to 66.92? 3. From 61.68 to 69.54? 7. Above 66.92 and below 59.06 combined? 8. Above 69.54 and below 59.06 combined? 9. What is the difference in the percent of data that lies above 66.92 and below 61.98? 10. What is the difference in the percent of data that lies above 59.06 and above 64.3? 72.16 69.54 66.92 64.3 61.68 6. Above 61.98? 59.06 5. Below 72.16? 56.44 4. Above 64.3? Solution: Using the Empirical Rule What % of data falls in each interval? 1. From 59.06 to 61.68? 2. From 56.44 to 66.92? 83.85% 3. From 61.68 to 69.54? 7. Above 66.92 and below 59.06 combined? 8. Above 69.54 and below 59.06 combined? 9. What is the difference in the percent of data that lies above 66.92 and below 61.98? 10. What is the difference in the percent of data that lies above 59.06 and above 64.3? 72.16 69.54 66.92 64.3 61.68 6. Above 61.98? 59.06 5. Below 72.16? 56.44 4. Above 64.3? Solution: Using the Empirical Rule What % of data falls in each interval? 1. From 59.06 to 61.68? 2. From 56.44 to 66.92? 3. From 61.68 to 69.54? 81.5% 7. Above 66.92 and below 59.06 combined? 8. Above 69.54 and below 59.06 combined? 9. What is the difference in the percent of data that lies above 66.92 and below 61.98? 10. What is the difference in the percent of data that lies above 59.06 and above 64.3? 72.16 69.54 66.92 64.3 61.68 6. Above 61.98? 59.06 5. Below 72.16? 56.44 4. Above 64.3? Solution: Using the Empirical Rule What % of data falls in each interval? 1. From 59.06 to 61.68? 2. From 56.44 to 66.92? 3. From 61.68 to 69.54? 7. Above 66.92 and below 59.06 combined? 8. Above 69.54 and below 59.06 combined? 9. What is the difference in the percent of data that lies above 66.92 and below 61.98? 10. What is the difference in the percent of data that lies above 59.06 and above 64.3? 72.16 69.54 66.92 64.3 61.68 6. Above 61.98? 59.06 5. Below 72.16? 50% 56.44 4. Above 64.3? Solution: Using the Empirical Rule What % of data falls in each interval? 1. From 59.06 to 61.68? 2. From 56.44 to 66.92? 3. From 61.68 to 69.54? 7. Above 66.92 and below 59.06 combined? 8. Above 69.54 and below 59.06 combined? 9. What is the difference in the percent of data that lies above 66.92 and below 61.98? 10. What is the difference in the percent of data that lies above 59.06 and above 64.3? © 2012 Pearson Education, Inc. All rights reserved. 72.16 69.54 66.92 64.3 6. Above 61.98? 61.68 99.85% 59.06 5. Below 72.16? 56.44 4. Above 64.3? Solution: Using the Empirical Rule What % of data falls in each interval? 1. From 59.06 to 61.68? 2. From 56.44 to 66.92? 3. From 61.68 to 69.54? 7. Above 66.92 and below 59.06 combined? 8. Above 69.54 and below 59.06 combined? 9. What is the difference in the percent of data that lies above 66.92 and below 61.98? 10. What is the difference in the percent of data that lies above 59.06 and above 64.3? © 2012 Pearson Education, Inc. All rights reserved. 72.16 69.54 66.92 64.3 84% 61.68 6. Above 61.98? 59.06 5. Below 72.16? 56.44 4. Above 64.3? Solution: Using the Empirical Rule What % of data falls in each interval? 1. From 59.06 to 61.68? 2. From 56.44 to 66.92? 3. From 61.68 to 69.54? 16% 2.5% 7. Above 66.92 and below 59.06 combined? 18.5% 8. Above 69.54 and below 59.06 combined? 9. What is the difference in the percent of data that lies above 66.92 and below 61.98? 10. What is the difference in the percent of data that lies above 59.06 and above 64.3? © 2012 Pearson Education, Inc. All rights reserved. 72.16 69.54 66.92 64.3 61.68 6. Above 61.98? 59.06 5. Below 72.16? 56.44 4. Above 64.3? Solution: Using the Empirical Rule What % of data falls in each interval? 1. From 59.06 to 61.68? 2. From 56.44 to 66.92? 3. From 61.68 to 69.54? 2.5% 2.5% 7. Above 66.92 and below 59.06 combined? 8. Above 69.54 and below 59.06 combined? 5% 9. What is the difference in the percent of data that lies above 66.92 and below 61.98? 10. What is the difference in the percent of data that lies above 59.06 and above 64.3? © 2012 Pearson Education, Inc. All rights reserved. 72.16 69.54 66.92 64.3 61.68 6. Above 61.98? 59.06 5. Below 72.16? 56.44 4. Above 64.3? Solution: Using the Empirical Rule What % of data falls in each interval? 1. From 59.06 to 61.68? 2. From 56.44 to 66.92? 3. From 61.68 to 69.54? 16% 16% 7. Above 66.92 and below 59.06 combined? 8. Above 69.54 and below 59.06 combined? 9. What is the difference in the percent of data that lies above 66.92 and below 61.98? 0% 10. What is the difference in the percent of data that lies above 59.06 and above 64.3? © 2012 Pearson Education, Inc. All rights reserved. 72.16 69.54 66.92 64.3 61.68 6. Above 61.98? 59.06 5. Below 72.16? 56.44 4. Above 64.3? Solution: Using the Empirical Rule What % of data falls in each interval? 1. From 59.06 to 61.68? 50% 2. From 56.44 to 66.92? 3. From 61.68 to 69.54? 7. Above 66.92 and below 59.06 combined? 8. Above 69.54 and below 59.06 combined? 9. What is the difference in the percent of data that lies above 66.92 and below 61.98? 10. What is the difference in the percent of data that lies above 59.06 and above 64.3? 47.5% © 2012 Pearson Education, Inc. All rights reserved. 72.16 69.54 66.92 61.68 59.06 6. Above 61.98? 56.44 5. Below 72.16? 64.3 97.5% 4. Above 64.3?