Positional Averages

Positional average determines the position or place of central values or variables in the series. There are two important types of positional averages:

Median Mode

Median

According to Prof. L.R.Conner“The median is that value of

variable which divides the group into two equal parts, one part comprising all values greater and the other all values less than median.”

Calculation Of Median

Individual Series1. Arrange the terms in ascending order.2. Count the no. of terms (N)

When no. of terms is odd:

3. Calculate (N+1)/2 th term. This term is the median.

When no. of terms is even:3. Calculate N/2 and N/2 +1.4. Mean of these two terms is the median.

Discrete Series

1. Arrange the data in ascending order.2. Find cumulative frequencies.3. Find the value of middle item by using

the formula, Median = Size of (N+1)/2 th item.

4. Find that total in the cumulative frequency column which is equal to(N+1)/2 th or nearer to that value.

5. Locate the value of the variable corresponding to that cumulative

frequency. This is the value of median.

X: 10 20 30 40 50 60 70

f: 4 7 21 34 25 12 3

Cf: 4 11 32 66 91 103 106

Terms:

0-4 5-11 12-32 33-66 67-91 92-103 104-106

X: 4 8 12 16 20 24 28

f: 7 18 25 18 15 10 7

Cf: 7 25 50 68 83 93 100

Terms:

0-7 8-25 26-50 51-68 69-83 84-93 94-100

Depending Measures on median

Quartiles: Quartiles divide the series in four equal parts. For any series there are three quartiles.First Quartile:

For Individual and Discreet series:Q1 = Size of (N+1)/4 th term

For Continuous series:Q1 = L + (N1 – Cf) i /f ; where N1 = N/4

Second Quartile:Q2 = Median = P50 = D5

Third Quartile:For Individual and Discreet series:

Q1 = Size of 3(N+1)/4 th termFor Continuous series:

Q3 = L + (N1 – Cf) i /f ; where N1 = 3N/4

Deciles:Decile divides the series into ten equal parts. There are 9 deciles for any series, from D1 to D9.

For Individual and Discreet series:Dn = Size of n(N+1)/10 th term

For Continuous series:Dn = L + (N1 – Cf) i/ f ; where N1 = nN/10

Percentiles:Percentiles divide the series in 100 equal parts. For any series there are 99 percentiles.

For Individual and Discreet series:Pn = Size of n(N+1)/100 th term

For Continuous series:Pn = L + (N1 – Cf) i/f ; where N1 =

nN/100

Q. Find D2, P40 and Q3 for the individual series:

21, 17, 18, 11, 27, 24, 22, 19, 14

Q. Find D2, P71 and Q1 for the discrete series:

Q. Median marks of a class of 50 students is 48. But two terms 46 and 73 were miss read as 64 and 37. Find correct value of Median?

X: 2 4 10 14 18 28

f: 3 7 12 14 9 5

•Continuous Series M = L + N1 – Cf x i

fWhere, M= Median

N1 = N/2

L = Lower limit of class interval in which frequency lies.

Cf = Cumulative frequency f = Frequency of that interval i = Length of that class interval

Calculate Median for the following data (Exclusive series)

X f

0-10 7

10-20 18

20-30 34

30-40 50

40-50 35

50-60 20

60-70 6

N = 170

Cf

7

25

59

109

144

164

170

Uses Of Median It can be easily calculated and is easy to

understand. Unlike mean, median is not affected by

the extreme values. For open end intervals, it is also a suitable

one. As taking any value of the intervals, value of median remains the same.

Median can also be used for other statistical devices such as Mean Deviation and Skewness.

It can be located graphically. Some items may not be available to

get median. Even if the number of terms is known, We can get the median.

Limitations Of Median Even if the value of extreme items is too large,

it does not affect too much, but due to this, sometimes median does not remain the representative of the series.

It’s affected much more by fluctuations of sampling than A.M.

If the no. of terms in series is even, we can only make an estimate, as the A.M. of two middle terms is taken as Median.

Mode

According to Zizek“The mode is the value occurring most

frequently in a series of items and around which the other items are distributed most densely.”

Calculation Of Mode: Individual series:

1. Arrange the terms in ascending or descending order.2. Note the term occurring maximum times.3. This term is Mode.

Discrete series: Here the variable with the highest

frequency is the Mode. This method is known as inspection method and has its own limitations.

Eg:

X: 5 10 15 20 25 30 35 40 45

f: 1 3 4 9 11 12 3 2 2

Grouping Table Frequencies are taken. Frequencies are added in twos. Leaving first item frequencies are added

in twos. Frequencies are added in threes. Leaving first frequency, frequencies are

added in threes. Leaving first two frequencies, frequencies

are added in threes.

Analysis Table

Note highest total in each column. Note the variable in each column

corresponding to that total. Check if that total is of individual term or

more ( 2 or 3) terms. If the total consists of 2 or more

frequencies, all such variables have to be marked as √.

Count √ marks in each column. Variable with max. marks denotes mode.

Continuous Series :Z = L + f1 – f0 x i

2f1 – f0 – f2

Where, L = Lower limit of Modal interval f = Frequency corresponding to

modal interval f = Frequency preceding interval f = Frequency succeeding interval i = Length of interval

Calculate mode for the following data:

X: 0-10 10-20 20-30 30-40 40-50 50-60 60-70

f: 4 13 21 44 33 22 7

Uses Of Mode Mode is the term that occurs most in

the series hence it is not an isolated value like mean that may not be there in the series.

It is not affected by extreme values hence is a good representative of series.

It can be found graphically too. With only a single glance at data we

can find its value. It is simplest.

Limitations Of Mode Mode can not be determined if the series

is bimodal or multimodal. Mode is most affected by fluctuation in

sampling. It is not capable of further algebraic

treatment. It is impossible to find the combined mode of some series as in case of mean.

If the no. of terms is too large only then we can consider it as the representative value.

Relation Between X, M and Z:

Z = 3 M – 2 X Symmetrical: In case of symmetrical

series, the mean, median and mode coincide. i.e. Z = M = X.

Positive skewed: If the tail is towards right. i.e. Z > M > X.

Negative Skewed: If the tail is towards left. i.e. Z < M < X