A Closer Look at the NEW High School Statistics

StandardsFocus on A.9 and AII.11

K-12 Mathematics InstitutesFall 2010

Fall 2010

Vertical Articulation5.16 The student will

b) describe the mean as fair share6.15 The student will

a) describe mean as balance pointAlgebra 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.

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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.

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Before we start – just a little reminder about sigma notation

and subscript notation

654321

6

1

xxxxxxxi

i

87654321i8

1i

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Mean of a Data Set Containing n Elements = µ

nxxxxx

n

xn

n

ii

...43211

x = Sample mean

µ = Population mean5

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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.

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Solve for x:

857

729791838085

x

What score will “balance” the number line ?

72 8380 91 97

85

13 5 2 6 12

7

87

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A student counted the number of players playing basketball in the Central Tendency Tournament

each day over its two week period.

Data Set#110, 30, 50, 60, 70, 30, 80,

90, 20, 30, 40, 40, 60, 20

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A student counted the number of players playing basketball in the Dispersion Tournament each day

over its two week period.

Data Set#250, 30, 40, 50, 40, 60, 50,

40, 30, 50, 30, 50, 60, 50

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How are the two data

sets similar and how are

they different?

Mean

Data Set #145

Data Set #245

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How are the two data

sets similar and how are

they different?

X Data Set #1 Data Set #2

10 1 020 2 030 3 340 2 350 1 660 2 270 1 080 1 090 1 0

Frequency

Frequency (x)

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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

0 10 20 30 40 50 60 70 80 90 100

x

x x x

x

x

x

x

x

x

x x x

Data Set #2

Line Plot

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Mean = 45

10

30 30

50

60

70

8090

20

30

40 40

20

60

Data Set#1Distance from the mean

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Mean = 45

10

30 30

50

60

70

8090

20

30

40 40

20

60

What if we find the average of the difference between each data value and the mean?

ixn

-35

-15

5

15 25

-15

3545

-25 -15

-5 -5

15

-25

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=0

ixn

-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?

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Fall 2010

Mean = 45

10

30 30

50

60

70

8090

20

30

40 40

20

60

What if we find the average of the DISTANCES from each data value to the mean?

ixn

35

15

5

15 25

15

3545

25 15

5 5

15

25

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ixn

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?

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Mean Absolute Deviation

1

n

ii

x

n

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Calculate the Mean

Absolute Deviation of Data Set #2

X | X - μ |50 530 1540 550 540 560 1550 540 530 1550 530 1550 560 1550 5Sum = 120

20

μ=45

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Mean Abs. Dev. = 57.814120

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Mean = 45

10

30 30

50

60

70

8090

20

30

40 40

20

60

What if we find the average of the squares of the difference from each data value to the mean?

n

x 2i

35

15

5

15 25

15

3545

25 15

5 5

15

25

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352+152+52+152+252+152+352+452+252+152+52+52+152+252=7550

7550 14 =539.28

6

Called the VARIANCE

n

x 2i What if we find the

average of the squares of the difference from each data value to the mean?

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Standard Deviation of a Population Data Set

2

1

n

ii

x

n

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Standard Deviation of Data Set #1

539.286 23.222

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Mean = 45

10

30 30

50

60

70

8090

20

30

40 40

20

60

One Standard Deviation on either side of the Mean

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Population vs. Sample Standard Deviation for Data Set #1

This is if the data set is the population.

Casio Texas Instruments

Population Standard DeviationSample Standard Deviation

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“Sample Standard Deviation” and Bessel Adjustment

11

2

n

xxs

n

ii

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Standard Deviation Notation Recap

µ = mean of a populationσ = population standard deviation s = sample standard deviation

(estimation of a population standard deviation based upon a sample)

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How do the 2 data sets compare?

Data Set #1 Data Set #2

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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.

ix

z-score standard measureix

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

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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).

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Whose achievement was higher on the mathematics portion of their national achievement test?

Consider this problem…

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Using z-scores to compare

Amy

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?

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Fall 2010

So what do Algebra I student need to be able to do?

A.9 DOE ESSENTIAL KNOWLEDGE AND SKILLSThe 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.

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Let’s gather some dataand calculate some statistics.

Report your heightto the nearest inch.

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Length of Boys’ Name

Summary

#letters freq1 02 13 104 715 1376 1537 898 269 9

10 211 212 013 014 0

total 500

http

://w

ww

.ssa

.gov

/OA

CT/

baby

nam

es/

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Statistics

Mean = 5.746

Population Standard Deviation = 1.3044

Sample Standard Deviation=1.3057

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Distribution

0 110

71

137

153

89

26

92 2 0 0 0

0

20

40

60

80

100

120

140

160

180

1 2 3 4 5 6 7 8 9 10 11 12 13 14

Num

ber o

f bab

ies

Number of letters

Length of most popular Boy names in 2009

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Length of most popular Boy names in 2009

0

20

40

60

80

100

120

140

160

180

1 2 3 4 5 6 7 8 9 10 11 12 13 14

Number of letters

Num

ber o

f bab

ies

50071

500137 500

153

50089

50026 500

9

5002

5002

5001 500

10

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 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

prob

abili

ty

height43

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Height as Continuous Data

0.1995

71”

0.0648

μ=68”

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Algebra II & Normal Distributions

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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.

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Examples of Normally Distributed Data

SAT scoresHeight of 10-year-old boysWeight of cereal in each

24 ounce boxTread life of tiresTime it takes to tie your shoes

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The probability density function for normally distributed data can be written as a function of the mean, standard deviation,

and data values.

2

2

2)(

22

1

x

ey

(x,y)=(data value, relative likelihood for that data value to occur)

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Area under curve – up to a data value

( ) 0.5P x

Area under curve – from a data value to ∞

1( ) ???P x x 1x

1x 2x

1 2( ) ??P x x x

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

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68-95-99.7 Rule – Empirical RuleA 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?

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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?

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Can you represent this as area under a normal curve?

12.412

Area=0.02275

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

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Standard Normal Distribution

0

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StandardNormalCurve

0

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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

59College Board State Profile Report – Virginia (college bound seniors March 2010)

Fall 2010

MappingSAT Score Standard score

512 ( ) 0512+110 ( ) 1512 –110 ( ) -1512+(2)110 ( ) 2512 – (2)110 ( ) -2

Xixi – μ

σ

2 2

z-score =

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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=

1875.22.36861

So, 1.43% of the population will be less than to 61”

Round to -2.19 for the z-table lookup.

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z-scores above the mean

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Using z-scores to compare (revisited)

Amy

Stephanie

0.978697th percentile

0.961696th percentile

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So what do Algebra II students need to be able to do?A2.11 DOE ESSENTIAL KNOWLEDGE AND SKILLSThe 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.

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Fall 2010

Resources2009 Mathematics SOL and related resources

http://www.doe.virginia.gov/testing/sol/standards_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

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