Features of gaussian distribution curve

Features of Gaussian
distribution curve
Represented To: Dr. IU khan
Represented By: Farzeen Javaid
0188-BH-CHEM-11
Course Title: Separation Techniques (Compulsory)
Course Code: CHEM-4201

DEFINITON
1) The area under a normal curve has a normal distribution
(a.k.a., Gaussian distribution)
2) The normal probability distribution (Gaussian distribution)
is a continuous distribution which is regarded by many as
the most significant probability distribution in statistics
particularly in the field of statistical inference.
Symbols Used
 “z” – z-scores or the standard scores. The table that transforms
every normal distribution to a distribution with mean 0 and
standard deviation 1. This distribution is called the standard
normal distribution or simply standard distribution and the
individual values are called standard scores or the z-scores.
 “µ” – the Greek letter “mu,” which is the Mean, and
 “σ” – the Greek letter “sigma,” which is the Standard Deviation

CHARACTERISTICS
 The normal curve is bell-shaped and has a single peak at the
exact center of the distribution.
 The arithmetic mean, median, and mode of the distribution
are equal and located at the peak.
 Half the area under the curve is above the peak, and the
other half is below it.
 The normal distribution is symmetrical about its mean.
 The normal distribution is asymptotic - the curve gets closer
and closer to the x-axis but never actually touches it.
 Unimodal - a probability distribution is said to be normal if the
mean, median and
 mode coincide at a single point
 Extends to +/- infinity- left and right tails are asymptotic with
respect to the horizontal lines
 Area under the curve = 1

Characteristics of a Normal
Distribution

THE NORMAL DISTRIBUTION OF
MATHEMATICAL FUNCTION (pdf)

The Normal Distribution
The normal curve is not a single curve but
a family of curves, each of which is
determined by its mean and standard
deviation.

PARAMETER
 The normal distribution can be completely specified by two
parameters:
1. Mean
2. Standard deviation
 If the mean and standard deviation are known, then one
essentially knows as much as if one had access to every
point in the data set.

Mean (µ)
 Mean : Measure of Central tendency
 Center or middle of data set around which observations are
lying
 Assuming : frequency in each class is uniformly distributed
and representable by mid point
 Mean for grouped data is given by
x̄ =
𝟏
𝒏
𝜮 𝒇𝒊 ∗ 𝒙𝒊
where
n = no of observations
fi = frequency of each (ith) class interval
xi = mid point of each class interval

Standard Deviation (σ)
 Standard Deviation : Measure of Dispersion
 Average deviation of observations around the mean
 Compactness or variation of data
 SD = root mean square deviation
 SD =  variance = 
𝟏
𝒏
𝜮 (xi – x̄ )²
where
n = no of observations
x̄ = mean of the frequency distribution
xi = mid point of each class interval

Curves with different means, same standard deviation
Curves with different means, different standard deviations

PROPERTIES OF A NORMAL
DISTRIBUTION
 As the curve extends farther and farther away from the
mean, it gets closer and closer to the x-axis but never
touches it.
 The points at which the curvature changes are called
inflection points. The graph curves downward between
the inflection points and curves upward past the
inflection points to the left and to the right.

Properties of normal curve
1. The function f(x) defining the normal distribution is a proper
p.d.f., i.e. f(x)≥0 and the total area under the normal curve
is unity.
2. The mean and variance of the normal distribution are µ
and σ2 respectively.
3. The median and the mode of the normal distribution are
each equal to µ, the mean of the distribution.
4. The mean deviation of the normal distribution is
approximately 4/5 of its standard deviation.
5. The normal curve has points of infection whih are
equidistant from the mean.
6. For the normal distribution, the odd order moments about
the mean are all zero.

7. if X is N (µ, σ2 ) and if Y a+bx, then Y is N (a+bµ, b2 σ2 ).
8. The sum of independent normal variables is a normal
variable. Stated differently, if X1 is N (µ1, σ1
2) and X2 is N
(µ1, σ2
2), then for independent X1 and X2, X1 + X2 is N (µ1 +
µ2, σ1
2 + σ2
2 ).
9. The normal curve approaches, but never really touches the
horizontal axis on either side of the mean towards plus and
minus infinity, that is the curve is asymptotic to the
horizontal axis as x ±∞
10. The height of any normal curve is maximized at x = µ.
11. All normal curves are positive for all x. That is, f(x) > 0 for
all x.
12. The area under an entire normal curve is 1.

13. No matter what the value of µ and σ are, area under
normal curve remain in certain fixed proportions within a
specified number of standard deviation on either side of µ. For
example the interval
 µ ± σ will always contain 68.26%
 µ ± 2σ will always contain 95.44%
 µ ± 3σ will always contain 99.73%

References
 C. M Sher. Introduction To Statistical Theory part 1, 8th
Edition. Markazi Kutab Khana Publisher, 2002
 http://www.unt.edu/rss/class/Jon/ISSS_SC/Module004/isss_
m4_normal/node4.html
 https://onlinecourses.science.psu.edu/stat414/node/149
 https://www3.nd.edu/~rwilliam/stats1/x21.pdf

Features of gaussian distribution curve

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