Applying the Normal Distribution: ZScores

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Applying the Normal Distribution Z-Scores

• Chapter 3.5 Tools for Analyzing Data
• Mathematics of Data Management (Nelson)
• MDM 4U
• Author Gary Greer (with K. Myers)

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Comparing Data

• Consider the following two students
• Student 1
• MDM 4U, Mr. Greer, Semester 1
• Mark 84,
• Student 2
• MDM 4U, Mr. Greer, Semester 2
• Mark 83,
• Can we fairly compare the two students when the
  mark distributions are different?

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Mark Distributions for Each Class
Semester 1
Semester 2
74
58
50
82
90
98
66
99.4
89.6
79.8
70
60.2
50.4
40.6
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Comparing Distributions

• it is difficult to compare two distributions when
  they have different characteristics
• for example, the two histograms have different
  means and standard deviations
• z-scores allow us to make the comparison

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The Standard Normal Distribution

• a distribution with a mean of zero and a standard
  deviation of one XN(0,1²)
• each element of any normal distribution can be
  translated to the same place on a standard normal
  distribution using the z-score of the element
• the z-score is the number of standard deviations
  the piece of data is below or above the mean
• if the z-score is positive, the data lies above
  the mean, if negative, below

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Standardizing

• the process of reducing the normal distribution
  to a standard normal distribution N(0,1) is
  called standardizing
• remember that a standardized normal distribution
  has a mean of 0 and a standard deviation of 1

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Example

• for the distribution XN(10,2²) determine the
  number of standard deviations each value lies
  above or below the mean
• a. x 7
• z 7 10
• 2
• z -1.5
• 7 is 1.5 standard deviations below the mean
• 18.5 is 4.25 standard deviations above the mean

• b. x 18.5
• z 18.5 10
• 2
• z4.25

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Example continued
99.7
95
34
34
13.5
13.5
2.35
2.35
10
12
14
8
6
16
7
18.5
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Standard Deviation

• a recent math quiz offered the following data
• the z-scores offers a way to compare scores among
  members of the class, find out how many had a
  mark greater than yours, indicate position in the
  class, etc.
• mean 68.0
• standard deviation 10.9

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Example 2

• compare your mark to the rest of the class
• suppose your mark was 64
• z (64 68.0)/10.9 -0.367
• (using the z-score table on page 398)
• we get 0.359 or 35.9
• so 35.9 of the class has a mark less than or
  equal to yours

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Example 3 Percentiles

• the kth percentile is the data value that is
  greater than k of the population
• if another student has a mark of 75, what
  percentile is this student in?
• z (75 -68)/10.9 0.642
• from the table on page 398 we get 0.739, so the
  student is in the 74th percentile

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Example 4 Ranges

• now find the percent of data between a mark of 60
  and 80
• for 60
• z (60 68)/10.9 -0.733 gives 23.2
• for 80
• z (80 68)/10.9 1.10 gives 86.4
• 86.4 - 23.2 63.2
• so 63.2 of the class is between a mark of 60 and
  80

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Back to the two students...

• Student 1

• Student 2

• Student 2 has the lower mark, but a higher
  z-score!

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Exercises and Assignment

• read through the examples on pages 180-185
• try page 186 2-5, 7, 8, 10

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Mathematical Indices

• Chapter 3.6 Tools for Analyzing Data
• Mathematics of Data Management (Nelson)
• MDM 4U
• Author Gary Greer (ideas from K. Myers)

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What is an Index?

• an index is an arbitrarily defined number that
  provides a measure of scale
• Body Mass Index (BMI) is an example
• these are used to indicate a value, but do not
  actually represent some actual measurement or
  quantity so that we can make comparisons

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Exercises

• read p189-192
• describe
• Body Mass Index
• Slugging Percentage
• Moving Averages
• try page 193 1 - 4, 8, 9

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References

• Halls, S. (2004). Body Mass Index Calculator.
  Retrieved October 12, 2004 from
  http//www.halls.md/body-mass-index/av.htm
• Wikipedia (2004). Online Encyclopedia. Retrieved
  September 1, 2004 from http//en.wikipedia.org/wik
  i/Main_Page