02 Frequency Distribution

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Frequency and SamplingDistribution

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Frequency distributionsRaw dataare collection of that have not been organised numerically. An

example is the set of weight of say 100 male students obtained from an

alphabetical listing from university records.

Arrayis an arrangement of raw numerical data in an ascending or

descending order of magnitude.

Useful data are distributed into classes or categories. The tabulararrangement of data by classes together with corresponding frequency is

called frequency distribution

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Example frequency tableWeights kg Number of students

60-62 5

63-65 18

66-68 42

69-71 27

72-74 8The frequency distribution of weights of 1 male students at !"#

uni$ersity is gi$en abo$e%

The first class consists of weights from &'&()g% The data organised

as in the abo$e frequency distribution are often called grouped data


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*ar +hart *ar +hart

A set of rectangles

having base on thehorizontal axis with the

centres at the class mar!

and length equal class

interval size and areasproportional to class

frequencies


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Frequency ,olygon

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,olygon

"s a line graph of the

class frequency plotted

against the class mar!.

"t can be obtained by

connecting the midpoints


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+umulati$e frequency

Weights No of students

59.5 0

62.5 5

65.5 23

68.5 65

71.5 92

74.5 100

A graph showing the cumulative frequency distribution of all valugreater than or equal to the lower class boundary of each class int


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,ower ,lant Example#requency distribution of mill availability $1%%%&'00()

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Number of mills unit 1

1 0

2 0

3 1

4 55

5 204

6 506

7 1064

8 413


1 0

2 0

3 4

4 48

5 188

6 615

7 1149

8 286


1 0

2 0

3 10

4 35

5 67

6 411

7 1322

8 457


1 0

2 0

3 2

4 9

5 29

6 278

7 1220

8 780


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+umulati$e Frequency

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-as Turbine Data ./ years0

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*as

turbine

*as

generator +ubrication

,ontrol

monitoring

-ower

turbine

tarting

system359 17 8 1 1 6

360 29 7 0 1 4

361 20 4 2 1 5

362 40 15 3 0 4

363 39 25 13 2 4

Summary of number of pre$enti$e maintenance acti$ities

There are many preventive maintenance activities at different intervals. An initial question to ask is

whether or not they were carried out only upon the failure of other items. Some information about

condition monitoring activities was gathered following a number of preventive maintenance and

corrective maintenance activities


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-as Turbine Data ./ years0

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*asturbine

*asgenerator +ubrication

,ontrolmonitoring

-owerturbine

tartingsystem

359 6 10 4 0 3

360 8 8 1 3 2

361 12 5 2 2 1

362 38 22 10 1 4

363 27 22 7 10 1

Summary of number of correcti$e maintenance actions

Dual redundancy with spares was experienced over the observation period of five years. Now it

remains to be seen what levels of corrective maintenance are performed on the oil platform with lots

of redundancy as opposed to with no redundancy. The corrective maintenance actions were referred

to as failure repairs or replacements while others were classified as periodic replacements!


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, and + Acti$ities

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The differences between the gas turbines and their sub"units are clearly evident. Among gas turbines

#$%& and #$%$ there are a lot of failures and maintenance activities. The question here is whether or

not these gas turbines are identical. 'f not there might be no particular reason for such similarities.

Similarly units #$() #$%* and #$%+ display roughly equal numbers of failures and might have

some commonality with each other.


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Frequency of , and +

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The history data indicate that a substantial amount of preventive maintenance activities

which consist of minor periodic service tasks inspections and periodic condition monitoring

activities are performed but the failure frequency of the gas generators does not improve.

This might possibly be due to imperfect maintenance or the interval period of ,- activities

may not be appropriate since similar failures were repeatedly observed.


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Sampling Distribution

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A sampling distributionshows how a statistic

would vary with repeated random sampling of the

same size and from the same population.

A sampling distribution therefore is a probability

distribution of the results of an infinitely largenumber of such samples.

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Descripti$e easures/hen data is clustered or grouped around a central point this central point is

often used to describe the data or the population and is used as a reference.

The mean $average) median and mode are measures of central tendency.

ean .or a$erage0is the sum of all the observations $) divided by the

number of observations $n).

2ean 3

edianis the middle value of an ordered set of data.

ode is the value which occurs most frequently in a set of data

( )

n

x

.

n

i

i== 1


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2ariance

#or a sample and a population the equations are4

ample 5ariance 3

/here is the sample variance is the sample mean x is a data value and n

is the number of values $sis the sample standard deviation).

-opulation 5ariance 3

/here is the population variance is the population averageis a data

value andNis the number of values $/is the population standard deviation).

( )

)1$

'

1'

=

=

n

xx

S

n

i

i

s'

.

( )

N

x

N

i

i

'

1' = =

'


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Standard De$iation

Standard De$iationis the square root of the variance. The standard deviation is the

most useful measurement of the spread of data in statistical analysis.

ample tandard 6eviation

-opulation tandard 6eviation

The standard deviation is the measure of spread or scatter in the population expressed in

the original units.

( ))1$

'

1'

== =

n

xx

SS

n

ii

( )

N

x

N

i

i

'

1'

=

==


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A population distributionof a random variable is

the distribution of its values for all members of the

population.

Thus a population distribution is also the

probability distribution of the random variable

when we choose one individual $i.e. observation

or sub7ect) from the population at random.

,opulation Distribution


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Sampling distribution of a

sample mean

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ampling distribution of a sample mean4 if a population

has a normal distribution then the sampling distribution of

a sample mean ofxfor nindependent observations willalso have a normal distribution.

*eneral fact4 any linear combination of independent

normal random $ariables is normally distributed.


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Standard de$iation of a sample mean3

4Standard error5

The standard error is calculated by dividing the

standard deviation of the sample mean by the

square root of sample size&n.

6oing so anchors the standard deviation to the

sample8s size&n4 the sampling distribution of the

sample mean across relatively small samples has

larger spread and across relatively large samples

has smaller spread.

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Sampling distribution of a

sample mean3

"f a population distribution

The sampling distribution of the sample mean is

9ormal if the population distribution is normal $i.e. a sample mean is a

linear combination of independent normal random variables).

The sampling distribution is approximately normal for large samples

in any case $according to the ,entral +imit Theorem).

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)$ N

):$ nNx


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6ormal ,opulation

/e can apply the ,entral +imit Theorem4

;ven if the population is not normal


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Sampling Distributions

6on'6ormal ,opulation

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-opulation 6istribution

Sampling Distribution.becomesnormal as n increases0

x

x

7arger

sample

si8e

Smaller sample si8e

x0

0


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+entral limit theorem

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,opulation mean 9 1%1/

.%/:%;/:1%0


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+entral limit theorem

ample size means everything? The more samples we collects

the closer we obtain information on the population itself?

Average conditions become more prominent.

The variability about the mean becomes less prominent.

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Sampling Dilemma

ampling does a good 7ob of accepting very good

lots and re7ecting bias lots. Unfortunately a large

area of indecision lies in the middle.

The sampling rule is based on probability and the

application of probability predicts the acceptance

of lots with substandard quality.


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02 Frequency Distribution