Normal Distribution, Skewness and kurtosis

1. Suresh Babu G
Normal distribution,
Skewness and Kurtosis.
Suresh Babu G
Assistant Professor
CTE Paippad, Kottayam

2. Suresh Babu G
Normal Distribution
The normal probability curve is based upon the
law of probability discovered by French
Mathematician Abraham Demoivre in 18th century.
The normal distribution is a probability function
that describes how the values of a variable are
distributed. It is a symmetric distribution where
most of the observations cluster around the central
peak and the probabilities for values further away
from the mean taper off equally in both directions.
Normal distribution, also known as the Gaussian
distribution.

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Properties of Normal Probability
Curve
• The normal probability curve is symmetrical –
one side of the curve is identical to that of the
other.

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• The normal probability curve is bell shaped –
This means that its peak is in the middle.

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• Mean = Median = Mode

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• The normal probability curve is unimodal, ie.
only one mode.
Mode value

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• It is a continuous frequency distribution.
• The height of the normal curve is at its
maximum at the mean.
Mean

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• It is asympototic to the base line on its either
sides – As the distance of the curve from the
mean increases, the curve comes closer and
closer to the axis but never touches it.

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• Normal curve is mesokurtic ie, β2 = 3
• The first and third quartiles are equi-distant from
median ie, Median – Q1 = Q3 – Median.
• It never touch the base line.
• No portion of the curve lies below the x-axis
since P(x) being the probability can never be
negative.
• Theoretically the range of normal curve is -3σ to
+3σ

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Importance of Normal Probability Curve
• The normal distribution is the
most important probability distribution in statistics
because it fits many natural phenomena. For
example, heights, blood pressure, measurement
error, and IQ scores follow the normal distribution.
• Normal curve distributions are
very important in education and psychology
because of the relationship between the mean,
standard deviation, and percentiles. In all normal
distributions 34 per cent of the scores fall between
the mean and one standard deviation of the mean

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Uses of Normal Distribution Curve
 To determine the percentage of cases within
given score.
 To determine the percentage of cases that are
above or below a given score or reference point.
 To determine the limits of scores which include
a given percentage of cases.
 To determine the percentile rank of a student in
his own group.

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 To find out the percentile value of a students
percentile rank.
 To compare two distributions in terms of
overlapping.
 To determine the relative difficulty of test items.
 To dividing a group into sub-groups according to
certain ability and assigning the grades.
Uses of Normal Distribution Curve

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Divergence in Normality
Two types of divergence in normal Probability
curve . They are
1) Skewness
2) Kurtosis

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Skewness
A distribution is said to be skewed when the
mean and median fall at different points in the
distribution and the balance, ie, the point of
center of gravity is shifted to one side or the
other to left or right.
According to Morris Hamburg, “skewness
refers to the asymmetry or lack of symmetry in
the shape of a frequency distribution”

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Positive Skewness
 Mean>Median>Mode
 The right tail of the curve is
longer than its left tail when the
data are plotted on a graph.
 The formula of skewness and its
co-efficient give positive figures.

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Negative Skewness
• Mean<Median<Mode
• The left tail of the curve is
longer than its right tail when
the data are plotted on a
graph.
• The formula of a skewness
and its co-efficient give
negative figures.

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Kurtosis
The term kurtosis refers to the divergence in the
height of the curve, specially in the peakedness.
According to Simpson and Kafka, “The degree
of kurtosis of a distribution is measured relative
to the peakedness to a normal curve”
Leptokurtosis
Platykurtic
MesokurticKurtosis

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Leptokurtic
 When a frequency curve is
more peaked than the normal
curve it is called leptokurtic.
 K > 0

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Mesokurtic
A mesokrutic curve is a
normal frequency curve one
which is neither too peaked
nor too flat. It is the normal
curve.
K = 0

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Platykurtic
When a frequency curve is
more flat topped then the
normal curve, it is called
platykurtic.
K < 0

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