1. 1 Lesson 5.1.2 Mean and Range

2. 2 Lesson 5.1.2 Mean and Range California Standard: Statistics, Data Analysis, and Probability 1.1 Compute the range, mean, median, and mode of data sets. What it means for you: You’ll learn how to find the mean and range for a set of data. Then you’ll see how the mode, mean, and median don’t all make sense in every situation. Key words: mean measure of central tendency sum range mode median

3. 3 Lesson 5.1.2 Mean and Range The mean (like the mode and median) is another way of finding a “typical value” of a data set. The range, on the other hand, gives a different kind of information — it tells you how spread out the data is.

4. 4 The Mean Is Also a Measure of Central Tendency Lesson 5.1.2 Mean and Range The mean is often referred to as the average. You have to do a calculation to find it: The Mean The mean of a data set is found by: (i) adding up all the items in a data set, and then (ii) dividing by the number of items in the data set.

5. 5 Example 1 Solution follows… Lesson 5.1.2 Mean and Range Find the mean of the data in the set {2, 7, 4, 8, 10, 24, 57, 39, 8}. Solution First, find the sum of the numbers in the set: Sum = 2 + 7 + 4 + 8 + 10 + 24 + 57 + 39 + 8 = 159 Next divide the sum by the number of values in the set. Mean = 159 ÷ 9 = 17.7 There are 9 values in the set. The exact answer was 17.666666... (You often have to round mean values to a sensible level of accuracy.)

6. 6 Guided Practice Solution follows… Lesson 5.1.2 Mean and Range Calculate the mean of each data set in Exercises 1–4. Round your answers to the nearest hundredth. 5. In a college course, a student’s final score is the mean of three tests. To earn a B grade, students need a final score of at least 80. A student earns 78, 91, and 73 points on the three tests. Will this student get a B? 1. {34, 67, 65, 45, 78, 35, 90} 2. {2, 3, 3, 3, 2, 4} 3. {4.2, 4.5, 4.8, 4.9, 4.5} 4. {12, 12, 24, 23, 26, 32, 14, 18, 18} 414 ÷ 7 = 59.14 Yes – the student’s mean score is: (78 + 91 + 73) ÷ 3 = 80.67 179 ÷ 9 = 19.89 22.9 ÷ 5 = 4.58 17 ÷ 6 = 2.83

7. 7 Use the Mean to Find Missing Values Lesson 5.1.2 Mean and Range On its own, the mean doesn’t tell you anything about individual values in a data set. Suppose you’ve taken five tests, but can’t find one result. If you know your mean score, you can use it to figure out the missing grade. But you can use it to find one missing value if you know the rest.

8. 8 Example 2 Solution follows… Lesson 5.1.2 Mean and Range Solution After taking five tests, Tiffany earns a grade average of 85%. She can only find four of her test papers. On these, she scored 90%, 82%, 87%, and 98%. Use the mean to find out her missing test score. You have to work backward to solve this sort of problem. You know that the mean is the total of all five test scores divided by the number of tests (5). In other words: mean = sum of all five scores ÷ 5 mean = 85 This means you can multiply 85 by 5 to find the sum of the 5 scores. Sum of all five scores = 85 × 5 = 425 Now compare this sum to the sum of the four test scores she already knows: 90 + 82 + 87 + 98 = 357 357 is 68 less than 425. So the missing test score is 68%.

9. 9 Guided Practice Solution follows… Lesson 5.1.2 Mean and Range 6. The mean of two numbers is 76. If one number is 15, what is the second number? 7. Max’s final grade is the mean of four tests. He wants to earn a final grade of 90. He got 83, 95, and 88 on the first three tests. What must he get on the fourth test to earn an average of 90? (76 × 2) – 15 = 137 83 + 95 + 88 = 266 (90 × 4) – 266 = 94

10. 10 The Range: Subtract the Least from the Greatest Lesson 5.1.2 Mean and Range The range tells you how spread out the data is. A set of data that has a big difference between the highest and lowest values will have a much bigger range than one in which the values are all fairly similar. The Range The range of a data set is the difference between the greatest and least values in the set.

11. 11 Example 3 Solution follows… Lesson 5.1.2 Mean and Range Solution Find the range of heights of these basketball players: Player 1: 195 cm; Player 2: 210 cm; Player 3: 202 cm; Player 4: 180 cm Subtract the least value from the greatest: 210 cm – 180 cm = 30 cm The range of the heights is 30 cm.

12. 12 Example 4 Solution follows… Lesson 5.1.2 Mean and Range Solution A data set contains 7 values. The range of these values is 9. Suggest a data set that meets this description. You know that the difference between the highest and lowest values is 9. Pick any number for the highest value — 11, say. Then the lowest value must be 11 – 9 = 2. But you can pick any other values for the remaining 5 items, as long as none of them is less than 2 or greater than 11. So one possible set of values is {11, 8, 3, 3, 2, 2, 4}. There are many different possibilities.

13. 13 Guided Practice Solution follows… Lesson 5.1.2 Mean and Range Calculate the range of the data sets in Exercises 8–9. 10. The range of the set of ages of people at a party is 12. Suggest possible ages for the youngest person and the oldest. 9. {1.2, 1.4, 1.1, 2.5, 3, 2.2} 8. {90, 120, 80} 120 – 80 = 40 3 – 1.1 = 1.9 For example: the youngest might be 18, and the oldest 30. Or the youngest 19, and the oldest 31, and so on.

14. 14 Mean and Range Independent Practice Solution follows… Lesson 5.1.2 Calculate the mean and range for each data set in Exercises 1–2. For Exercises 3–5, use the data set {19, 13, 7, 2, 1, 1, 25, 4}. State how many values in the set are: 3. greater than the mean. 4. less than the mean. 5. equal to the mean. 1. {5, 9, 3, 12} 2. {–5, 7, 18, –1, 6} Mean = 7.25, Range = 9 Mean = 5, Range = 23 3 5 none

15. 15 Mean and Range Independent Practice Solution follows… Lesson 5.1.2 Each set in Exercises 6–9 has a mean of 5. Find the missing values. 11. Which one data value could be removed from the data set to give it a range less than 9? {1, 2, 4, 6, 7, 7, 9, 11} 10. The range of a data set is 12. If the greatest value is 188, what is the least value? 6. Set A {1, 9, 7, 8, 2, __} 7. Set B {1, 4, 10, 5, 6, __} 8. Set C {5, 4, 6, 5, 4, __} 9. Set D {10, –2, 0, __} 3 4 6 12 176 11

16. 16 Mean and Range Lesson 5.1.2 Round Up Now you know about the range and three measures of central tendency — mean, mode, and median. You’ve also seen a measure of spread — the range. There’s more about these next Lesson.