The bell shape curve

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The bell shape curve

• Normal distribution FETP India

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Competency to be gained from this lecture

• Use the properties of normal distributions to
  estimate the proportion of a population between
  selected values

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Key issues

• Normal distribution
• Properties of the normal distribution
• Z score

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Frequency distribution

• For a continuous variable, the values taken by
  the variable may be listed
• One can examine how commonly the variable will
  take specific values
• The relative frequency with which the variable is
  taking selected values is called a frequency
  distribution

Normal distribution
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Distribution

• We observe the frequency distribution of values
• If we smoothen the distribution, we obtain a
  curve
• If the curve can correspond to a mathematical
  formula, we can apply formula that allow
  predicting a number of parameters

Normal distribution
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The normal curve presented as an histogram
Normal distribution
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Observation

• Many naturally occurring events follow a rough
  pattern with
• Many observations clustered around the mean
• Few observations with values away from the mean
• This bell-shaped curve was named Normal
  distribution by a mathematician called Gauss

Normal distribution
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The normal distribution

• Normal distribution
• The symmetrical clustering of values around a
  central location
• Normal curve
• The bell-shaped curve that results when a normal
  distribution is graphed

Normal distribution
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Properties of the normal distribution of a
continuous variable

• Symmetric about its mean
• The median the mode the mean
• The entire distribution is known if two
  parameters are known
• The mean
• The standard deviation

Properties
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Additional properties of the normal distribution

• 68 of the values lie between
• Mean one standard deviation
• Mean - one standard deviation
• 95 of the values lie between
• Mean two standard deviations
• Mean - two standard deviations
• gt 99 of the values lie between
• Mean three standard deviations
• Mean - three standard deviations

Properties
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Distribution of the values according to the
standard deviation
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Characterizing a normal distribution

• The mean specifies the location
• The standard deviation specifies the spread
• Hence
• For different values of mean or standard
  deviation or both,
• We get different normal distributions.

Properties
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Usefulness of the normal distribution

• Many statistical tests are based on the
  assumption that the variable is normally
  distributed in the population
• Using the standard deviation it is possible to
• Describe the normal range between x-standard
  deviations
• Compare the degree of variability in the
  distribution of a factor
• Between two populations
• Between two different variables in the same
  population

Properties
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Distributions that are approximately normal

• For distributions that are approximately normal
• Unimodal (One mode)
• Symmetrical
• Having a bell shaped curve
• The standard deviation and the mean together
  provide sufficient information to describe the
  distribution totally

Properties
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z-score

• Every normal distribution can be standardized in
  terms of a quantity called the normal deviate
  (z)
• The z score is an index of the distance from the
  mean in units of standard deviations

Z-score
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Standardizing a normal distribution

• Z is defined as
• Observation - Mean
• Z -----------------------------------------
• Standard deviation
• The probabilities associated with normal
  distribution are obtained from the knowledge of z

Z-score
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Representing a normal curve on a standard
deviation scale
Mean
One standard deviation
Minus one standard deviation
The x-axis expresses the data values in a
standardized format
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Knowing what proportion of the values lies
between two values
Between the mean and 1 standard deviation,
there is 68 / 2 34 of the values
Z-score
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Area under the curve and Z-score

• What proportion of the population is between
  between 0 and 1.96?

Z-score
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First example of use of the normal distribution
Heights

• We are examining a population of persons with
  heights that are normally distributed
• Consider the normal distribution of heights
• Mean height X 65"
• Standard deviation SD 2"

Z-score
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What is the proportion of persons whose height
exceeds 68?

• Normal deviate
• Z (x-x)/SD (68-65)/2 1.5
• The area under the curve from Z 1.5 to ?
• 0.0668
• 6.68
• 6.68 of persons have a height that exceeds 68"

Z-score
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What is the proportion of persons whose height is
less than 60?

• Normal deviate
• Z (x-x)/SD (60 - 65 ) / 2 - 2.5
• The area under the curve from z - ? to z -2.5
  is equal to the one from Z 2.5 to z ?
• The area under the curve from Z 2.5 to ?
• 0.0062
• 0.62
• 0.62 of persons have a height below 60

Z-score
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What is the proportion of persons whose height is
between 64 " and 67 (1/2)?

• Normal deviate for x64
• Z1 (65-64)/2 - 0.5
• Area under the curve
• From Z1 - ? to - 0.5
• From Z1 0.5 to ?
• Proportion of the population with height less
  than 64
• 0.3085
• 30.8

Z-score
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What is the proportion of persons whose height is
between 64 " and 67 (2/2)?

• Normal deviate for X67" Z2 (67-65)/2
• Area under the curve from Z2 1 to ?
• 0.1587
• 15.8 of the population has a height exceeding
  67"
• Heights between 64" and 67
• 1 - 0.3085 - 0.1587 0.5328 53.28

Z-score
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Second example of use of the normal distribution
Cholesterol level

• We are examining a population of persons with
  cholesterol levels that are normally distributed
• Consider the normal distribution of cholesterol
  levels
• Mean cholesterol 242 mg
• Standard deviation 45 mg

Z-score
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What is the cholesterol level exceeded by 10 of
men?
Example 2

• What is the Z corresponding to an area of 10
  (0.1) on the right?
• The Z value from the table is
• 1.282
• Z (x-242)/45 1.3
• X - 242 1.3 x 45 58.5
• X 58.5 242 300.5 mg

Z-score
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What is the cholesterol level that is exceeded by
2.5 of men ?

• What is the Z corresponding to 2.5 of the area
  (0.025) on the left?
• From the table,
• Z corresponding to an upper area of 0.025 1.96
• By symmetry, the lower value of Z is -1.96
• (x-x) / SD z
• (x-242) /45 -1.96
• (x-242) /45 -1.96
• X 242 -1.96 x 45 - 88.2
• X 242 - 88.2 153.8 mg

Z-score
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How does one know the distribution is normal?

• Is the distribution symmetrical?
• A normal distribution is symmetrical
• Is the distribution skewed?
• A normal distribution is not skewed
• What is the kurtosis of the distribution?
• The normal distribution is neither too sharp nor
  too shallow
• Computer programmes are available to test the
  normality of the distribution

Z-score
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Scores and normal curve
Z-score
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Key messages

• The symmetrical clustering of values around a
  central location is called a normal distribution
• Normal distributions are symmetric, have a common
  value for the mean, the median and the mode and
  are solely characterized by their mean and their
  standard deviation
• Z-scores allow estimating the proportion of the
  population lying between selected values