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Introduction
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Dispersionis statistical measure whichgives information about the extent ofthe scatterednessor variationof various
items of a series from their mean value.
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It gives information about the composition of
series and facilitates the comparison of variabilityof two series.
It helps in finding average variation of values ofitems from mean value of data
It helps in determining thevariability of mean.
Various statistical measures like Skewness,Kurtosis, etc. can be calculated with the help ofdispersion which helps in studying further
characteristics of the data.
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Different Measures of dispersion are :
Range
Quartile Deviation
Mean Deviation
Standard Deviation
Rangeand Quartiledeviations measures are locationdevices whereasMean deviationand Standard deviationare measuresof calculation.
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Range:
The difference of the largest and the smallestitem of a series is Range.
Range = Largest value Smallest value
Coefficient of Range = (L - S) / (L + S)
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PARTITION VALUES
Median divides a statistical series into two equal parts.Like Median there are other partition values whichdivides a series into four, ten and hundred equal parts.
Partition values dividing a series into fourequalparts areknown as Quartile,partition which divides series into ten equalparts are
called Decilesandthose divides series into hundred equalparts are knownas Percentiles.
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Quartile: There are three quartiles
Q1divides the distribution in such a way that
th of items lie below it and th above it.
Q2divides the distribution into approximatelytwo halves i.e. 50% items below it and 50%above it. (Q2= Median)
Q3divides the distribution in such a way that
3/4th
of items lie below it and 1/4th
lie above it.
In symmetrical distribution, the curve is bellshaped and Q1 and Q2 are equi-distant from the
median which lies in the centre..
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Deciles: There are nine deciles denoted by D1,
D2, D3, D4, D5, D6, D7,D8 and D9
The fifth decile D5 is equal to Median or Q2
Percentile: There are ninety-nine percentileswhich are denoted by P1, P2, , P99
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Quartile Deviation
Quartile deviation is also called semi Inter- QuartileRange
Q.D. = (Q3 Q1) / 2
m for Q3= 3(N+1)/4 th place item in the seriesm for Q1= (N+1)/4 th place item in the seriesQ1 = L1 + (L1 + L2)/f X (m-c)
Q2 = L1 + (L1 + L2)/f X (m-c)
Coefficient of Q.D. = (Q3 Q1) / (Q3 + Q1)
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Mean Deviation:
The Arithmetic mean of the deviations of all thevalues taken from some statistical average (mean,median, mode) of the series is known as MeanDeviation or Average deviation.
N
DDM
||..
For Discrete Series
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For ungrouped Data For grouped Data
fi
DfiDM
||..
fi
DfiDM
||..
Where xi are individualobservations
Where xi are the mid-pt.of class- interval
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Calculation of Mean Deviation
1. Calculate the statistical average to be used( Mean, Mode, Median)
2. Calculate Deviation of all the values of seriesfrom average used.
3. Sum of deviation divided by N
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Standard deviationsquared
score mean deviation* deviation
8 9.67 - 1.67 2.79
25 9.67 +15.33 235.01
7 9.67 - 2.67 7.135 9.67 - 4.67 21.81
8 9.67 - 1.67 2.79
3 9.67 - 6.67 44.49
10 9.67 + .33 .11
12 9.67 + 2.33 5.43
9 9.67 - .67 .45sum of squared dev= 320.01
Standard Deviation = Square root(sum of squared deviations / (N-1)
= Square root(320.01/(9-1))
= Square root(40)
= 6.32
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Variability provides a quantitative measureof the degree to which scores in adistribution are spread out or clustered
together.
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The larger the standard deviation figure, the wider therange of distribution away from the measure of centraltendency
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Standard Deviation
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Deviation: deviation of one score from the mean
Variance: Square of deviation( from mean, mode ormedian) is variance.
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Standard Deviation:
The positive square root of variance is calledStandard deviation
S.D.= Sqrt(Variance)
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S.D. enables us to determineas to how far
individual items in a distribution deviate from itsmean.In a symmetrical , bell shaped curve such asone below
Uses of Standard Deviation
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Relative Dispersion: Coefficient Of Variation
)100(var. XCof
Coefficient of variance is used for comparingtwo different groups of data.
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Skewness
When the median, mean and mode do not have samevalue in a distribution, then it is known as skeweddistribution.
Skewness indicate lack of symmetry.When the frequency distribution is elongated to right, itis , having a longer tail to right and is said to bepositively skewed.In contrast, if distribution has a longer tail to the left, itis said to be negatively skewed.
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Skewness
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Positively Skewed(Skewed to Right)
Negatively Skewed(Skewed to Left)
Mean > Median> Mode Mode > Median> Mean
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The main difference between skewness and
dispersion is that
skewness measures the magnitude as well as
direction of variation in the data whereas
dispersion only measures the magnitude of
variation
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Relative Measure of skewness
Karl Pearsons Measure :
Skewness = Mean Mode
Coefficient of skewness = (Mean Mode)/ S.D.
When we are unable to calculate mode,Coefficient of skewness = 3(Mean Median)/ S.D.
The difference between mean and mode indicates theextent of departure from symmetry.
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Kurtosis
Kurtosis measures the degree of peakednessof a frequency distribution