6.Concepts of Statistics

8/7/2019 6.Concepts of Statistics

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Fabric quality assurance

Basic concepts of Statistics


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What is statistics?

` Basically, statistics is the science of data.

` There are three main tasks in statistics:` (A) collection and organization,` (B) analysis, and` (C) interpretation of data.


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` (A) Collection and organization of data: We will seeseveral methods of organizing data :

` Graphically (through the use of charts and graphs) and

Numerically (through the use of tables of data).

` The type of organization we do depends on the type of analysis we wish to perform.

` Qu ick Example Let u s collect the stat u s of a gro u p of test result of yarn count.

` We could then organize the data in any of the above ways.


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` (B) Analysis of data:

` Once the data is organized, we can go ahead andcomp u te vario u s quantities (called statistics or parameters) associated with the data.


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` (C) Interpretation of data: Once we haveperformed the analysis,

` we can use the information to make assertions about thereal world.


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Ar ithmetic Mean o r ave r age ( X )

` The simplest statistic is the mean or average.

` The most generally useful statistical measure is the mean or average.` This can be obtained by adding together the individual value of the

variable & dividing the sum by the number of individual.

` For example, given the following ten results of yarn count` - 90, 91, 89, 84, 88, 93, 80, 90, 85, 87

` the mean or Xbar is 877/10 or 87.7.

` [The term Xbar refers to a symbol having a line or bar over the X]


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Ar ithmetic Mean o r ave r age ( X )` The mean value characterizes the "central tendency" or

"location" of the data.

` Although the mean is the value most likely to be

observed, many of the actual values are different than themean.

` When as saying yarn count, it is obvious thattechnologists will not achieve the mean value each andevery time a yarn count is analyzed.

` The values observed will show a dispersion ordistribution about the mean,


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M edianM edian

` The median is the middle value of a series of valuesarranged in order of magnitude .

` The median divides the area under the fre quency curve

into two e qual parts.


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M edianM edian` Example seven threads have been tested for single

thread strength in grams & the value noted down inorder of increasing strength .

` 150,152,160,170,172,175,180--gramsThe median is the 4 th value i.e. =170------------------------------------------------` 150,152,160,170,172170,172,175,180,181

The median is the 4th

& 5th values divided by two=170+172/2 =171


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M ode` The mode is the most fashionable value in the sense that

it is the value which occurs most often. That is the modeis the value of the variable corresponding to themaximum fre quency of the fre quency curve.

` Where the curve is moderately


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S tanda r d deviation

` The dispersion of values about the mean is predictableand can be characterized mathematically through a seriesof manipulations,


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M ean deviation` This measure of dispersion uses every observed value in its

calculation & is therefore a more accurate indication of variationthan the range.

` The deviation is the difference between each value & thearithmetic mean.

Mean deviation =sum of deviation from mean i.e (x-x )

Total number of observation i.e n

Percentage mean deviation P.M.D= mean deviation X 100mean


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Col u mn A( test res u lts)

Col u mn B

d= x-xDeviation from mean

Col u mn C(d) 2

X val u e X val u e-Xbar (X-Xbar) 2

90 90 - 87.7 = 2.30 (2.30) 2 = 5.2991 91 - 87.7 = 3.30 (3.30) 2 = 10.8989 89 - 87.7 = 1.30 (1.30) 2 = 1.6984 84 - 87.7 = -3.70 (-3.70) 2 = 13.6988 88 - 87.7 = 0.30 (0.30) 2 = 0.0993 93 - 87.7 = 5.30 (5.30) 2 = 28.0980 80 - 87.7 = -7.70 (-7.70) 2 = 59.29

90 90 - 87.7 = 2.30 (2.30)2

= 5.2985 85 - 87.7 = -2.70 (-2.70) 2 = 7.2987 87 - 87.7 = -0.70 (-0.70) 2 = 0.49

X = 877 (X-Xbar) = 0 (X-Xbar) =132.10

or=


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` The first mathematical manipulation is to sum ( ) the individual points andcalculate the mean or average, which is 877 divided by 10,

`

i.e X =87.787.7.

` The second manipulation is to subtract the mean value from each controlvalue, as shown in column B.

` This term, shown as X value - Xbar, is called the difference score(Deviationfrom mean).

` As can be seen here, individual difference scores (d) can be positive ornegative and the sum of the difference scores is always zero.and the sum of the difference scores is always zero.

` The third manipulation is to s quare the difference score (d) 2 to make allthe terms positive, as shown in Column C.

` Next the s quared difference scores are summed. ( i.e 132.10)

` Finally, the predictable dispersion or standard deviation (SD or s)predictable dispersion or standard deviation (SD or s) can becalculated as follows :

`

= [132.10/(10-1)]1/2 = 3.83


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coefficient of variation

` The coefficient of variation (CV) is defined as the ratio of the standard deviation to the mean :

` coefficient of variation = standard deviationMean

This is only defined for non-zero mean, and is most usefulfor variables that are always positive. It is sometimesexpressed as percentage, so that the value here ismultiplied by 100


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` The coefficient of variation should only be computed fordata measured on a ratio scaleratio scale


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CV CV C omparison to standard deviationC omparison to standard deviation` Advantages` The coefficient of variation is useful because the standard

deviation of data must always be understood in the context of the mean of the data. The coefficient of variation is adimensionless number . So when comparing between data setswith different units or widely different means, one should usethe coefficient of variation for comparison instead of thestandard deviation.

` Disadvantages` When the mean value is close to zero, the coefficient of

variation is sensitive to small changes in the mean, limiting itsusefulness.

` Unlike the standard deviation, it cannot be used to constructconfidence intervals for the mean.


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R ange` The range is simply the difference between the highest &

the lowest values.


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C lass interval


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6.Concepts of Statistics