(8b) grouped data_central tendency and dispersion

7/30/2019 (8b) Grouped Data_central Tendency and Dispersion

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Applied Statistics and Computing Lab

CENTRAL TENDENCY AND DISPERSION:For grouped data


Indian School of Business


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Learning goals

Understanding data with class intervals

Learning to evaluate various measures ofcentral tendency and dispersion, for grouped

data

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Introduction We studied measures of central tendency and

dispersion for discrete data The data was represented in form of a list

How do we deal with data with class intervals?

Can we find a value that represents a given classinterval?

Class intervals could emerge from both discrete as well

as continuous data

We would look at a dataset consisting of Nobservations, distributed across n classes

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Class mark Class mark is the midpoint of a class interval

Calculated as the arithmetic mean of the class limits E.g. if we are looking at the number of students whose scores lie

between 60 and 70, (60 is the lower limit and 70 is the upper limit)6 0 + 7 0

2= 65

is the class mark or the midpoint of the class interval 60-70

Class mark cannot be determined for a data with open classes

(intervals indicated by open bracket on either sides)

In case of overlapping classes (where the upper limit of a class and

the lower limit of the next one are equal), we assign that

overlapping value to that class where the value is the lower limit

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Cumulative frequency Cumulative frequency is the frequency of values up to the upper limit of the

corresponding class interval

For the class, denote its frequency as , cumulative frequency as and classmark as

Class # Class interval Frequency Cumulative frequency Class mark

1 =

=

+

22 = +

= +

2

3 = + + =

+

2

i = + ++ =

+

2

n = + + ++ = = +

2


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Example Weights from the body measurement data used earlier

Weight values are given up to one decimal point

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Class interval Frequency Cumulative frequency Class mark

40-49.9 27 27 44.95

50-59.9 124 151 54.95

60-69.9 120 271 64.95

70-79.9 115 386 74.95

80-89.9 87 473 84.95

90-99.9 25 498 94.95

100-109.9 8 506 104.95

110-119.9 1 507 114.95

Total 507 - -


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Summation of all values In ungrouped data set, the value of each observation is

considered

In grouped data, that is not possible How can we account for all the values of a dataset? As the class mark or the midpoint is considered to

represent every value belonging to that particular classinterval, that value holds to be a proxy for all the values

Can repeat the class mark as many times as thenumber of values belonging to that class interval;nothing but the frequency

Hence, =

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Means For data consisting of N observations distributed across n distinct

class intervals, =

= =

=

=

For the weights data, Arithmetic mean = 69.15

Geometric mean = 67.88

Harmonic mean = 66.64

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Median How do we determine the value that has 50% of the data on each of its

two sides?

Initially we can at least determine the class interval in which the valuewould lie, the median class

Let be the upper limit and be the lower limit of the median class

Let indicate the frequency of the median class and indicate thecumulative frequency of the class preceding the median class, then

= +( )(

2 )

This is obtained under the assumption that cumulative frequencyincreases from every class to another

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Median (contd.) Total 507 observations

= 253.5 observation splits the data into 2 equal halves

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Class interval Frequency Cumulativefrequency

40-49.9 27 27

50-59.9 124 151

60-69.9 120 271

70-79.9 115 386

80-89.9 87 473

90-99.9 25 498

100-109.9 8 506

110-119.9 1 507

Total 507 -

Median class, as the

253.5th observation

would lie in thisinterval

= +

( )(2

)

= 60 +

(69.9 60)(507

2 151)

120 = 68.46


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Quantiles Suppose k is the number of quantiles

k=4 for quartiles, k=10 for deciles and k=100 for percentiles

The quantile is the (

) value of the data

Must note that (

) is not the numerical value of the quantile, it is only the position corresponding

to the quantile when the data is organised in an ascending order

For median i.e. the 2nd quartile, it was the (

)= (

) value

Using cumulative frequencies, we can then determine the class to which the given quantile belongs

As per the notations used earlier,

= + [

]

= 4 and = 2 gives the median

= = { +

34

} { +

14

}

Where, the s and s refer to the lower and upper limits of the corresponding quantile classes

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Mode Can easily identify the class interval with the highest frequency; the

modal class

How do we determine the value which has the highest density? Formula given by:

= + [

] where;

u

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Applied Statistics and Computing Lab13

Mode (contd.)

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Class interval Frequency

40-49.9 27

50-59.9 124

60-69.9 120

70-79.9 115

80-89.9 87

90-99.9 25

100-109.9 8

110-119.9 1

Total 507

Modal class, class

interval with the

highest frequency

= +

+ = 50 + 59.9 50

124 27

124 27 + 124 120= 59.51


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Absolute deviations

For data consisting of N observations

distributed across n distinct class intervals,

=

=

where, is the class mark

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Central moments For grouped data consisting of N observations

distributed across n distinct class intervals, = =

( )

where, is the class mark

= = ()

= Coefficient of skewness and kurtosis can be

calculated accordingly

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Conclusion We can verify that the values obtained with the

formulae for grouped data, are very close to the valuesobtained by considering the data as ungrouped

In many situation, describing data using class intervalsis more insightful

Therefore these formulae can be useful for quick handcalculation In this age of extensive computational power, these

measures can be calculated without dividing the data

into class intervals Yet, these formulae are important from theoretical

point of view

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Thank you

(8b) grouped data_central tendency and dispersion

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