Intro to LSP 121

1
LSP 121

• Intro to LSP 121
• Normal Distributions

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Welcome to LSP 121

• Quantitative Reasoning and Technological Literacy
  II
• Continuation of concepts from LSP 120
• Topics we feel you will need to make it through
  college and into a career
• Normal distributions
• Descriptive statistics and correlation
• Probability and risk
• Databases
• Algorithms
• If you feel you know this material, take the test
• See Syllabus under Prerequisites

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What is a Normal Distribution?

• Very common, very special type of distribution
• Most data values are clustered near the mean (a
  single peak)
• Distribution is symmetric
• Tapering tales as you move away from the mean
• Looks like a bell curve

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The 68-95-99.7 Rule

• About 68 (68.3), or just over 2/3, of the data
  points fall within 1 standard deviation ( or -)
  of the mean
• About 95 (95.4) of the data points fall within
  2 standard deviations of the mean
• About 99.7 of the data points fall within 3
  standard deviations of the mean

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Pop-Quiz How many percent lie between mean -1
standard deviation and mean 1 standard
deviation? 68 How many percent lie between
mean 1 stdev and mean 3 stdev? 15.85 Ho
w many percent lie greater than mean 3 stdev?
0.15
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Example

• In the real world, SAT exams typically produce
  normal distributions with a mean of 500 and a
  standard deviation of 100.
• Thus, 68 of the students score between 400 and
  600
• 95 of the students score between 300 and 700
• 99.7 score between 200 and 800
• What if someone scored 720 on the SAT? What
  percentage of students scored less than or equal
  to 720?
• Use Excels NORMDIST function
• In a cell type NORMDIST(X, mean, stdev, true)
• For our problem NORMDIST(720, 500, 100, TRUE)
• Answer 0.986097, or 98.6097
• What percentage scored greater than 720?

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Another Example

• A survey finds that prices paid for two-year-old
  Ford Explorers are normally distributed with a
  mean of 16,500 and a standard deviation of 500.
  Consider a sample of 10,000 people who bought
  two-year-old Ford Explorers.
• How many people paid between 16,000 and 17,000?
• NORMDIST(16000,16500,500,true) yields 0.158655
• NORMDIST(17000, 16500, 500, true) yields
  0.841345
• Subtract 0.841345 0.158655 yields 0.682689
• Or use the graph two slides back

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Another Example

• How many paid less than 16,000?
• NORMDIST(16000, 16500, 500, true) yields
  0.158655, or 15.8655
• Or use the graph
• What is another way of saying What percentage of
  values are less than or equal to some value X?
  (see next slide)

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Percentiles

• The nth percentile of a data set is the smallest
  value in the set with the property that n of the
  data values are less than or equal to it.
• In a normal distribution, a z score of 0 is the
  mean. At the mean, 50 (or 0.50) of all the
  values are less than or equal to the mean. The
  mean is the 50th percentile.

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Example

• Cholesterol levels in men 18 to 24 years of age
  are normally distributed with a mean of 178 and a
  standard deviation of 41.
• In what percentile is a man with a cholesterol
  level of 190?
• Using Excels normdist function
• normdist(190,178,41,true) returns 0.61, or 61st
  percentile

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Standard Scores

• The standard score is the number of standard
  deviations a value lies above or below the mean.
  
• aka Standard score, z-score, z
• The standard score of the mean is z0
• Recall that mean is a better word for average
• Example The standard score of a data value 1.5
  standard deviations above the mean is z1.5
• Example What is the standard score for a student
  who scores 300 on an exam with a mean of 400,
  standard deviation of 100?
• This student scored exactly 1 SD below the mean,
  so z -1

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Standard Scores

• The standard score of a data value 2.4 standard
  deviations below the mean is z -2.4
• In general
• z (data value mean) / standard deviation
• the data value is typically called x

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Example

• The Stanford-Binet IQ test is designed so that
  scores are normally distributed with a mean of
  100 and a standard deviation of 16. What are the
  z-scores for IQ scores of 95 and 125?
• z (95 - 100) / 16 -0.31
• z (125 - 100) / 16 1.56 Thus, an IQ score of
  125 lies 1.56 standard deviations above the mean.

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Inverse Normal Distribution Function

• What if you know the mean, standard deviation,
  and percentile, and want to know the actual value
  (X)?
• Recall z (x mean) / standard deviation
• You can also use Excels NORMINV
• Know how to use BOTH. On an exam, youll use the
  formula.
• Example If a set of exam scores has a mean of
  76, a standard deviation of 12, and one score is
  at the 86th percentile, what was the students
  exact numeric score?
• Answer x 88.9