 ## Measures of Dispersion

Subject: Optional Mathematics #### Overview

The measures of any set of data show the range of the data or the amount of variation is known as variability or dispersion. It is already mentioned earlier that if the values of data in a set are widely dispersed then the different measures of central tendency cannot represent the values of the data accurately.

### Measures of Dispersion

The measures of any set of data show the range of the data or the amount of variation is known as variability or dispersion.
It is already mentioned earlier that if the values of data in a set are widely dispersed then the different measures of central tendency cannot represent the values of the data accurately. Likewise, it has also been mentioned that if the data are close together then the measures of central tendency may represent each value of data in the set more precisely. While working on various tasks using statistics, it is necessary to find the different measures of dispersion. Here, the following four measures of dispersion are included:

• Range
• Mean deviation
• Quartile Deviation
• Standard Deviation

#### Range

The difference between the highest and the lowest values of the variable of any set of data is called the range. The range tells us how far the values of data are spread out.  In any variation,
If, the Largest item = L
the smallest item = S, then
Range (R) = L - S

Coefficient of Range
A range is an absolute value. Hence, to compare variations between two distributions we generally use the relative value which is calculated using the following formula. This relative value is called the coefficient of range.
Coefficient of Range = $\frac{L - S}{L + S}$

Merits of range

• Rigidly identified
• Simple understand and easy to estimate
• Required minimum time for variability

Demerits of range

• Changed by variation of sampling
• Changed by extreme value
• Cannot be used for open-end distributions
• Not suitable for more mathematical treatment

#### Quartile Deviation

The difference between the first quartile (Q1) and the third quartile (Q3) is known as the interquartile range. The half of the interquartile range is called semi- interquartile range or quartile deviation.
Thus, if Q1and Q3 represent the first and the third quartile respectively,
(Q.D.) = $\frac {Q_3 - Q_1}2$

To find the quartile deviation according to this formula, we need to find Q1and Q3. For this, let us recall the formulas to find Q1 and Q3.
For individual and discrete series:
Q1 = ($\frac{N+1}{4}$)th term
Q3 = [$\frac {3(N+1)}{4}$]th term
where, N = no. of terms in the series.

Similarly,
for continuous series, the quartiles are found as follows:
Q1 = ($\frac N4$)th term
Q3 = ($\frac {3N}4$)th term
From this the class interval in which Q1 and Q3 lies can be found. To find the quartiles the following formulas are used:
Q1 = L + $\frac {\frac N4- c.f.}f$ × i
where, L = lower limit of Q1 class
c.f. = cumulative frequency of the class preceding the class containing Q1
f = frequency of the class
i = class size

Similarly,
Q3 = L + $\frac {\frac {3N}4- c.f.}f$ × i

Coefficient of Quartile Deviation
As the quartile deviation is an absolute value, we need to find the coefficient of quartile deviation to compare this with other data.
Coefficient of Quartile Deviation = $\frac {Q_3 - Q_1}{Q_3 + Q_1}$

Merits of Quartile Deviation

• Rigidly described
• Simple to understand and easy to estimate
• Not changed by ultimate values
• Useful to study dispersion in open-end series

Demerits of Quartile Deviation

• Not based on all observation
• Affected by the fluctuations of sampling
• Not competent of extra mathematical practice
##### Things to remember
1. The measures of any set of data show the range of the data or the amount of variation is known as variability or dispersion.
2. The difference between the highest and the lowest values of the variable of any set of data is called the range.
3. The difference between the first quartile (Q1) and the third quartile (Q3) is known as the interquartile range. The half of the interquartile range is called semi- interquartile range or quartile deviation.
4. Coefficient of Range = $\frac{L - S}{L + S}$
5. Quartile deviation (Q.D.) = $\frac {Q_3 - Q_1}2$
• It includes every relationship which established among the people.
• There can be more than one community in a society. Community smaller than society.
• It is a network of social relationships which cannot see or touched.
• common interests and common objectives are not necessary for society.
##### Videos for Measures of Dispersion

Soln:
Here, Largest item (L) = 42

Smallest item (S) = 28

∴ Range(R) = L - S = 42 kg - 28 kg. Ans.

Soln:
Here, Largest item (L) = Rs. 400

Smallest item (S) = Rs. 100

∴ Range(R) = L - S = Rs. 400 - Rs. 100 = Rs. 300 Ans.

Soln:
Here, Largest mark (L) = 60

Smallest mark (S) = 40

∴ Range (R) = L - S = 60 - 40 = 20 Ans.

Soln:
Here, Largest number (L) = 14

Smallest number (S) = 10

∴ Range (R) = L - S = 14 - 10 = 4 Ans.

Soln:
Here, Largest C.I. (80 - 90) and

Smallest C.I (30 -40)

So, Range (R) = upper limit of largest class - lower limit of smallest class

= 90 - 30 = 60 Ans.

Soln:

Here, Largest C.I (40-50)

Smallest C.I(0 - 10)

So, Range (R) = upper limit of largest class - lower limit of smallest class

= 50 - 0 = 50 Ans.

Soln:
Here, Largest term (L) = 210mm

Smallest term (S) = 150mm.

∴ Range (R) = L - S = 210mm - 150mm = 60mm.

∴Cofficient of range

=$\frac{L - S}{L + S}$ =$\frac{210kg - 150kg}{210kg + 150kg}$ =$\frac{60kg}{360kg}$ =$\frac{1}{6}$ Ans.

Soln:

Here, Largest weight (L) = 41kg

Smallest weight (S) = 30kg

∴ Range (R) = L - S = 41kg. 30kg. = 11kg.

∴ Coefficient of Range:

$\frac{L - S}{L + S}$ =$\frac{41kg - 30kg}{41kg + 30kg}$ = $\frac{11kg}{71kg}$ =$\frac{11}{71}$

Soln:

Here, Largest C. I = 40 - 50

Smallest C. I = 0 - 10

So, Range (R) = Upper limit of largest class - Lower limit of smallest class.

= 50 - 0 = 50 Ans.

∴ Coeff. of Range =$\frac{L - S}{L + S}$ =$\frac{50 - 0}{50 + 0}$ = 1 Ans.

Soln:

Here, arranging the given data into ascending order:

12, 14, 17, 18, 22, 26, 30, 32, 34, 35, 41.

No. of terms (n) = 11

∴ First quartile (Q1) =$\frac{n + 1}{4}$ nth term =$\frac{11 + 1}{4}$nth term.

= 3th term = third term = 17

Third quartile (Q3) =$\frac{3}{4}$(n + 1) nth term =$\frac{3}{4}$(11 + 1)nth term

= 9 nthterm = 34

∴ Quartile deviation (Q. D) =$\frac{Q3 - Q1}{2}$ =$\frac{34 - 17}{2}$ =$\frac{17}{2}$ = 8.5

∴Coeff. of Q. D. =$\frac{Q3 - Q1}{Q3 + Q1}$ =$\frac{34 - 17}{34 + 17}$ =$\frac{17}{51}$ =$\frac{1}{3}$. Ans.

Soln:

Here, no of terms (n) = 10

∴ First quartile (Q1) =$\frac{n + 1}{4}$nth term =$\frac{10 + 1}{4}$nth term

=$\frac{11}{4}$nth term = 2.75 nth term

= 2nd term + (3rd term - 2nd term)× 0.75

=30 +(45 - 30)×0.75 = 30 +15× 0.75

= 41.25

∴ Third quartile (Q3) =$\frac{3}{4}$(n + 1) nth term =$\frac{3}{4}$ (10 + 1) nth term

=$\frac{33}{4}$nth term = 8.25 nth term.

= 8th term + (9th term - 8th term)× 0.25

= 115 + (118 - 115)× 0.25

= 115 + 3× 0.25

= 115 + 0.75

= 115.75

∴ Quartile deviation (Q. D) =$\frac{Q_3 - Q_1}{2}$ =$\frac{115.75 - 41.25}{2}$ = 37.25 Ans.

∴ Coeff of Q. D.

$\frac{Q_3 -Q_1}{Q_3 +Q_1}$=$\frac{115.75-41.25}{115.75+41.25}$ =$\frac{74.5}{157}$ =$\frac{149}{314}$. ans

Soln:

Frequency table:

 Age(years) (X) Frequency (f) No. of student Cumulative frequency (c. f) 12 5 5 13 5 5+5=10 14 5 10+5=15 15 6 15+6=21 16 1 21+1=22 18 1 22+1=23 N = 23

No. of term N = 6

First quartile (Q1) = nth term $\frac{N + 1}{4}$ th term =$\frac{23 + 1}{4}$ th term =$\frac{24}{4}$ th term = 6 th term = 13.

∴ Third quartile (Q3) =$\frac{3}{4}$ (N + 1) th term =$\frac{3}{4}$× 24 th term = 18 th term = 15.

∴ Quartile deviation (Q. D) =$\frac{Q_3 - Q_1}{2}$ =$\frac{15 - 13}{2}$ =$\frac{2}{2}$ = 1.

∴ Coeff. of Q .D =$\frac{Q_3 - Q_1}{Q_3 + Q_1}$ =$\frac{15 -13}{15 +13}$ =$\frac{2}{28}$ =$\frac{1}{14}$ = 0.07 Ans.

Soln:

From the given data:

 Marks (x) No. of students (f) Cumulative frequency (c. f) 50 16 16 60 12 28 75 8 36 82 5 41 90 3 44 91 1 45 N = 45

No. of terms N = 45

First quartile(Q3) =$\frac{N + 1}{4}$ th term =$\frac{45 + 1}{4}$ th term =$\frac{23}{4}$ th term = 11.5 th term = 50

[ Here of greater then 11.5 in 16 which mark is 50.]

Third quartile (Q3) =$\frac{3}{4}$ (N + 1) th term =$\frac{3}{4}$ (45 + 1) th term

=$\frac{69}{2}$ th term = 34.5 th term = 75.

∴ Quartile deviation (Q. D) =$\frac{Q_3 - Q _1}{2}$ =$\frac{75 - 50}{2}$ =$\frac{25}{2}$ = 12.5 Ans.

Soln:

 Marks (x) No. of students (f) Cumulative frequency (c. f) 20-30 5 5 30-40 15 20 40-50 10 30 50-60 8 38 60-70 6 44 70-80 2 46 N = 46

C. I lying Q1 is

=$\frac{N}{4}$ th term =$\frac{46}{4}$ th term = 11.5 th term

= 30-40

[ Here C. F. just greater than 11.5 is 20 which is in 30-40]

Using formula,

Q1 =L +$\frac{\frac{N}{4} - c.f.}{f}$× i

Here, L = 30, $\frac{N}{4}$ = 11.5, c. f. = 5, f = 15 and i = 10

∴ Q1 = 30 +$\frac{11.5 - 5}{15}$× 10 = 30 + 4.33 = 34.33

C. I. lying Q3 is

=$\frac{3}{4}$ N th term =$\frac{3}{4}$× 46 th term =$\frac{69}{2}$ th term = 34.5 th term = 50-60

Using formula, Q3 = L +$\frac{\frac{N}{4} - c.f.}{f}$× i

Here, L = 50, $\frac{3}{4}$×N =$\frac{3}{4}$× 46 =$\frac{69}{2}$

c. f. = 30, f = 8, i = 10

∴Q3 = 50 +$\frac{\frac{69}{2} - 30}{8}$× 10 = 50 +$\frac{9}{16}$× 10 =50 +$\frac{9}{8}$×5 =$\frac{445}{8}$ = 55.625

∴ Quartile deviation (Q. D) =$\frac{Q_3 - Q_1}{2}$ =$\frac{55.62 -34.33}{2}$ =$\frac{21.29}{2}$ = 10.645 Ans.