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Main Ideas / Questions: Algebra 1 Name: ________________________________ Notes Period: _______Date: _________________ TOPIC Mean Absolute Deviation and Standard Deviation Sample Population Use this column to Examples: write down a) You want to know the average income of families in Loudoun County. What would be the: any extra Sample group? _____________________________________________________________ thoughts you have while Population? ________________________________________________________________ watching the b) You need to know the average height of 5 year olds. What would be the: video. Sample group? _____________________________________________________________ Population? ________________________________________________________________ It is many times impossible to know for sure information on the population, but we use the information on the sample to predict what the same information would be for the entire population Deviation of a data point Subtract data point value minus mean value. Dispersion of a data set We will discuss TWO ways to measure dispersion of a data set: – – 1) MEAN ABSOLUTE DEVIATION (MAD) Say you have the sample data set: 2, 8, 3, 4, 6, 1 a) What is the mean ? _______________ b) Recall that deviation is _____________________________________________________ Subtract each data point value minus the mean value. Distances are always ____________. * Remember: Data point, x 2 8 3 4 6 1 Deviation x Absolute Deviation x c) What would be a good way to summarize the numbers you got for all the absolute deviations? Find the mean of all the absolute deviations. So, the Mean Absolute Deviation for this data set is: _______________ MAD = _______________. NOW THINK: This MAD answer says that on average, the points in that data set should be 2 numbers away from the mean (i.e., 2 numbers higher or 2 numbers lower than the mean). What would you think about a data point in that set that is twice the MAD away from the mean? (i.e., 4 numbers higher or lower than the mean) That number is a little unusual in this group of points. What would you think about a data point in that set that is three times the MAD away from the mean? (i.e., 6 numbers higher or lower than the mean) That number is definitely very unusual in this group of points. An outlier, maybe? How can you use the Mean Absolute Deviation of a sample data set to describe the dispersion of that sample data set? The smaller the MAD, the less disperse a data set is away from the mean. The larger the MAD, the more disperse a data set is away from the mean. You can predict that dispersion of the population will be about the same as the dispersion you got for the sample group. Example 1: Calculate the MAD of each data set. Then, describe the dispersion of the data sets. Height of students in Group ‘A’ (inches) Height of students in Group ‘B’ (inches) 65, 63, 61, 62, 64, 63 55, 71, 67, 59, 69, 57 Calculate MAD for Group ‘A’ Data point, x Deviation x Group ‘A’ MAD = Calculate MAD for Group ‘B’ Absolute Deviation x Data point, x Deviation x Absolute Deviation x Group ‘B’ MAD = Describe dispersions: Data for Group ‘A” is not very disperse. Data for Group ‘B’ is much more disperse. 2) STANDARD DEVIATION ( ) This is another way to summarize the deviation from the mean of all the points in a data set. It’s the same as for MAD, except instead of using absolute value to make all the deviations positive, this method now squares the deviations to make them all positive! Let’s find the standard deviation, , of the same data set we used to learn about MAD: 2, 8, 3, 4, 6, 1 a) What is the mean ? _______________ b) Now calculate the deviation from the mean of each point but this time square those results in order to make them positive: Data point, x 2 8 3 4 6 1 Why have both MAD and ? Square of Deviation Deviation x x 2 c) Summarize: find the mean of all the square of the deviations _______________ The mean of all the square of the deviations is called the ___________________. d) Remember you squared all the deviations? Our last step must be to “undo” all that squaring. What is the inverse operation of squaring a number? Find its square root. = So, the Standard Deviation, , for this data set is: = _______________ NOW THINK: This standard deviation, , says that the points in that data set are expected to be 2.4 numbers away from the mean (i.e., 2.4 numbers higher or 2.4 numbers lower than the mean). What would you think about a data point in that set that is twice the away from the mean? (i.e., 4.8 numbers higher or lower than the mean) That number is a little unusual in this group of points. What would you think about a data point in that set that is three times the away from the mean? (i.e., 7.2 numbers higher or lower than the mean) That number is definitely very unusual in this group of points. An outlier, maybe? How can you use the standard deviation of a sample data set to describe the dispersion of that sample data set? The smaller the, the less disperse a data set is away from the mean. The larger the , the more disperse a data set is away from the mean. How can you use the standard deviation, , of a sample data set to describe the dispersion of the population? You can predict that the dispersion of the population will be abou Below are the calculator steps for Finding Standard Deviation. If you have a graphing calculator, feel free to practice this. IF not, don’t worry! We will use the calculators in class to practice!! Calculator Steps for Standard Deviation Fortunately, your calculator can find the standard deviation of a data set for you. Follow the same calculator steps you did last class to find the quartiles for the box-and-whisker plot. The list of answers you get also includes the standard deviation. REVIEW: Step 1: Press STAT , 1 Step 2: Enter your data set as a list under L1. (Data points do not have to be in order) Step 3: Press STAT , (scroll to CALC), 1 Step 3: Press ENTER . Your stats results on your screen as shown below. The mean, is the first number The standard deviation is here 10.2222222 2 92 1080 4.17665469 3.93778781 5 9 Example 2: Find the mean and standard deviation of the following set of test grades (use calculator): 95, 76, 85, 92, 78, 84, 71, 80, 65 a) mean, _____________ _____________ b) What test grade would be one standard deviation above the mean? ________ c) What test grade would be one standard deviation below the mean? ________ d) What test grades would be two standard deviations away from the mean? MAD predicts population dispersion better when data has outliers or is skewed data better for symmetrically distributed data that does not have outliers. Summary: What did you learn??