LSD - Statistical Concepts

7/30/2019 LSD - Statistical Concepts

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Limit State Design-

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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LSD - Statistical Concepts

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