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

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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. 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

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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

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.
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.
Mean of a Data Set Containing n Elements = µ

x = Sample mean

µ = Population mean

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.

Solve for x:

13

12

5

6

2

72

80

83

91

97

85

What score will “balance” the number line ?

2

87

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

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

Data Set #1

Data Set #2

Mean

Frequency (x)

Frequency

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

Data Set#1

Distance from the mean

90

80

70

60

60

50

40

40

30

30

30

20

20

10

Mean = 45

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

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

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

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

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

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

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

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

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)
How do the 2 data sets compare?

Data Set #1

Data Set #2

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.
Why do that?

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

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).
Consider this problem…
• Whose achievement was higher on the mathematics portion of their national achievement test?
Using z-scores to compare
• Amy
• Stephanie

1.89 vs. 1.77

What Does This Mean?

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?
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.
Length of Boys’ Name Summary

http://www.ssa.gov/OACT/babynames/

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

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?

What is the probability of selecting a name with exactly 6 letters?

Make up a problem:

What is the probability of ________________?

Let’s look at a distribution of heights for a population.

0.1995

probability

0.0648

71”

μ=68”

height

Height as Continuous Data

0.1995

0.0648

μ=68”

71”

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

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)

68-95-99.7 Rule – Empirical Rule

Do not underestimate the power of the quick sketch.

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 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?

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

Standard

Normal

Curve

0

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

College Board State Profile Report – Virginia

(college bound seniors March 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.

Using z-scores to compare (revisited)
• Amy
• Stephanie

0.9786

97th percentile

0.9616

96th percentile

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.
Resources
• 2009 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