3.MMS Lec3 (Dispersion)

8/12/2019 3.MMS Lec3 (Dispersion)

1/27

Introduction


2/27

Dispersionis statistical measure whichgives information about the extent ofthe scatterednessor variationof various

items of a series from their mean value.


3/27

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.


4/27

Different Measures of dispersion are :

Range

Quartile Deviation

Mean Deviation

Standard Deviation

Rangeand Quartiledeviations measures are locationdevices whereasMean deviationand Standard deviationare measuresof calculation.


5/27

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)


6/27

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.


7/27

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..


8/27

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


9/27

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)


10/27

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


11/27

For ungrouped Data For grouped Data

fi

DfiDM

||..

fi

DfiDM

||..

Where xi are individualobservations

Where xi are the mid-pt.of class- interval


12/27

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


13/27


14/27

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


15/27

Variability provides a quantitative measureof the degree to which scores in adistribution are spread out or clustered

together.


16/27

The larger the standard deviation figure, the wider therange of distribution away from the measure of centraltendency


17/27

Standard Deviation


18/27

Deviation: deviation of one score from the mean

Variance: Square of deviation( from mean, mode ormedian) is variance.


19/27

Standard Deviation:

The positive square root of variance is calledStandard deviation

S.D.= Sqrt(Variance)


20/27

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


21/27

Relative Dispersion: Coefficient Of Variation

)100(var. XCof

Coefficient of variance is used for comparingtwo different groups of data.


22/27

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.


23/27

Skewness


24/27

Positively Skewed(Skewed to Right)

Negatively Skewed(Skewed to Left)

Mean > Median> Mode Mode > Median> Mean


25/27

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


26/27

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.


27/27

Kurtosis

Kurtosis measures the degree of peakednessof a frequency distribution

3.MMS Lec3 (Dispersion)

Documents