Variation, Mean Absolute and Standard Deviation (12-3) Objective: Calculate and interpret variation in real world context through the use of mean absolute deviation, standard deviation, and variance.

Statistical Analysis Data that involve only one variable are called univariate data. This kind of data can be represented by measures of central tendency, such as the mean, median, and mode. Univariate data can also be represented by measures of variation, such as range, quartiles, and interquartile range.

Statistical Analysis TypeDescriptionWhen Best Used Range The difference between the greatest and least values. To describe which numbers are included in the data set. Quartile The values that divide the data set into four equal parts. To determine values in the upper and lower portions of a data set. Interquartile Range The range of the middle half of a data set; the difference between the upper and lower quartiles. To determine what values lie in the middle half of the data set.

Statistical Analysis The mean absolute deviation is the average of the absolute values of the differences between the mean and each value in the data set. Recall that absolute value makes all numbers positive. Mean Absolute Deviation =

Example 1 A rescue agency records the number of pets adopted each month: 14, 18, 12, 17, 15, 20. Find the mean absolute deviation.

Steps for Calculating Mean Absolute Deviation 1. Go to STAT on your calculator and choose 1:Edit. 2. Enter your data into L Go to STAT, right arrow over to CALC, and choose 1:1-Var Stats. Press ENTER. 4. Find the mean (x). For Mean Absolute Deviation, we will call this μ.

Example 1 A rescue agency records the number of pets adopted each month: 14, 18, 12, 17, 15, 20. Find the mean absolute deviation. ◦ μ = 16

Steps for Calculating Mean Absolute Deviation 5. Go to STAT and choose 1:Edit. 6. Scroll up so that L 2 is highlighted. Enter the expression |x – μ | by pressing MATH, right arrowing over to NUM, and choosing 1:abs(. Follow that with L 1 minus the number you found for μ in step 4 and close the parenthesis. Press ENTER and L 2 will automatically be filled in with data values.

Steps for Calculating Mean Absolute Deviation 7. Go to STAT, right arrow over to CALC, and choose 1:1-Var Stats. Press L 2 before you press ENTER. 8. Find the Σ x value and the n value. 9. Calculate the Mean Absolute Deviation by

Example 1 A rescue agency records the number of pets adopted each month: 14, 18, 12, 17, 15, 20. Find the mean absolute deviation. ◦ μ = 16 ◦ Σ x = 14 ◦ n = 6 ◦.◦. Mean Absolute Deviation = 2.3

Check Your Progress Choose the best answer for the following. ◦ Giles and his father record the number of catfish they catch each month out of the river behind their house: 6, 8, 10, 4, 5, 9. Find the mean absolute deviation. A.12 B.7 C.2.5 D.2 μ = 7 Σ x = 12 n = 6 MAD = 12 / 6

Standard Deviation The standard deviation is a calculated value that shows how the data deviates from the mean of the data. The standard deviation is represented by the lower-case Greek symbol sigma, σ. Standard Deviation = The variance of the data is the square of the standard deviation. Variance is represented by σ 2.

Example 2 Find the mean, variance, and standard deviation of 5, 7, 8, 14, 16. ◦ μ = 10 ◦ σ = 4.2 ◦ σ 2 = 17.64

Check Your Progress Choose the best answer for the following. ◦ Find the variance and standard deviation of 22, 24, 19, 18, 17. A.8.3; 2.9 B.6.8; 2.6 C.5.4; 2.3 D.3.4; 5.8

Standard Deviation The standard deviation illustrates the spread of the data. For example, when the mean is 75 and the standard deviation is 3, we know that almost all of the data values are very close to the mean. When the mean is 75 and the standard deviation is 15, then the data are more spread out and there may be an outlier.

Example 3 Daisy kept track of the number of text messages she sent each month for 6 months. Find the standard deviation of the data set. MonthMessages January985 February1005 March1100 April950 May1200 June1010 σ = 84.1

Check Your Progress Choose the best answer for the following. ◦ The city of Hardesty keeps track of the number of fans the local baseball team draws to the games each month. Find the standard deviation of the data set. A B C D MonthBaseball Fans January965 February805 March1120 April1006 May1225 June1310