Measures of Dispersion

Transcription

1 Lesson 7.6 Objectives Find the variance of a set of data. Calculate standard deviation for a set of data. Read data from a normal curve. Estimate the area under a curve. Variance Measures of Dispersion The set S 5 {15, 25, 35, 45, 55} represents the scores on a standardized test. Compute the variance. Statistics also can describe how the data of a set are spread out. These values are called measures of dispersion. One measure of dispersion is range, which is the difference between the greatest value of a set and the least value of that set. Another measure of dispersion is variance. The variance of a set of numbers is defined as the mean of the squared difference between each number in the set and the mean of all numbers in the set. The formula for variance is v 5 (a 1 m)2 1 (a 2 m) (a n m) 2, n where m represents the mean, n represents the number of values in the set, and a 1, a 2,, a n represent the data set. Activity Finding Variance Use the set from the introductory paragraph of this lesson. Compute the variance. 1 How many numbers are in the set? 5 2 Find the mean of set S Complete a table for the remaining elements of S. 25, 10, 100; 35, 0, 0; 45, 10, 100; 55, 20, 400 Element of the set Difference between element and mean Square of the Difference a n a n m (a n m) ( 20) Find the mean of the data in the third column of the table What is the variance of set S? Measures of Dispersion 421

2 Standard Deviation Standard deviation, denoted (sigma), of a data set is the square root of the variance for that set. Example 1 Calculating Standard Deviation Find the standard deviation of test scores used in the activity on the previous page. Solution The variance equals 200. The standard deviation is or In statistics the mean and standard deviation are important properties of a graph called the normal curve. Normal curves accurately represent many frequency distributions drawn from large populations. For example, heights of people, test scores, and life expectancy of cars are all represented by normal curves when the data are taken from large populations. Graphs of normal curves are by far the most important types of graphs in statistics. The curve is shaped like a bell; hence, the reason for the common name, bell curve. The mean, x, corresponds to the maximum of the curve, because half the data are above this value and half are below. In a normal curve, the mean, median, and mode are all equal. In a normal curve about 68% of the data are within one standard deviation of the mean. About 95% of the data are within two standard deviations of the mean and 99.7% are within three standard deviations of the mean. Example 2 Reading a Normal Curve Use the normal curve of 10th grade test scores to answer the following questions. The standard deviation of the scores is Chapter 7 Statistics

3 a. What is the mean test score? b. If 350 students taking this test are selected at random, how many would you expect to have test scores between 72 and 80? c. How many would you expect to have test scores between 68 and 84? Solution a. The mean score is 76. b. Because 68% of the data are within one standard deviation of the mean, multiply 350 by About 238 of the selected students should have scores between 72 and 80. c. Because 95% of the data are within two standard deviations of the mean, multiply 350 by About 333 students should have scores between 68 and 84. Many statisticians use an area model to study the normal curve. Using the midpoints of each interval marked by the standard deviation, draw a rectangle that is the width of the standard deviation and the height of the curve. The area under the curve between two points on the horizontal scale represents the probability that a given value falls between those two points. Example 3 Estimating the Area Under a Curve The chart on the right shows a normal weight distribution of a given population of males, ages The standard deviation for this data is 20 lb. What is the probability that a male chosen at random in this group weighs between 150 and 180 pounds? 7.6 Measures of Dispersion 423

4 Solution Because this a normal curve, you know 68% of the males weight falls between 130 lb and 170 lb, so half of that or 34% is between 150 lb and 170 lb. P(150 # x # 170) 0.34 Because 95% of the data fall within two standard deviations, subtract the 68% to find that 27% fall between 110 lb to 130 lb and 170 lb to 190 lb. Half of that or 13.5% fall between 170 lb to 190 lb. But the shaded region does not include all of the weights from 170 lb to 190 lb, only half of them. P(170 # x # 180) So, P(150 # x # 180) An estimated 41% of males in the given population weigh between 150 lb and 180 lb. Problem Solving Using the Four-Step Plan see margin The scores on a standardized test have a normal distribution. The x 40 and the 6. What is the probability that someone chosen at random has a test score between 46 and 52? Step 1 Understand the Problem What scores are given? What do you have to find? Step 2 Develop a Plan Problem-solving strategy: Use a diagram. A general diagram of a normal distribution shows the expected probabilities (expressed as percentages) associated with each standard deviation from the mean. How can you particularize a general diagram for a normal distribution to help answer the question? Step 3 Carry Out the Plan Draw a general diagram for a normal distribution, indicating the percentages associated with each interval. Substitute the given values to locate the required interval. Step 4 Check the Results Check the accuracy of your diagram. Does the result seem reasonable? 424 Chapter 7 Statistics

5 Lesson Assessment Think and Discuss 1. Explain why variance is a measure of dispersion. 2. Which of these two curves might represent the scores for a group of seniors taking a first grade math quiz? 3. Describe a set of data that has a variance of 0. Practice and Problem Solving Calculate the variance for each set. Round to nearest hundredth , 68, 86, 60, 76, 80, 91, , 25, 19, 32, , 3.5, 1.8, 2.5, 1.1, , 7, 7, Calculate the standard deviation for each set. Round to the nearest hundredth , 205, 188, 214, , 9.5, 9.4, 8.5, 8.8, , 15.4, 18.9, 17.2, , 301, 296, 284, 274, Measures of Dispersion 425

6 12. On a history test taken by a class of 32, the mean of the scores was The standard deviation was 4.6. Assume that the class results modeled a bell curve. a. What was the approximate sum of all the scores? 1,987 b. Name the range of scores for 68% of the class to 91 c. Name the range of scores for 95% of the class to 95.6 d. How many students scored within two standard deviations of the mean? about In a plant growth experiment, the heights of 450 bean sprouts are measured ten days after planting. The data can be modeled with a normal curve, as shown below. The standard deviation is 2 cm. a. What is the average height of the bean sprouts? 10 cm b. How many sprouts have a height greater than average? about 225 c. How many sprouts have a height between 8 cm and 12 cm? about 306 d. How many sprouts have a height between 12 cm and 14 cm? about 61 Mixed Review Solve each equation x 12 2(3x 4) y 40 8y y 10 4x (3 w) 3(w 12) m 18. 4(c 1 5) 2c m 426 Chapter 7 Statistics