This presentation talks about Normal Probability Curve. It also talks about its uses in solving problems. It explains its main properties. Lastly it also talks about skewness and kurtosis of the curve in reference to NPC.

1. Normal Distribution
Vikramjit Singh, PhD
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
0.045
0 10 20 30 40 50 60 70
Normal Probablity Curve

2. © V. Singh

3. © V. Singh
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
0.045
0 10 20 30 40 50 60 70
Normal Probablity Curve
-3σ -2σ -1σ 0 1σ 2σ 3σ
50%
MEAN
B
50%
68.26%
A

4. © V. Singh
𝒚 =
𝑵
𝝈 𝟐𝝅
ⅇ
−
𝒙𝟐
𝟐𝝈𝟐
The equation of Normal Probability Curve
Here
x - scores(expressed as deviations from the
mean) laid off along the base line or X-
axis.
y – the height of the curve above X-axis or
base line or the frequency of a given x
value.
N - Number of Participants/ Observations
σ – standard deviation of the entire
distribution
Π – 3.1416 (ratio of circumference of a
circle to its diameter)
e – 2.7183 (base of the Napierian System of
logarithms)

5. © V. Singh
PROPERTIES /CHARACTERSTICS OF NORMAL PROBABLITY CURVE
1. Normal distribution
curve was separately worked
upon by Gauss and Laplace.
They also named it as
‘curve of error’ where
‘error’ is used to denote the
deviations from the mean or
normal.

6. © V. Singh
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
0.045
0 10 20 30 40 50 60 70
Normal Probablity Curve
2. The height of
the curve declines
symmetrically in
both the direction .
PROPERTIES /CHARACTERSTICS OF NORMAL PROBABLITY CURVE

7. © V. Singh
PROPERTIES /CHARACTERSTICS OF NORMAL PROBABLITY CURVE
3. The shape of this curve is bell
shaped. The curve as it is visible here is
symmetrical and bilateral from the
centre as referenced from the line
segment AB. Each of the 50 % of the
cases of total 100% are towards the
left and right of the line segment AB.
50 %
50 %

8. © V. Singh
PROPERTIES /CHARACTERSTICS OF NORMAL PROBABLITY CURVE
4. The Mean, Mode and Median of the
normal distribution curve coincides or
falls at the same point. This point is
represented by 0. This makes it clear
that it has.
Mean = 0
Median = 0
Mode = 0 (Unimodal)
Mean
=Median
=Mode =0

9. © V. Singh
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
0.045
0 10 20 30 40 50 60 70
Normal Probablity Curve
PROPERTIES /CHARACTERSTICS OF NORMAL PROBABLITY CURVE
5. The Points of
Influx occur at
point±1 Standard
Déviation (± 1 σ).
The height of the
ordinates are same at
both the influx
points.
-1σ 1σ

10. © V. Singh
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
0.045
0 10 20 30 40 50 60 70
Normal Probablity Curve
+ ∞
- ∞
PROPERTIES /CHARACTERSTICS OF NORMAL PROBABLITY CURVE
6 . The normal curve is
Asymptotic to the X-
Axis. The curve height
continues to decrease
on both the side but
never touches the X-
axis.

11. © V. Singh
0.3989
PROPERTIES /CHARACTERSTICS OF NORMAL PROBABLITY CURVE
7. The maximum
ordinate occurs at the
centre and height of this
ordinate is 0.3989 unit.
Ordinate at point ‘0’
from the formaula
𝒚𝟎 =
𝑵
𝝈 𝟐𝝅
ⅇ
−
𝒙𝟐
𝟐𝝈𝟐
is 0.3989

12. © V. Singh
PROPERTIES /CHARACTERSTICS OF NORMAL PROBABLITY CURVE
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
0.045
0 10 20 30 40 50 60 70
Normal Probablity Curve
8. As the curve never
touches base line hence
we start with the mean
as starting point of the
curve. Further SD are
used as unit of
measurement or
deviation from the
mean. The curve
extends on both side -3
σ distance on the left
to +3 σ on the right.
-3σ -2σ -1σ 0 1σ 2σ 3σ

13. © V. Singh
PROPERTIES /CHARACTERSTICS OF NORMAL PROBABLITY CURVE
9. The percentage
of cases falling in
Successive Standard
Deviation in both
the side of the
Mean is fixed and
are equal.

14. © V. Singh
PROPERTIES /CHARACTERSTICS OF NORMAL PROBABLITY CURVE

15. © V. Singh
PROPERTIES /CHARACTERSTICS OF NORMAL PROBABLITY CURVE
10. The scale of X-
axis in normal
curve is generalised
by Z deviates.
Z =
𝑋−𝑀
σ or
when M = 0 , Z =
𝑿
σ

16. © V. Singh
PROPERTIES /CHARACTERSTICS OF NORMAL PROBABLITY CURVE
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
0.045
0 10 20 30 40 50 60 70
Normal Probablity Curve 11. The curve is perfectly
symmetrical in both the
direction. In other way
the Skewness of the
curve is Zero.

17. © V. Singh
12. The normal curve
is a smooth curve, not
a histogram . It is
moderately peaked.
The kurtosis of the
normal curve is 0.263.
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
0.045
0 10 20 30 40 50 60 70
Normal Probablity Curve
PROPERTIES /CHARACTERSTICS OF NORMAL PROBABLITY CURVE

18. © V. Singh
Skewness in Short
Skewness refers to a distortion or asymmetry
that deviates from the symmetrical bell
curve, or normal distribution, in a set of
data. If the curve is shifted to the left or to
the right, it is said to be skewed. -
Investopedia

19. © V. Singh
Positive Skew
Positive Skewed is that when
the curve is tailed in the right
side of the curve. Here
Median and Mean is more than
the Mode.
Ex- When we have a hard test.
Income Distribution.

20. © V. Singh
Negative Skew
Negative Skewed is that when
the curve is tailed in the left
side of the curve. Here Median
and Mean is less than the
Mode.
Ex- Very Easy Test,
Distribution of Age of Death

21. © V. Singh
Kurtosis
The Kurtosis of a distribution refers to its
'curvedness' or ' peakedness ' . The
distributions may have the same mean and the
same variance and may be equally skewed, but
one of them may be more peaked than the other.
A normal curve have a kurtosis value equal to
0.263

22. © V. Singh
Leptokurtic, Platykurtic , Mesokurtic
Normal Probablity Curve is called as
Mesokurtic (Ku=0.263) , when the
Kurtosis is greater than 0.263 then
the curve is more peaked and is
called as Leptokurtic. When the
Kurtosis is lesser than 0.263 then the
curve is less peaked or plateaued and
is called as Platykurtic.

23. © V. Singh