A Closer Look at the NEW High School Statistics Standards Focus on A.9 and AII.11

1. A Closer Look at the NEW High School Statistics Standards Focus on A.9 and AII.11 K-12 Mathematics Institutes Fall 2010

2. Vertical Articulation • 5.16 The student will b) describe the mean as fair share • 6.15 The student willa) describe mean as balance point • Algebra I SOL A.9 The student, given a set of data, will interpret variation in real-world contexts and interpret mean absolute deviation, standard deviation, and z-scores.

3. Vertical Articulation • AFDA.7 The student will analyze the normal distribution. • Algebra II SOL A.11 The student will identify properties of the normal distribution and apply those properties to determine probabilities associated with areas under the standard normal curve.

4. Before we start – just a little reminder about sigma notation and subscript notation

5. Mean of a Data Set Containing n Elements = µ x = Sample mean µ = Population mean

6. Mean Problem • Joe has the following test grades: • 85, 80, 83, 91, 97 and 72. In order to make the academic team he needs to have an 85 average. With one test yet to take, he wants to know what score he will need on that to have an 85 average.

7. Solve for x: 13 12 5 6 2 72 80 83 91 97 85 What score will “balance” the number line ? 2 87

8. A student counted the number of players playing basketball in the Central Tendency Tournament each day over its two week period. Data Set#1 10, 30, 50, 60, 70, 30, 80, 90, 20, 30, 40, 40, 60, 20

9. A student counted the number of players playing basketball in the Dispersion Tournament each day over its two week period. Data Set#2 50, 30, 40, 50, 40, 60, 50, 40, 30, 50, 30, 50, 60, 50

10. How are the two data sets similar and how are they different? Data Set #1 Data Set #2 Mean

11. Frequency (x) How are the two data sets similar and how are they different? Frequency

12. Line Plot x x x x x x x x x x x x x x x x x x x x x x x 0 10 20 30 40 50 60 70 80 90 100 0 10 20 30 40 50 60 70 80 90 100 x x x x Data Set #1 Data Set #2

13. Data Set#1 Distance from the mean 90 80 70 60 60 50 40 40 30 30 30 20 20 10 Mean = 45

14. 90 80 70 60 60 50 40 40 30 30 30 20 20 10 What if we find the average of the difference between each data value and the mean? 45 35 25 15 15 5 Mean = 45 -5 -5 -15 -15 -25 -25 -15 -35

15. What if we find the average of the difference between each data value and the mean? -35-15+5+15+25-15+35+45-25-15-5-5+15-25 14 =0

16. 90 80 70 60 60 50 40 40 30 30 30 20 20 10 What if we find the average of the DISTANCES from each data value to the mean? 45 35 25 15 15 5 Mean = 45 5 5 15 15 25 25 15 35

17. 280 14 35+15+5+15+25+15+35+45+25+15+5+5+15+25= 14 What if we find the average of the DISTANCES from each data value to the mean? = 20

18. Mean Absolute Deviation

19. Calculate the Mean Absolute Deviation of Data Set #2 μ=45 Sum = 120

20. Mean Abs. Dev. =

21. What if we find the average of the squares of the difference from each data value to the mean? 90 80 70 60 60 50 40 40 30 30 30 20 20 10 45 35 25 15 15 5 Mean = 45 5 5 15 15 25 25 15 35

22. What if we find the average of the squares of the difference from each data value to the mean? Called the VARIANCE 7550 14 352+152+52+152+252+152+352+452+252+152+52+52+152+252=7550 = 539.286

23. Standard Deviation of a Population Data Set

24. Standard Deviation of Data Set #1

25. One Standard Deviation on either side of the Mean 90 80 70 60 60 50 40 40 30 30 30 20 20 10 Mean = 45

26. This is if the data set is the population. Population vs. Sample Standard Deviation for Data Set #1 Casio Texas Instruments Population Standard Deviation Sample Standard Deviation

27. “Sample Standard Deviation” and Bessel Adjustment

28. Standard Deviation Notation Recap • µ = mean of a population • σ= population standard deviation • s = sample standard deviation (estimation of a population standard deviation based upon a sample)

29. How do the 2 data sets compare? Data Set #1 Data Set #2

30. Describing the position of data relative to the mean. • Can measure in terms of actual data distance units from the mean. • Measure in terms of standard deviation units from the mean.

31. Why do that? So we can compare elements from two different data sets relative to the position within their own data set.

32. Consider this problem… • Amy scored a 31 on the mathematics portion of her 2009 ACT® (µ=21 σ=5.3). • Stephanie scored a 720 on the mathematics portion of her 2009 SAT® (µ=515 σ=116.0).

33. Consider this problem… • Whose achievement was higher on the mathematics portion of their national achievement test?

34. Using z-scores to compare • Amy • Stephanie 1.89 vs. 1.77 What Does This Mean?

35. By the end of Algebra I, we have asked and answered the following BIG questions…. • How do we quantify the central tendency of a data set? • How do we quantify the spread of a data set? • How do we quantify the relative position of a data value within a data set?

36. So what do Algebra I student need to be able to do? • A.9 DOE ESSENTIAL KNOWLEDGE AND SKILLS • The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to • Analyze descriptive statistics to determine the implications for the real-world situations from which the data derive. • Given data, including data in a real-world context, calculate and interpret the mean absolute deviation of a data set. • Given data, including data in a real-world context, calculate variance and standard deviation of a data set and interpret the standard deviation. • Given data, including data in a real-world context, calculate and interpret z-scores for a data set. • Explain ways in which standard deviation addresses dispersion by examining the formula for standard deviation. • Compare and contrast mean absolute deviation and standard deviation in a real-world context.

37. Let’s gather some dataand calculate some statistics.Report your heightto the nearest inch.

38. Length of Boys’ Name Summary http://www.ssa.gov/OACT/babynames/

39. Statistics • Mean = 5.746 • Population Standard • Deviation = 1.3044 • Sample Standard Deviation=1.3057

40. Distribution

41. What is the probability of selecting a name greater than or equal to 3 letters, but less than or equal to 9 letters? What is the probability of selecting a between 1 and 13 letters? What is the probability of selecting a name with exactly 6 letters? Make up a problem: What is the probability of ________________?

42. Let’s look at a distribution of heights for a population. 0.1995 probability 0.0648 71” μ=68” height

43. Height as Continuous Data 0.1995 0.0648 μ=68” 71”

44. Algebra II & Normal Distributions

45. 5 Characteristics of a Normal Distribution • The mean, median and mode are equal. • The graph of a normal distribution is • called a NORMAL CURVE. • 3. A normal curve is bell-shaped and • symmetrical about the mean. • 4. A normal curve never touches, but gets • closer and closer to the x-axis as it gets • farther from the mean. • 5. The total area under the curve is equal to • one.

46. Examples of Normally Distributed Data • SAT scores • Height of 10-year-old boys • Weight of cereal in each 24 ounce box • Tread life of tires • Time it takes to tie your shoes

47. The probability density function for normally distributed data can be written as a function of the mean, standard deviation, and data values. (x,y)=(data value, relative likelihood for that data value to occur)

48. Area under curve – up to a data value

49. Area under curve – from a data value to ∞

50. Area under curve – between two data values.