Statistics: How We View the World

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Statistics How We View the World

• Day 1
• Center and Spread

2
One-Minute Question

• Find the mean of the following grades
• 70, 70, 80, 92, 98, 100

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One-Minute Question

• Arithmetic Mean average (7070809298100)/6
  
• 85

4
Review

• Find all the other Measures of Central Tendency
  that you know
• 70, 70, 80, 92, 98, 100

5
Review

• Did you name
• Median 86
• Mode 70
• 70, 70, 80, 92, 98, 100

6
Review

• Besides measures of Central Tendency, what else
  do we care about?

7
Review

• Find all the Measures of Dispersion that you
  know
• 70, 70, 80, 92, 98, 100

8
Review

• Range 30
• Mean Absolute Deviation
• 11.67
• Inner-Quartile Range (IQR)
• 28
• 70, 70, 80, 92, 98, 100

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New Concept

• Mean Absolute Deviation is rarely used, but
  Standard Deviation is used quite often
• 70, 70, 80, 92, 98, 100

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To find Standard Deviation

• Find each numbers difference
• from the mean.
• 2. Square these differences.
• Find the average of these
• squared differences.
• 4. Take the square root of your result.
• 70, 70, 80, 92, 98, 100

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Standard Deviation

• In other words
• Standard Deviation s
• s
• 70, 70, 80, 92, 98, 100

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Standard Deviation

• In other words
• Standard Deviation
• 70, 70, 80, 92, 98, 100

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Variance (Standard Deviation)2
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So, Who Cares??

• All of us who are Normal!

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So, Who Cares??

• Remember that histograms are graphs of the
  distribution of data???
• Well, think about other more general
  distributions.

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So, Who Cares??

• Suppose we roll a single die 1,000,000 times.
• What should the distribution of number of dots on
  the top face look like?
• Uniform distribution
• (Close to constant!)

Frequency
1 2 3 4 5 6
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So, Who Cares??

• What should the distribution be for a VERY, VERY
  easy test?
• It should be skewed to the left.
• (There should be lots of data to the right of the
  graph.)

Frequency
F D C B A
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So, Who Cares??

• But the heights of 20 year old males is normal.
• That means that the data forms a bell-shape that
  is symmetric about the mean.

Frequency
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Furthermore, the percentage of the area covered
by each standard deviation from the mean is shown
by this graph.
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So, suppose the mean of a normal distribution is
100 with a standard deviation of 15. What
percentage of the scores lie between 85 and 115?
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So, suppose the mean of a normal distribution is
100 with a standard deviation of 15. What
percentage of the scores lie between 70 and 115?
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In order to find how many standard deviations
away an x value is from the mean we use a
z-score.
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If the mean of a set of test grades is 84 with a
standard deviation of 6, what is the z-score for
a test grade of 96?
What percent of the other students scored lower
that a 96?
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For homework,work all the odd problems on pages
262 and 266