Measures of Position

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Measures of Position
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Z Score

• Also called the standard score

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Z Score

• Also called the standard score
• Represents the number of standard deviations a
  score is from the mean

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Z Score

• Also called the standard score
• Represents the number of standard deviations a
  score is from the mean
• Always round value to 2 decimal places.

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Formulas

• Sample
• Population

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Example

• Human body temperatures have a mean of 98.20
  degrees and a standard deviation of 0.62 degrees.
• Find the z score for temperatures of
• 100 degrees
• 97 degrees

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Solution

• Z (100 98.20)/0.62
• Z 2.90

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Solution

• Z (100 98.20)/0.62
• Z 2.90
• Z (97 98.20)/0.62
• Z -1.94

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Significance of Z

• Z scores above 2 or below -2 are considered to be
  UNUSUAL.
• Z scores above 3 or below -3 are considered to be
  VERY UNUSUAL.

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Conclusion about temperatures

• The temperature of 100 degrees is UNUSUAL.
• The temperature of 97 degrees is ordinary.

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Another use of z scores

• Z scores can also be used to compare relative
  position for different data sets.

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Example page 100 10
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Example page 100 10

• Z (144 128)/34 0.47
• Z (90 86)/18 0.22
• Z (18 15)/5 0.60
• The third score is the largest, so that is the
  test result with the highest relative score.

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Percentiles

• A percentile tells the percent of scores that are
  lower than a given score.

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Percentiles

• A percentile tells the percent of scores that are
  lower than a given score.
• Write P90 (or whatever number we need)

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Percentiles

• A percentile tells the percent of scores that are
  lower than a given score.
• Write P90 (or whatever number we need)
• We will not be calculating percentiles as the
  data sets should be quite large in order for the
  percentile to be meaningful.

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Example

• A pediatrician reports that a child is in the
  90th percentile for heights among children of
  that age. This is P90.
• That means 90 of all children of that age are
  shorter than the given child. The child is
  taller than average.

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Quartiles

• Quartiles divide the data set into 4 groups, each
  of which has the same number of members.
• Q1 corresponds to P25
• Q2 corresponds to P50 or the median
• Q3 corresponds to P75

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Quartiles
Q1, Q2, Q3 divides ranked scores into four
equal parts
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Finding quartiles

• Sort the data.

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Finding quartiles

• Sort the data.
• Locate the median.

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Finding quartiles

• Sort the data.
• Locate the median.
• Q1 is the median of the group of scores starting
  at the minimum value and going up to but not
  including the true median.

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Finding quartiles

• Sort the data.
• Locate the median.
• Q1 is the median of the group of scores starting
  at the minimum value and going up to but not
  including the true median.
• Q3 is the median of the group of scores starting
  just past the true median and going up to the
  maximum value.

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Example

• Use Harry Potter data found on page 69 2

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Median

• The median is the average of the 6th and 7th
  scores.
• (80.2 82.5)/2
• 81.35

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Q1

• Find the median of the first 6 scores
• (78.6 79.2)/2
• 78.9

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Q3

• Find the median of the last 6 scores
• (84.384.6)/2
• 84.45

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Another Example

• Weights of regular coke

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Another Example

• Weights of regular coke
• Median (0.81810.8192)/2
• 0.81865

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Another Example

• Weights of regular coke
• Median (0.81810.8192)/2
• 0.81865
• Q1
• 0.8163

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Another Example

• Weights of regular coke
• Median (0.81810.8192)/2
• 0.81865
• Q1
• 0.8163
• Q3
• 0.8211

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Using the TI

• We can check results with the TI calculator.
• Put the data into a list.
• Press STAT, CALC, One-Var stats
• Enter the name of the list
• Scroll down to see the values

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Harry Potter results

• Here is the screen output