2. 1.Assemble the following tools: Graphing calculator z-tables (modules 3 - 5)z-tables Paper and pencil Reference for calculator keystrokes 2.Complete the diagnostic to assess your prior knowledge or at least review the diagnostic to help you frame the content within this module. Before beginning this module

3. 1.Access solutions if you completed the diagnostic.solutions 2.If you did not score 100%, continue with this module 3.If you score 100%, you may proceed to the next module. Next Steps

4. Previously Learned Defined the mean in terms of a balance point Represented the mean graphically Recognized and interpreted sigma notation Entered data into a graphing calculator and calculated 1-variable statistics Recognized x and µ as symbolic representations of the mean

5. Outcomes Compare and contrast mean absolute deviation, variance, and standard deviation. Calculate and apply the three measures of dispersion. Understand the properties of three measures of dispersions. Interpret the mean absolute deviation and standard deviation. Determine when to use Mean Absolute Deviation and Standard Deviation

6. How do you describe a data set? Measures of Central Tendencies –Mean (μ, x) –Median –Mode Measures of Spread –Range only uses the minimum and maximum. Extremely sensitive to outliers. –Interquartile Range (IQR) Represents the middle 50%. It is relative to the median. –Mean Absolute Deviation –Variance –Standard Deviation These three new measures of variability are each relative to the mean.

7. Measures of Dispersion Measures of Central Tendency (Mean, Median, Mode) give a typical value for a data set. Measures of Dispersion(Variability or Spread) provide measures to describe the data set’s variation from the typical value. Are the data points clustered tightly around the mean or median, somewhat near the mean or median, or very spread out from the mean or median? What value is the average, which is the middle value, which occurs the most, or which is most typical?

8. Measures of Dispersion (Spread) Relative to the Mean

9. In Central Tendency Park, a student counted the number of players playing basketball each day over a two week period and gathered the following data. 10 30 50 60 70 30 80 90 20 30 40 60 20 Data Set#1 Example: Comparing Data Sets

10. In Dispersion Park, another student counted the number of players playing basketball over a two week period and gathered the following data. 50 30 40 50 40 60 50 40 30 50 30 50 60 50 Data Set #2 Example: Comparing Data Sets

11. How are the two data sets similar and how are they different? Data Set #1 Data Set #2 In this case, one measure does not tell us very much about the data sets.

12. Value Data Set #1 Data Set #2 1010 2020 3033 4023 5016 6022 7010 8010 9010 Let’s look at the frequency of the values. Compare and contrast the data sets

13. Data Set #1 0 10 20 30 40 50 60 70 80 90 100 X X X X X X X X X X X X X X Data Set #2 What do you observe? 0 10 20 30 40 50 60 70 80 90 100 X X X X X X X X X X X X X X How does the spread of the data compare between the two data sets? Where is the data relative to the mean?

14. One way to analyze a data set is to look at the deviations between each data value and the mean. We could then compare the deviations of one data set with the deviations of another comparable data set. How can we do this?

15. Deviation from the mean: Data Set #1 (sorted) -35 -25 -15 -5 +5 +15 +25 +35 +45 10 30 50 60 70 80 90 20 30 40 20 60 Deviation is the observed value minus the mean. It can be positive or negative depending on whether the observed value is greater than or less than the mean.

16. Which data set has data values further from the mean or is more spread out? If we compare the sums of the deviations of the two data sets, we would find that both have a sum of 8. While the sum of the deviations are the same, the variation of data set is quite different. A B 3 3 Each data set has a mean equal to 3.

17. Data Set #1 (sorted) What if we found the average of all the deviations? = 0 The sum of the deviations below the mean is -140. The sum of the deviations above the mean is +140. This makes sense because we know mean is the balance point. -35 -25 -15 -5 +5 +15 +25 +35 +45 10 30 50 60 70 80 90 20 30 40 20 60

18. What if we found the average of all the DISTANCES? 35 25 15 5 5 5 25 35 45 Data Set #1 (sorted) 10 30 50 60 70 80 90 20 30 40 20 60 Remember that distance is positive!

19. Compare the Mean Absolute Deviations Data Set #1 (sorted) 10 30 50 60 70 80 90 20 30 40 20 60 How do you expect the mean absolute deviation of data set #1 to compare to data set #2 ? In other words, compare the average distance of the data values from the mean for each data set. Data Set #2 (sorted) 60 30 40 50

20. Calculate the Mean Absolute Deviation for Data Set #2 Value |x i - µ| 30 15 30 15 30 15 40 5 5 5 505 5 5 5 5 5 6015 6015 Total120 60 Data Set #2 (sorted) 30 50 60 30 40 50

21. Data Set #1 Data Set #2 How does Mean Absolute Deviation help us differentiate between the data sets? 0 10 20 30 40 50 60 70 80 90 100 X X X X X X X X X X X X X X X X X X X X X X X X X X X X Having these two measures tell us more about how different or similar the data sets are. The larger the value, the more spread out the data or the more variablity exists in the data.

22. Practice with a Small Data Set The ages (years) of 10 grooms at their first marriage are given below. Determine the mean absolute deviation. 24.3, 46.6, 41.6, 32.9, 26.8, 39.8, 21.5, 45.7, 33.9, 35.1 Value|x i - µ| 24.310.52 46.611.78 41.66.78 32.91.92 26.88.02 39.84.98 21.513.32 45.710.88 33.90.92 35.10.28 Total69.4

23. Measures of Dispersion (Spread) Relative to the Mean The variance and standard deviation are the most common and useful measures of variability.

24. Variance is the average of the deviations squared. Data Set #1 (sorted) -35 -25 -15 -5 +5 +15 +25 +35 +45 10 30 50 60 70 80 90 20 30 40 20 60 To do the formula substitution, notice there is one -35, two -25’s, three -15’s, etc. players 2 By squaring the deviations, all squares will be positive. Therefore, the sum will not be zero.

25. Calculate the Variance for Data Set #2 Value (x i - µ) 2 30 225 30 225 30 225 40 25 40 25 40 25 5025 5025 5025 5025 5025 5025 60225 60225 Total1350 60 Data Set #2 (sorted) 30 50 60 30 40 50

26. Data Set #1 How does variance help us differentiate between the data sets? 0 10 20 30 40 50 60 70 80 90 100 X X X X X X X X X X X X X X Data Set #2 X 0 10 20 30 40 50 60 70 80 90 100 X X X X X X X X X X X X X Variance can be very cumbersome to use especially with large data values and large spread. Squaring the deviation can result in large values. Just look at the value for data set #1. The unit for variance is the original data’s unit squared. Not always very meaningful when trying to interpret. Again, the larger the value for variance, the more variability in the data.

27. Practice with a Small Data Set The ages (years) of 10 grooms at their first marriage are given below. Calculate the variance. 24.3, 46.6, 41.6, 32.9, 26.8, 39.8, 21.5, 45.7, 33.9, 35.1 Value(x i - µ) 2 24.3110.67 46.6138.77 41.645.97 32.93.69 26.864.32 39.824.80 21.5177.42 45.7118.37 33.90.85 35.10.09 Total684.936

28. Measures of Dispersion (Spread) Relative to the Mean

29. Standard Deviation of a Population Data Set Is just the positive square root of the variance. So for data set # 1 Mean Absolute Deviation, Variance and Standard Deviation will always be POSITIVE!!!!!!

30. Calculate the Standard Deviation for Data Set #2 Value (x i -µ) 2 30 225 30 225 30 225 40 25 40 25 40 25 5025 5025 5025 5025 5025 5025 60225 60225 Total1350 60 Data Set #2 (sorted) 30 50 60 30 40 50

31. How do you do this on the calculator? For the TI, –enter the data into a list, STAT, EDIT –Select STAT, CALC, 1-Var Stats, and then list you entered the data into. For the Casio, –Enter the data into a list, Select MENU, STAT –Set up the calculator, Select F6 (Set) and verify the 1- Var List Xlist has the specified list, Exit. –Calculate the mean and standard deviation, select 1- Var Stats. σ x is the standard deviation of the population. Calculate variance by squaring the standard deviation. Take standard deviation out to 4 decimal places when calculating the variance. This way the answers calculated by the formula will be closer to those calculated from squaring the standard deviation.

32. Data Set #2 How does Standard Deviation help us differentiate between the data sets? 0 10 20 30 40 50 60 70 80 90 100 X X X X X X X X X X X X X X Data Set #1 0 10 20 30 40 50 60 70 80 90 100 X X X X X X X X X X X X X X Just like mean absolute deviation, the more the data is spread out from the mean, the larger the value.

33. Practice with a Small Data Set The annual number of deaths from tornadoes in the United States from 1990 through 2000 is given. 53, 39, 39, 33, 69, 30, 25, 67, 130, 94, 40. Determine the standard deviation and variance.

34. How would you compare and contrast each pair of data sets? Problem 1 Data Set 1 Data Set 2 μ = 212 μ = 208 σ = 14 σ = 23 Problem 2 Data Set 1 Data Set 2 μ = 55 μ = 65 Mean.Abs.Dev. = 4.9 Mean.Abs.Dev. = 1.5

35. How would you compare the variability of the data of the two data sets? Problem 1 Data Set 1 Data Set 2 μ = 212 μ = 208 σ = 14 σ = 23 Problem 2 Data Set 1 Data Set 2 μ = 55 μ = 65 Mean.Abs.Dev. = 4.9 = 1.5 Problem 1-Answer Data Set 1 has a higher mean but it’s data is less spread out than Data Set 2 since its standard deviation is less than data set 2. Problem 2-Answer Data Set 1 has a lower mean but it’s data has more variability than Data Set 2 since its mean absolute deviation is more than Data Set 2.

36. Without calculating the standard deviation, order each figure’s standard deviation from smallest to largest. 1 2 3 4 5 6 7 8 A.A. 1 2 3 4 5 6 7 B.B. Answer: B, C, A C.C. 1 2 3 4 5 6 7 8 µ = 4.5, σ = 2.29

37. Describe the standard deviation you would expect for the situation given. You’re in charge of a carnival game, where the player will win a prize by being the first to shoot 3 ping pong balls through a hole. 1.What type of standard deviation for the accuracy of the ping pong pistols would you want to have? 2. What would the participants want? A larger standard deviation would indicate less pistol accuracy and less chance of getting the 3 ping pong balls in the hole. Thus, no prize. The participants would benefit from a small standard deviation, indicating a greater accuracy with the pistol.

38. Describe the standard deviation you would expect for the situation given. You are having laser eye surgery. Describe the relative size of the standard deviation you would want in the precision of the laser. You have been hired by a company that has employees with years of service ranging from 0 to 27 years. What type of standard deviation would you expect in the salaries of the employees? What would the data look like if the standard deviation is 0?

39. “Sample Standard Deviation” Bessel Adjustment Sample standard deviation formula subtracts one from the size. This results in dividing by a smaller number, therefore getting a larger value. Since it is a sample of the population, it is better to err with a larger standard deviation. You could see though, that as the sample size gets larger and larger, subtracting 1 has less and less effect. (example, dividing by 1000 versus 999) Teacher Note:

40. How do the 2 data sets compare? Data Set #1 Data Set #2 Screen shots from a TI calculator. (Remember on a Casio, the sample and population standard deviation are in reverse order.

41. Population vs. Sample Standard Deviation for Data Set #1 μ if data set is population Sample Standard Deviation Population Standard Deviation Throughout Algebra I and Algebra II, we will be dealing with population only!

42. By examining the formulas, which of these would be the least sensitive to outliers?

43. Mean absolute deviation is more resistant to outliers than variance or standard deviation. With variance and standard deviation, the deviation of the outlier is being squared, adding a significantly larger value to the sum. Let’s test the sensitivity of outliers, by adding a single value of 100 to data set #2. We would expect the mean to increase. How much did each of our measures of spread increase? Which had the least amount of change? Which had the most amount of change? Value |xi- µ |(x i - µ ) 2 3015225348.57 3015225348.57 3015225348.57 4052575.17 4052575.17 4052575.17 505251.77 505251.77 505251.77 505251.77 505251.77 505251.77 6015225128.37 6015225128.37 100552634.80 Total 120/175 1350 4173.33

44. Symbol Review PopulationSample Number of Entries Nn Mean Deviation Variance Standard Deviation

45. Summary The sum of the deviations,, is equal to 0. Mean absolute deviation, variance, and standard deviation are all measures of variability (dispersion or spread) relative to the mean. Mean Absolute Deviation (MAD) –It’s more resistant to outliers than variance and standard deviation. –If the data has outliers, the MAD will be a better indicator of spread. –The greater the value, the more dispersed the data will be from the mean. –It’s always positive.

46. Summary Variance –It’s the square of Standard Deviation. –Since the units of variance are units 2, this measure is used less frequently than standard deviation. –It’s always positive. Standard Deviation –It’s the square root of variance. –It’s more sensitive to outliers than Mean Absolute Deviation. –The units are the same as the data. –The greater its value, the more dispersed or spread out the data will be from the mean. –The closer it is to 0, the more clustered the data will be about the mean.

47. Revisit your diagnostic problem set that you completed prior to starting this module or take it for the first time. Assess your knowledge and understanding of the targeted outcome skills. Go to Solutions.Solutions Go to final slide for a quick survey.

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