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:
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
Demerits of range
The difference between the first quartile (Q_{1}) and the third quartile (Q_{3}) is known as the interquartile range. The half of the interquartile range is called semi- interquartile range or quartile deviation.
Thus, if Q_{1}and Q_{3} 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 Q_{1}and Q_{3}. For this, let us recall the formulas to find Q_{1} and Q_{3}.
For individual and discrete series:
Q_{1} = (\(\frac{N+1}{4}\))^{th} term
Q_{3} = [\(\frac {3(N+1)}{4}\)]^{th} term
where, N = no. of terms in the series.
Similarly,
for continuous series, the quartiles are found as follows:
Q_{1} = (\(\frac N4\))^{th} term
Q_{3} = (\(\frac {3N}4\))^{th} term
From this the class interval in which Q_{1} and Q_{3} lies can be found. To find the quartiles the following formulas are used:
Q_{1} = L + \(\frac {\frac N4- c.f.}f\) × i
where, L = lower limit of Q_{1} class
c.f. = cumulative frequency of the class preceding the class containing Q_{1}f = frequency of the class
i = class size
Similarly,
Q_{3} = 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
Demerits of Quartile Deviation
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 item(L) = 15^{o}C
Smallest item (S) = 6^{o}C
∴ Range(R) = L -S = 15^{o}C - 6^{o}C = 9^{o}C.
Find the range:
Marks Obtained | 40 | 45 | 50 | 55 | 60 |
No of students | 2 | 2 | 5 | 3 | 2 |
Soln:
Here, Largest mark (L) = 60
Smallest mark (S) = 40
∴ Range (R) = L - S = 60 - 40 = 20 Ans.
Find the range:
Number | 10 | 11 | 12 | 13 | 14 |
Frequency | 3 | 12 | 13 | 15 | 1 |
Soln:
Here, Largest number (L) = 14
Smallest number (S) = 10
∴ Range (R) = L - S = 14 - 10 = 4 Ans.
Find the range of the following grouped data:
Age(yrs) | 30-40 | 40-50 | 50-60 | 60-70 | 70-80 | 80-90 |
No of students | 6 | 10 | 16 | 14 | 10 | 5 |
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.
Find the range of following data:
Marks obtained | 0-10 | 10-20 | 20-30 | 30-40 | 40-50 |
No of students | 10 | 20 | 18 | 32 | 21 |
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.
Find the range and the coefficient of follwing data:
Weight(kg) | 30 | 32 | 35 | 38 | 32 | 38 | 41 |
No. of Students | 5 | 5 | 6 | 10 | 8 | 4 | 2 |
Soln:
Here, Largest weight (L) = 41kg
Smallest weight (S) = 30kg
∴ Range (R) = L - S = 41kg. 30kg. = 11kg.
∴ Coefficient of Range:
\(\frac{L - S}{L + S}\) =\(\frac{41kg - 30kg}{41kg + 30kg}\) = \(\frac{11kg}{71kg}\) =\(\frac{11}{71}\)
Find the range and its coefficient:
Marks obtained | 0-10 | 10-20 | 20-30 | 30-40 | 40-50 |
No of students | 1 | 3 | 5 | 20 | 3 |
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 (Q_{1}) =\(\frac{n + 1}{4}\) n^{th} term =\(\frac{11 + 1}{4}\)n^{th} term.
= 3^{th} term = third term = 17
Third quartile (Q_{3}) =\(\frac{3}{4}\)(n + 1) n^{th} term =\(\frac{3}{4}\)(11 + 1)n^{th} term
= 9 n^{th}term = 34
∴ Quartile deviation (Q. D) =\(\frac{Q_{3} - Q_{1}}{2}\) =\(\frac{34 - 17}{2}\) =\(\frac{17}{2}\) = 8.5
∴Coeff. of Q. D. =\(\frac{Q_{3} - Q_{1}}{Q_{3} + Q_{1}}\) =\(\frac{34 - 17}{34 + 17}\) =\(\frac{17}{51}\) =\(\frac{1}{3}\). Ans.
Soln:
Here, no of terms (n) = 10
∴ First quartile (Q_{1}) =\(\frac{n + 1}{4}\)n^{th} term =\(\frac{10 + 1}{4}\)n^{th} term
=\(\frac{11}{4}\)n^{th} term = 2.75 n^{th} term
= 2nd term + (3rd term - 2nd term)× 0.75
=30 +(45 - 30)×0.75 = 30 +15× 0.75
= 41.25
∴ Third quartile (Q_{3}) =\(\frac{3}{4}\)(n + 1) n^{th} term =\(\frac{3}{4}\) (10 + 1) n^{th} term
=\(\frac{33}{4}\)n^{th} term = 8.25 n^{th} term.
= 8^{th} term + (9^{th} term - 8^{th} 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
The smallest term of a data is 27 and range is 58.Find the coefficient of range.
The largest term of a data is 60 and the range is 20.Find the coefficient of range of the data.
In a data, range is 17 and coefficient of range is 0.2. Find the smallest term of the data.
In a data, range is 60 and coefficient of range is 0.5. Find the smallest term of the data.
In a data, range is 20 and coefficient of range is 0.2. Find the greatest term of the data.
In a data, range is 58 and coefficient of range is (frac{29}{56}). Find the greatest term of the data.
Find the range and the coefficient of range.
35,27,32,40,50,55,75,72,82,85,67
In a data the smallest term is 20% of the greatest term.If their sum is 60 then find the range and coefficient of the range.
In a data the ratio of the smallest term to the greatest term is 1:5.If their range is 40, find the coefficient of range and also,the numbers.
The following data are in ascending order.If range and coefficient of this data are 40 and(frac{8}{9}) respectively, the values of a and b.
a,10,15,20,25,35,40,b
Find the first quartile from the following data:
40,20,30,10,16,12,8
12
10
9
6
Find the first quartile from the following data:
40,20,30,10,16,12,8
9
10
6
12
Find the first quartile from the following data:
16,25,10,30,35,8,12
25
20
10
15
Find the third quartile from the given data:
50,40,55,60,61,70,49
61
65
55
60
Find the third quartile from the given data:
18,20,17,24,19,21,23
14
23
35
30
Out of a total observation arranged in ascending order, the 12^{th} and 13^{th} observations are 25 and 30 respectively.What is the value of Q_{1}?
25
28
26
27
Out of a total 31 observation arranged in ascending order, the 7^{th},8^{th} and 9^{th} observations are 20,24 and 28 respectively.What is the value of Q_{1}?
30
24
12
20
In a data, the quartile deviation and its coefficient are 15 and 3/7 respectively.Find the first quartile.
25
20
25
10
In a data, the quartile deviation and its coefficient are 14 and 7/22 respectively.Find the first quartile.
10
30
25
15
The third quartile of a data is 15.If the coefficient of quartile deviation is (frac{1}{4}) find the first quartile and the inter-quartile range of the data.
2 and 8
15 and 20
5 and 2
13 and 2
The third quartile of a data is 58.If the coefficient of quartile deviation is (frac{7}{22}) find the first quartile and the inter-quartile range of the data.
40 and 13
25 and 35
20 and 12
30 and 28
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