15.1 Mean Absolute Deviation How can you determine and use the mean absolute deviation of a set of data points?

Texas Essential Knowledge and Skills The student is expected to: Measurement and data—8.11.B Determine the mean absolute deviation and use this quantity as the measure of the average distance data are from the mean using a data set of no more than 10 data points. Mathematical Processes 8.1.C Select tools, including real objects, manipulatives, paper and pencil, and technology as appropriate, and techniques including mental math, estimation, and number sense as appropriate, to solve problems.

ADDITIONAL EXAMPLE 1 A carpenter wants to hire an assistant. He is interviewing Robert and Tanya to see who is more precise at cutting lumber. Each person was told to cut 10 pieces of wood, and the lengths of their cuts in feet are given below. Whose cuts had less variability in length? Robert: 2.3, 2.4, 2.3, 2.5, 2.5, 2.8, 2.4, 2.5, 2.3, 2.5 Tanya: 2.0, 2.2, 2.1, 2.4, 2.3, 2.4, 2.6, 2.5, 2.4, 2.3 The MAD for Robert is 0.11 ft, and the MAD for Tanya is 0.14 ft, so Robert’s cuts had less variability in length.

ADDITIONAL EXAMPLE 2 A fudge factory is testing two machines. Each machine is programmed to add 1.5 ounces of frosting to each piece of fudge. The actual amounts of frosting in ounces on 10 pieces of fudge frosted by each machine are given below. Use a spreadsheet to determine which machine has less variability. Explain. Machine A: 1.489, 1.479, 1.505, 1.491, 1.499, 1.511, 1.477, 1.502, 1.491, 1.497 Machine B: 1.506, 1.506, 1.508, 1.494, 1.510, 1.502, 1.506, 1.492, 1.486, 1.515

ADDITIONAL EXAMPLE 2 The MAD for Machine A is 0.0087, and the MAD for Machine B is 0.0072, so Machine B has less variability.

15.1 LESSON QUIZ 8.11.B An engineer is testing two designs for a catapult. She wants the catapult to launch objects 150 feet. The engineer records the distances in feet for 10 trials for the two designs, shown below. Catapult A: 144, 154, 148, 147, 150, 150, 156, 151, 153, 151 Catapult B: 150, 148, 149, 151, 153, 152, 150, 144, 152, 153

Catapult A: 144, 154, 148, 147, 150, 150, 156, 151, 153, 151 Catapult B: 150, 148, 149, 151, 153, 152, 150, 144, 152, 153 1. Calculate the mean absolute deviation for Catapult A. 2.6 feet 2. Calculate the mean absolute deviation for Catapult B. 2.0 feet

Catapult A: 144, 154, 148, 147, 150, 150, 156, 151, 153, 151 Catapult B: 150, 148, 149, 151, 153, 152, 150, 144, 152, 153 3. Which catapult has the least variability? Explain. Catapult B has the smaller MAD, so Catapult B has the least variability.

A restaurant owner orders tomatoes from 2 different farms A restaurant owner orders tomatoes from 2 different farms. The weights of a random sample of 10 tomatoes from each farm are shown below in ounces. Culver Farm: 3.1, 4.3, 5.2, 4.9, 3.8, 4.1, 4.3, 3.9, 4.3 Jonesville Farm: 4.1, 5.5, 4.9, 5.4, 4.9, 5.1, 5.9, 5.2, 4.8, 5.1 4. Calculate the mean absolute deviation for tomatoes from Culver Farm. approx. 0.432 ounce

Culver Farm: 3.1, 4.3, 5.2, 4.9, 3.8, 4.1, 4.3, 3.9, 4.3 Jonesville Farm: 4.1, 5.5, 4.9, 5.4, 4.9, 5.1, 5.9, 5.2, 4.8, 5.1 5. Calculate the mean absolute deviation for tomatoes from Jonesville Farm. 0.332 ounce 6. Which tomatoes have the least variability? Explain. The tomatoes from Jonesville Farm have a lower MAD, so they have the least variability.

There are five numbers in a set: 10, 14, 15, 18, and 21 There are five numbers in a set: 10, 14, 15, 18, and 21. There are many subsets that can be made using exactly three of the numbers in the set. For example (10, 14, 21) is a three-number subset. Find all three-number subsets that have a mean absolute deviation that is a whole number.

(10, 14, 15), (10, 14, 21), and (15, 18, 21); the MAD for every three-number subset is shown below:

How can you determine and use the mean absolute deviation of a set of data points? Sample answer: The mean absolute deviation is the mean distance each data value is from the mean of the data set. The MAD can be used to quantify the spread in the data set.