Download Normal Distribution

The bell shape curve
Normal distribution
FETP India
Competency to be gained
from this lecture
Use the properties of normal distributions to
estimate the proportion of a population
between selected values
Key issues
• Normal distribution
• Properties of the normal distribution
• Z score
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
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
The normal curve presented as an
histogram
Num be r of obse rvations for e ach interval
Normal distribution
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
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
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
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
• > 99% of the values lie between
 Mean + three standard deviations
 Mean - three standard deviations
Properties
Distribution of the values according to
the standard deviation
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
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
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
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
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
Representing a normal curve on a
standard deviation scale
Minus one
standard
deviation
Mean
One
standard
deviation
The x-axis expresses the data values in a standardized format
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
Area under the curve and Z-score
• What proportion of the population is
between between 0 and 1.96?
Z-score
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
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
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
What is the proportion of persons whose
height is between 64 " and 67 ” (1/2)?
}
• Normal deviate for
x=64”:
 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
What is the proportion of persons whose
height is between 64 " and 67 ” (2/2)?
• Normal deviate for X=67" =
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
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
Example 2:
What is the cholesterol level exceeded
by 10% of men?
• 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
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
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
Scores and normal curve
Z-score
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