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

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

Vertical Articulation 5.16 The student will b) describe the mean as fair share 6.15 The student will a) 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. 5.16 The student will b) describe the mean as fair share 6.15 The student will a) 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. 2

Fall 2010 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. 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. 3

Fall 2010 Before we start – just a little reminder about sigma notation and subscript notation 4

Fall 2010 Mean of a Data Set Containing n Elements = µ x = Sample mean µ = Population mean 5

Fall 2010 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. 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. 6

Fall 2010 Solve for x : What score will “balance” the number line ? 7283809197 85 13 5 2 6 12 7 87 2

Fall 2010 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 8

Fall 2010 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 9

Fall 2010 How are the two data sets similar and how are they different? Mean Data Set #1 Data Set #2 10

Fall 2010 How are the two data sets similar and how are they different? X Data Set #1Data Set #2 1010 2020 3033 4023 5016 6022 7010 8010 9010 Frequency Frequency (x) 11

Fall 2010 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 x x x x x x x x x x x x x Data Set #2 Line Plot 12

Fall 2010 So what do you observe about the spread of the data? Data Set 1 is more spread out Data set 2 has data more clustered around the mean. Data Set 1 is more spread out Data set 2 has data more clustered around the mean. Can we quantify the spread of the data? 13

Fall 2010 Mean = 45 10 30 50 60 70 80 90 20 30 40 20 60 Data Set#1 Distance from the mean 14

Fall 2010 Mean = 45 10 30 50 60 70 80 90 20 30 40 20 60 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 15 -25 15

Fall 2010 =0 -35-15+5+15+25-15+35+45-25-15-5-5+15-25 14 What if we find the average of the difference between each data value and the mean? 16

Fall 2010 Mean = 45 10 30 50 60 70 80 90 20 30 40 20 60 What if we find the average of the DISTANCES from each data value to the mean? 35 15 5 25 15 35 45 25 15 5 5 25 17

Fall 2010 35+15+5+15+25+15+35+45+25+15+5+5+15+25= 14 280 14 = 20 What if we find the average of the DISTANCES from each data value to the mean? 18

Fall 2010 Mean Absolute Deviation 19

Fall 2010 Calculate the Mean Absolute Deviation of Data Set #2 X | X - μ | 50 5 30 15 40 5 50 5 40 5 60 15 50 5 40 5 30 15 50 5 30 15 50 5 60 15 50 5 Sum = 120 20 μ=45

Fall 2010 Mean Abs. Dev. = 21

Fall 2010 Mean = 45 10 30 50 60 70 80 90 20 30 40 20 60 What if we find the average of the squares of the difference from each data value to the mean? 35 15 5 25 15 35 45 25 15 5 5 25 22

Fall 2010 35 2 +15 2 +5 2 +15 2 +25 2 +15 2 +35 2 +45 2 +25 2 +15 2 +5 2 +5 2 +15 2 +25 2 =7550 7550 14 =539.286 Called the VARIANCE What if we find the average of the squares of the difference from each data value to the mean? 23

Fall 2010 Standard Deviation of a Population Data Set 24

Fall 2010 Standard Deviation of Data Set #1 25

Fall 2010 Mean = 45 10 30 50 60 70 80 90 20 30 40 20 60 One Standard Deviation on either side of the Mean 26

Fall 2010 Population vs. Sample Standard Deviation for Data Set #1 This is if the data set is the population. Casio Texas Instruments Population Standard Deviation Sample Standard Deviation 27

Fall 2010 “Sample Standard Deviation” and Bessel Adjustment 28

Fall 2010 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) µ = mean of a population σ = population standard deviation s = sample standard deviation (estimation of a population standard deviation based upon a sample) 29

Fall 2010 How do the 2 data sets compare? Data Set #1 Data Set #2 30

Fall 2010 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. -Can measure in terms of actual data distance units from the mean. -Measure in terms of standard deviation units from the mean. 31

Fall 2010 Why do that? So we can compare elements from two different data sets relative to the position within their own data set. 32

Fall 2010 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). 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

Fall 2010 Whose achievement was higher on the mathematics portion of their national achievement test? Consider this problem… 34

Fall 2010 Using z-scores to compare Amy Amy Stephanie Stephanie Amy Amy Stephanie Stephanie 35 1.89 vs. 1.77 What Does This Mean?

Fall 2010 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? 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

Fall 2010 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. 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

Fall 2010 Let’s gather some data and calculate some statistics. Report your height to the nearest inch. 38

Fall 2010 Length of Boys’ Name Summary #lettersfreq 10 21 310 471 5137 6153 789 826 99 102 112 120 130 140 total500 http://www.ssa.gov/OACT/babynames/ 39

Fall 2010 StatisticsStatistics Mean = 5.746 Population Standard Deviation = 1.3044 Sample Standard Deviation=1.3057 Mean = 5.746 Population Standard Deviation = 1.3044 Sample Standard Deviation=1.3057 40

Fall 2010DistributionDistribution 41

Fall 2010 What is the probability of selecting a name with exactly 6 letters? 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 name between 1 and 13 letters? Make up a problem: What is the probability of ________________?

Fall 2010 Let’s look at a distribution of heights for a population. μ=68” 0.1995 71” 0.0648 probability height 43

Fall 2010 Height as Continuous Data 0.1995 71” 0.0648 μ=68” 44

Fall 2010 Algebra II & Normal Distributions 45

Fall 2010 5 Characteristics of a Normal Distribution 1. The mean, median and mode are equal. 2. 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

Fall 2010 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 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

Fall 2010 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

Area under curve – from a data value to ∞

Area under curve – between two data values.

Fall 2010 68-95-99.7 Rule – Empirical Rule Do not underestimate the power of the quick sketch. 52

Fall 2010 68-95-99.7 Rule – Empirical Rule A normally distributed data set has µ=50 and σ=5. What percent of the data falls between 45 and 55? A normally distributed data set has µ=22 and σ=1.5. What would be the value of an element of this data set with z-score = 2? z-score = -2? A normally distributed data set has µ=50 and σ=5. What percent of the data falls between 45 and 55? A normally distributed data set has µ=22 and σ=1.5. What would be the value of an element of this data set with z-score = 2? z-score = -2? 53

Fall 2010 A machine fills 12 ounce Potato Chip bags. It places chips in the bags. Not all bags weigh exactly 12 ounces. The weight of the chips placed is normally distributed with a mean of 12.4 ounces and with a standard deviation 0.2 ounces. If you purchase a bag filled by this dispenser what is the likelihood it has less than 12 ounces? 54

Fall 2010 Can you represent this as area under a normal curve? 12.4 12 Area=0.02275 55

Fall 2010 What fraction of the bags have between 12.1 and 12.5 ounces? Shade the region that represents that amount. 56

Fall 2010 Standard Normal Distribution 57

Fall 2010 Standard Normal Curve 0 58

Fall 2010 Normal Distributions can be transformed into a Standard Normal Distribution using the z-score of corresponding data values. Example: 2010 SAT math scores for college bound seniors in VA Mean=512 Standard Deviation=110 59 College Board State Profile Report – Virginia (college bound seniors March 2010)

Fall 2010 MappingMapping SAT Score Standard score 512 ( ) 0 512+110 ( ) 1 512 –110 ( ) 512+(2)110 ( ) 2 512 – (2)110 ( ) -2 XiXiXiXi x i – μ σ z-score = 60

Fall 2010 61 z-scores below the mean

Fall 2010 Given the height of a population is normally distributed with a mean height = 68” with a standard deviation = 3.2”, what percent of the population is less than 61”? z-score= So, 1.43% of the population will be less than to 61” Round to -2.19 for the z-table lookup. 62

Fall 2010 z-scores above the mean 63

Fall 2010 Using z-scores to compare (revisited) Amy Amy Stephanie Stephanie Amy Amy Stephanie Stephanie 0.9786 97 th percentile 0.9616 96 th percentile 64

Fall 2010 So what do Algebra II students need to be able to do? A2.11 DOE ESSENTIAL KNOWLEDGE AND SKILLS The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to -Identify the properties of a normal probability distribution. -Describe how the standard deviation and the mean affect the graph of the normal distribution. -Compare two sets of normally distributed data using a standard normal distribution and z-scores. -Represent probability as area under the curve of a standard normal probability distribution. -Use the graphing calculator or a standard normal probability table to determine probabilities or percentiles based on z-scores. A2.11 DOE ESSENTIAL KNOWLEDGE AND SKILLS The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to -Identify the properties of a normal probability distribution. -Describe how the standard deviation and the mean affect the graph of the normal distribution. -Compare two sets of normally distributed data using a standard normal distribution and z-scores. -Represent probability as area under the curve of a standard normal probability distribution. -Use the graphing calculator or a standard normal probability table to determine probabilities or percentiles based on z-scores. 65

Fall 2010 ResourcesResources 2009 Mathematics SOL and related resources http://www.doe.virginia.gov/testing/sol/stan dards_docs/mathematics/review.shtml http://www.doe.virginia.gov/testing/sol/stan dards_docs/mathematics/review.shtml http://www.doe.virginia.gov/testing/sol/stan dards_docs/mathematics/review.shtml Instructional docs including the technical assistance documents for A.9 and AII.11 http://www.doe.virginia.gov/instruction/high _school/mathematics/index.shtml http://www.doe.virginia.gov/instruction/high _school/mathematics/index.shtml http://www.doe.virginia.gov/instruction/high _school/mathematics/index.shtml 2009 Mathematics SOL and related resources http://www.doe.virginia.gov/testing/sol/stan dards_docs/mathematics/review.shtml http://www.doe.virginia.gov/testing/sol/stan dards_docs/mathematics/review.shtml http://www.doe.virginia.gov/testing/sol/stan dards_docs/mathematics/review.shtml Instructional docs including the technical assistance documents for A.9 and AII.11 http://www.doe.virginia.gov/instruction/high _school/mathematics/index.shtml http://www.doe.virginia.gov/instruction/high _school/mathematics/index.shtml http://www.doe.virginia.gov/instruction/high _school/mathematics/index.shtml 66

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