Message to the user'''

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Message to the user...
Revised 2002 Statistics Show 3 of 3
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printing the OUTLINE VIEW will be helpful as
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The NORMAL Curve

• Properties
• Example

3
Properties of the Normal Curve

• Bell Shaped (symmetric) distribution
• Mean Median (? is the mathematical symbol used
  for this measure of center)
• The standard deviation(?) is used for the measure
  of spread.
• THE 68-95-99.7 RULE
• Approximately 68 of the data lies within 1
  standard deviation of the mean
• Approximately 95 of the data lies within 2
  standard deviations of the mean
• Approximately 99.7 of the data lies within 3
  standard deviations of the mean.

It is important to remember that this rule
applies only to NORMALLY DISTRIBUTED DATA. Using
the mean and standard deviation to describe the
spread of the data can be done for any set of
data. Just because you choose this measure of
spread does not mean that the rule applies!
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A brief explanation...Percentile ranks

• A percentile rank indicates a place in any set of
  data where a certain percentage of the data falls
  AT or BELOW.
• For example, the MEDIAN is the 50th percentile,
• which means that 50 of the data values are EQUAL
  TO or LESS THAN that median.
• It can also be said that the MEDIAN is greater
  than or equal to 50 of the data values in the
  set.

...SET OF DATA (a ranked list of all the values
in the set)

• Another example
• The 95th percentile (95ile) for a set of data
  would be...
• that data value that is greater than or equal to
  95 of all of the data values in the set.
• Notice, also that it is less than or equal to 5
  of the values in the set

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Percentile ranks and the Normal Distribution

• So, if you know that one property of a Normal
  Distribution is that 68 of the data lies within
  1 standard deviation from the mean...
• And you know that the mean median for this type
  of distribution...
• You can say something about percentile ranks for
  this set of data...

50ile
...SET OF Normally distributed DATA
M
Measure one s.d. to the left and one to the right
of M. This is where you find 68 of the data.
34
34
68 of the data
84ile

• You know that M 50ile
• and you know that a NORMAL DISTRIBUTION means
  that the data is SYMMETRIC about that data value
• So, 34 of the data is in that part of the set of
  values that is one standard deviation to the left
  of M
• and 34 is in the set to the right of M
• To find the percentile rank for the value to the
  LEFT of the mean, subtract 50 -34 16ile
• Similarly, on the other side, 50 34 84ile

16ile
Watch!
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Using the properties of the Normal Curve...
You can use the 68-95-99.7 rule to determine
percentile ranks for other points on the Normal
Curve...

• Bell Shaped (symmetric) distribution
• Mean Median (?) 50th percentile
• The standard deviation(?) is used for the measure
  of spread.
• 68-95-99.7 rule...
• DISTRIBUTION OF VALUES

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Normal Curve common notation

• DISTRIBUTION OF VALUES
• using STANDARD SCORES (Z-SCORES)
• (ex.) a standard score(z-score) of
• 1 an actual score of ??
• (ex.) a z-score of -3 an actual score of ?-3?

Another common notation that is used when
describing normally distributed data is called a
STANDARD VALUE (standard score, z-score) This
notation simply uses the standard deviation as a
standard unit, and describes how far away (in
standard deviations) each actual data value is
from the mean. The sign of the standard value
indicates its direction from the mean.
Each actual data value corresponds to a standard
value.
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Application of the Normal Curve

• DISTRIBUTION OF VALUES
• for weights of adult women whose height is
  approximately 5ft. 5 in.
• ? 134 pounds ? 6 pounds

134
140
146
152
128
122
116
Weights in pounds
Put the actual values on the graph and complete
the application
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Application of the Normal Curve

• DISTRIBUTION OF VALUES for weights of adult women
  whose height is approximately 5ft. 5 in.
  ? 134 pounds ? 6 pounds

75
84th ile
(a) A woman weighing 140 lb. will be in which
percentile? (b) A woman weighing 122 lb. will be
in which percentile? (c) A woman the 75th
percentile will weigh approximately?
2.5th ile
About 137 pounds
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Application Analysis

• DISTRIBUTION OF VALUES for weights of adult women
  whose height is approximately 5ft. 5 in.
  ? 134 pounds ? 6 pounds
• (a) A woman weighing 140 lb. will be in the 84th
  percentile
• 84 percent of women in this height category
• weigh 140 lb or less. (also, 16 weigh more than
  140)
• (b) A woman weighing 122 lb. will be in the 2.5th
  percentile
• 2.5 percent of women
• weigh 122 lb or less.(and 97.5 weigh more)
• (c) A woman in the 75th percentile will weigh
  approximately?
• The 75th percentile is between the 50th and 84th
  therefore, the weight is between 134 and 140 lb

There are charts that can help you estimate more
accurately between the standard percentiles but
we wont discuss them here.
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Analysis Using the Normal Distribution...

• Many distributions that occur naturally exhibit a
  normal (or nearly normal) tendency
• Weights of children
• Weights of adults in certain height categories
• Scores on state-wide or national tests
• etc
• The properties exhibited by normally distributed
  data allow us to make predictions about similar
  sets of data,
• and to draw conclusions about these sets without
  having to handle each piece of data.
• See your text for more examples of normal
  distributions and the uses of their properties!

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End of show 3

• This is the final show in the
• Statistics Unit

REVISED 2002 Prepared by Kimberly Conti, SUNY
College _at_ Fredonia Suggestions and comments to
Kimberly.Conti_at_fredonia.edu