AEB801_20222023-lecture_04 Normal Distribution

LECTURES 4
Normal Distribution or Gaussian Distribution,
Proportions of a Normal Distribution
(The Z - Scores),
Symmetry, Skewness and Kurtosis of Normal
Distributions
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Normal Distribution or Gaussian Distribution
What is a Normal Distribution?
It is a distribution
 with a preponderance of values around the means
 with progressively fewer observations towards the extremes of the range of values.
 Many of the variables in human population, such as blood pressure, age, height, and
weight, have an almost normal distribution.
 It is also called Gaussian distribution in honour of the British biometrician Karl F.
Gauss who developed it in the early nineteenth century (1777 – 1855 {78yrs}).
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In a normal distribution, most values fall near the average, with only a small
percentage of values falling far above or below the average.
 Normal distributions generally develop when the sample size or number of
observations is very large.
 Data that are normally distributed when plotted on a graph resembles the shape of
a bell. The curve formed by a normal distribution is called a normal curve, bell
curve, normal distribution curve, or normal probability curve,
 With the mean, mode and median at the centre
 Because of its symmetric nature, the normal distribution curve is unimodal,
meaning it possesses only one mode
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Normal Curve
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The total area under the normal
distribution curve is considered to
be 100% or 1. Half of this area
(50%) lies to the right of the mean,
while the remaining half (50%) is
to the left.
99% of the population lie within
three standard deviations.
 1sd – 68%
 2sd – 95%
 3sd – 99%

Use of a Normal Distribution Curve
Normal Distribution Curve gives an indication of how likely a specific data or range
of data are i.e. whether some data values or set of data values are unusual.
 Data values in dense areas of the curve (centre) are more likely while those far
from the centre are less likely.
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STANDARD NORMAL DISTRIBUTION
A normal curve with µ = 0 and δ = 1 is said to be a standard normal curve.
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Any normal distribution can be transformed to a standard normal distribution to find
the part of area under the curve that lies between two points (interval) or to obtain
the ratio of an area which is more or less than a point (a certain value). This
transformation is done using the following relation:
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PROPORTIONS OF A NORMAL DISTRIBUTION
(THE Z - SCORES)
The Z-score for an item indicates how far and in what direction that item
deviate from its distribution mean expressed in units of its distribution’s standard
deviation.
Z-Scores sometimes called “standard scores” or the “normal deviate” tells how
many standard deviations from the mean the particular value of Xi is located.
OR
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This calculation is known as normalizing or standardizing Xi.

Example
If mean iron intake (x) is 20.4 mg and SD is 4.0.
1. Find out the Z Value for individual iron intake (X) of 30.4.
2. Find also the percentage of individuals exceeding X.
Step 1: Z=
30.4 −20.4
4
=
10.0
4
= 2.50
Step 2: Z of 2.50; proportion exceeding X is 0.0062 or 0.62%
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CONTINOUS ASSESSMENT
Suppose the IQ has a normal distribution with a mean of 100 and a standard
deviation of 15.
1. What percentage of people has an IQ greater than 130?
2. What percentage of people has an IQ less than 85?
3. What percentage of people has an IQ between 85 and 130?
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What percentage of people has an IQ greater than 130?
Using the formula
µ = 100, IQ = 130, δ = 15
130 −100
15
=
30
15
= 2.0
Using the z-table, 2.0 translates to 0.0228 or 2.28%
i.e. P(x > 130) = P(Z > 2.0) = 0.0228.
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The amount by which values vary from one another defines the shape of the normal curve.
If most of the values are similar, then the normal curve is a tall and thin bell shape (A).
If there is a lot of variation among the values, then the normal curve is a short and wide
bell shape (B).
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A
B
C

In a normal distribution, for any given standard deviation (δ) there are an infinite number
of normal curves that are possible depending on the population mean (µ).
Example; Normal Curves for similar µ= 0 but different standard deviation (δ).
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Similarly, for any given standard deviation (δ), infinity of normal curves is possible each
with a different value of mean (µ).
Example; Normal Curves for µ= 52, 76 and δ=12.
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Symmetry
A symmetrical distribution is one in which
 the mean and the median are identical and
 the portion of the frequency polygon to the left of the mean is a mirror image of the portion to the
right of the mean.
 A symmetrical distribution can be unimodal or bimodal.
 Although all normal distributions are symmetrical not all symmetrical distributions are normal
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A=Unimodal
B=Bimodal

KURTOSIS
Kurtosis refers to the shapes of the symmetric distribution.
The kurtosis is a measure of the thickness of the tails of a distribution and the
sharpness of its peak.
They are of various types namely;
1. Mesokurtic (Normal),
2. Leptokurtic and
3. Platykurtic.
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MESOKURTIC
It is a Standard Normal Distribution Curve (With µ = 0 and δ = 1)
If the Kurtosis of a normal distribution is zero, the distribution is assumed to be mesokurtic (middle:
neither high nor low).
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LEPTOKURTIC
The prefix “Lepto” means “thin”, like the shape of its peak.
In a leptokurtic distribution, there are
 more values near the mean or at the extremes, and
 fewer values that are moderately above or below
the mean (compared to a normal distribution),
 Shape is skinny: that are thin and tall
This type of distribution may be produced by the
composite of two normal populations having the same µ
but with different δ.
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PLATYKURTIC
Platykurtic is derived from the prefix “platy” which means
“broad”, resembling its shape-flat, wide or broad.
In a platykurtic distribution
 most of the values share about the same frequency of occurrence.
 The points along the x-axis are highly dispersed,
 The low peak, with corresponding thin tails, means the
distribution is
 less clustered around the mean than in a mesokurtic or
leptokurtic distribution.
 As a result, the curve is very flat, or plateau-like.
 Uniform distributions are platykurtic.
 The values are more between the mean and tails than do normal
distribution.
Such a distribution might be the composite of two normal
populations with the same variance but different means (µ).
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Skewness
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Skewness is the degree of symmetry, or degree of departure from
symmetry.
Distributions that are not symmetrical are said to be skewed.
 This could be positive or negative.
 A distribution is said to be positively skewed (i.e. skewed to
the right) if the frequency distribution curve has a longer tail
to the right of the central maximum than to the left
 with the mean greater than the mode and median.
E.g mean of ages of milk consumption in humans

A distribution is said to be
negatively skewed (i.e. skewed to
the left) if the frequency distribution
curve has a longer tail to the left of
the central maximum than to the right
with the mean less than the mode
and median.
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E.g. Frequency of death against age groups in a
human population

Although there is lack of symmetry in many populations, they are all
treated as normal population or sample and statistical techniques could
be applied with accuracy.
However, the farther the curve shifts from normal the more the
probability of committing errors.
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AEB801_20222023-lecture_04 Normal Distribution

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