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7/30/2019 LSD - Statistical Concepts
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Limit State Design-
Statistical Concepts
7/30/2019 LSD - Statistical Concepts
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If an event occurs in m occasions in a total ofn
occasions.
Probability of an event to happen =
Probability of an event not to occur =
n
m
n
m1
Probability
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Frequency Distribution
Results of 100cylinder spli tting tensi le testson concrete are given
in Table1.
Numbers in Table areCharacter istic Values of Variate
Character istic values can be studied more conveniently if they arerearranged in the ascending order of magni tude
Character istic values are divided into class intervals
No. of values fall ing in each interval represents the frequency in
that interval.
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Frequency Distribution
Frequency distr ibution (of tensile strengths) is H istogram.
The principle of histogram is that the area (not the
ordinate)of each rectangle represents the proportion of
observations fall ing in that intervals
I f the number of observations increased and the class
interval reduced, histogram resembles a smooth curve
I f the number of observations is inf inite, histogram becomes
a smooth curve known asNormal Distribution Curve
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Characteristics of Distribution
Mean =
If numbers and frequencies
respectively are given, mean may be calculated as
nnxxxxx in x321
ixxx
21, ifff ,, 21
n
xf
fff
xfxfxfx
ii
i
ii
21
2211
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Standard Deviation
Standard deviation=
=Deviation of a number from the mean.
= root mean square of the deviation from the mean.
COV = Ratio of standard deviation to mean, as percentage.
If the values of the individual samples are known along with
their frequency, the standard deviation can be estimated by
n
xx2
xx
n
xx 2
n
xxf ii 2
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Normal Distribution
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Normal Distribution Curve
Normal distribution curve is defined by
= Standard Deviation
e = Exponential = 2.71828 (base of the natural logo)
= Mean of the variable.
Let x be expressed in terms of another variable z as below
z is the deviation from the mean expressed in multiple of the
standard deviation. The normal distribution form is
2
2
2
1
exp2
1
xx
y
x
xxz
zxx
2
2
1exp
2
1zy
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