THE STANDARD DEVIATION AS A RULER AND THE NORMAL MODEL

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THE STANDARD DEVIATION AS A RULER AND THE
NORMAL MODEL
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The womens heptathlon in the Olympics consists
of seven track and field events the 200-m the
800-m runs, 100-m hurdles, shot put, javelin,
high jump, and long jump. Somehow, the
performances have to be combined into one score.

• How can performances
• in such different events
• be compared?

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They dont even have the same units the races
are recorded in seconds and the throwing and
jumping events, in meters. In the 2000 Olympics,
the best 800-m time, run by Getrud Bacher of
Italy, was 8 seconds faster than the mean. The
winning long jump by the Russian Yelena
Prokhorova was 60 cm longer than the mean.
Which performance deserves more points?

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• The trick in comparing very different looking
  values
• is to use the standard deviation.
• How far is the value from the mean?
• How different are these two statistics?
• The standard deviation tells us how the
• collection of values varies, so its a natural
• ruler for comparing an individual to the
• group.

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• Bachers winning 800-m time of 129 seconds was 8
  seconds faster than the mean of 137 seconds. How
  many standard deviations better than the mean is
  that?
• The standard deviation of all 27 qualifying times
  was 5 seconds.
• In other words, and x 129
• or 1.6 standard deviations better than
  the mean.
• Calculate Prokhorovas performance in terms of
  standard deviations better than the mean if her
  winning long jump was 660 cm and the average long
  jump was 6-m with a standard deviation of 30 cm.
• Was Bachers winning 800-m sprint better or
  Prokhorovas winning long jump?

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STANDARDIZING WITH Z-SCORES

• To understand how an athlete performed in a
  heptathlon event, we standardized her result,
  finding out how many standard deviations from the
  event mean she performed.
• We call the results standardized values or
  z-scores

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• Standardized values have NO UNITS because
  z-scores measure the distance of each data value
  from the mean in standard deviations. A z-score
  of 2 tells us that a data value is 2 standard
  deviations above the mean. It doesnt matter
  whether the original variable was measured in
  inches, dollars, or seconds.
• What does having a z-score of -3 mean?
• Regardless of what direction, the farther the
  data value is from the mean, the more unusual it
  is.

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• Now we can compare values that are measured on
  different variables, with different scales, with
  different units, or for different populations.
• What method did we learn about in the last
  chapter that enabled us to change units or scales?

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ANSWER THIS

• Your statistics teacher has announced that the
  lower of your two tests will be dropped. You got
  a 90 on test 1 and an 80 on test 2.
• Youre all set to drop the 80 until she announces
  that she grades on a curve. She standardized
  the scores in order to decide which is the lower
  one.
• If the mean of the first test is 88 with a
  standard deviation of 4 and the mean of the
  second was a 75 with a standard deviation of 5,
  which one will be dropped and is this fair?

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LINEAR TRANSFORMATIONS

• SHIFTING DATA
• Adding or subtracting a constant to every data
  value adds (or subtracts) the same constant to
  measures of position, but leaves measures of
  spread unchanged.
• RESCALING
• When we multiply (or divide) all the data values
  by any constant, all measures of position (mean,
  median, percentiles), and measures of spread
  (range, IQR, std dev) are multiplied (or divided)
  by that same constant.

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IMPORTANT

• Standardizing data into z-scores is just shifting
• them by the mean and then rescaling them by the
• standard deviation.
• When we shift the data by subtracting the mean
  from every data value, we are shifting the mean
  to what value?
• Does this change the standard deviation?
• When we divide each of these shifted values by s,
  the standard deviation should be divided as well.
  Since the standard deviation was s to start
  with, the new standard deviation becomes what
  value?

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HOW DOES STANDARDIZING AFFECT THE DISTRIBUTION OF
A VARIABLE?

• Standardizing into z-scores
• does not change the shape of the distribution of
  a variable
• changes the center by making the mean 0
• changes the spread by making the standard
  deviation 1

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EXAMPLE 1

• Many colleges and universities require applicants
  to submit scores on standardized tests. The
  college you want to apply to says that while
  there is no minimum score required, the middle
  50 of their students have combined SAT scores
  between 1530 and 1850. Youd feel confident if
  you knew her score was in their top 25, but
  unfortunately, you took the ACT test. How high
  does your ACT need to be to make it into the top
  quarter of equivalent SAT scores?
• For college bound seniors, the average SAT score
  is about 1500 and the standard deviation is about
  250 points. For the same group, the ACT average
  is 20.8 with a standard deviation of 4.8.

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Quantitative Variable Condition

• scores for both tests are quantitative but have
  no meaningful units other than points
• If the middle 50 scores between 1530 and 1850,
  if you want to be in the top quarter, what score
  would you have to have?

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Calculate the corresponding z-score

• The SAT score of 1850 is 1.4 standard deviations
  above the mean of all test takers. For the ACT,
  1.40 standard deviations above the mean is
• 20.8 1.4(4.8) 27.52
• So to be in the top quarter of applicants, you
  need to have an ACT score of at least 27.52.

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• How far does a z-score have to be from zero in
  order to indicate that it is surprising or
  unusual?
• We need to MODEL the datas distribution. A
  model will let us say much more precisely how
  often wed be likely to see z-scores of
  different sizes.
• Models will be wrong they cant match reality
  exactly, but they will still be useful.

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AP STATISTICS

• DO NOW Check your homework solutions
• with your group (and the key on your desks).

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A COMPARISON
DISTRIBUTION MODEL
Real data Theoretical values
Observed Imagined
Histogram Mathematical curve
Statistics Parameters
Center Center
Spread s Spread
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• Bell shaped curves are called Normal Models.
  Normal models are appropriate for distributions
  whose shapes are unimodal and roughly symmetric.
  There is a normal model for every possible
  combination of mean and standard deviation. We
  write to represent a normal model
  with a mean of mu and a standard deviation of
  sigma.
• Why the Greek? This particular mean and standard
  deviation are not numerical summaries of the
  data. They are part of the model. Such numbers
  are called parameters of the model. We use greek
  letters for parameters and latin letters for
  statistics (numerical summaries of actual data).
  The z-score formula becomes
• Its easier to standardize the data first and
  then we can use the model N(0, 1) this normal
  model is called the STANDARD NORMAL MODEL or the
  STANDARD NORMAL DISTRIBUTION.
• Lets practice writing sigma!

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• Be careful you dont want to use the Normal
  model for any distribution.
• You must check the Normality Assumption or Nearly
  Normal Condition the shape of the datas
  distribution is unimodal and symmetric
• MAKE A PICTURE FIRST

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Does your picture look like this?
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EMPIRICAL RULE

• About 68 of the values fall within 1 standard
  deviation of the mean
• 95 fall within two standard deviations
• 99.7 fall within 3 standard deviations of the
  mean.

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The 68-95-99.7 Rule
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ANSWER THIS

• As a group, the Dutch are the tallest people in
  the world. The average Dutch man is 184 cm tall
  just over 6 ft tall. If a Normal model is
  appropriate and the standard deviation is 8cm,
  what percentage of all Dutch men will be over 2
  meters (66) tall?

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Example 2

• Suppose that it takes you 20 minutes, on average,
  to drive to school, with a standard deviation of
  2 minutes. Suppose a Normal model is appropriate
  for the distributions of driving times.
• How often will you arrive at school in less than
  22 minutes?
• How often will it take you more than 24 minutes?
• Do you think the distribution of your driving
  times is unimodal and symmetric?
• What does this say about the accuracy of your
  predictions?

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• The SAT Reasoning Test has three parts Writing,
  Math, Critical Reading (Verbal). Each part has a
  distribution that is roughly unimodal and
  symmetric and is designed to have an overall mean
  of 500 and a standard deviation of 100 for all
  test takers. In any one year, the mean and
  standard deviation may differ from these target
  values by a small amount, but they are good
  overall approximations. Suppose you earned a 600
  on one part of your SAT. Where do you stand
  among all students who took the test?

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• Suppose you scored a 680 on the test, now where
  would you stand?

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• What percentage of people score higher than a 725
  on any given section?
• What percentage of people score lower than 300?
• What percentage score between a 360 and a 510 on
  any given section?

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Problem 1

• Assume the cholesterol levels of Adult American
  women can be described by a normal model with a
  mean of 188 mg/dL and a standard deviation of 24.
  
• Sketch and label a normal model

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• What percent of adult women do you expect to have
  cholesterol levels over 200 mg/dL?
• What percent do you expect to have cholesterol
  levels between 150 and 170 mg/dL?
• Estimate the IQR of the cholesterol levels.
• Above what value are the highest 15 of womens
  cholesterol levels?

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188 mg/dL and s 24.

• What percent of adult women do you expect to have
  cholesterol levels over 200 mg/dL?