5.42 I FUNDAMENTALS OF BUSINESS MATHEMATICS AND STATISTICSMeasures of Central Tendency and Measures of Dispersion
- If first of two groups has 100 items and mean 45 and combined group has 250 items and mean 5/, find
the mean of second group. [Ans. 55] - Find the median of the following distribution
Weight (kg) : 65 66 67 68 [Ans. 67]
No. of students : 5 15 17 4 - Find G.M. of 3, 6, 24, 48 [Ans. 12]
- A.M. of two numbers is 25 and their H.M. is 9, find their G.M. [Ans. 15]
- The means of samples of sizes 50 and 75 are 60 and x respectively. If the mean of the combined group
is 54, find x. [Ans. 50] - Find the median of the given distribution :
Value (x) : 1 2 3 4 [Ans. 3]
Frequency (f) : 7 12 18 4 - If each of 3, 48 and 96 occurs once and 6 occurs twice verify that G.M. is greater than H.M.
- Find G M. of 1, 2, 3, 1 12 3,. What will be G.M. if ‘0’ is added to above set of velues?
[Ans. 1 ; 0] - The G.M. of a, 4, 6 is 6, find a [Ans. 9]
- A.M. of a variable x is 100, find the mean of the variable 2x – 50. [Ans. 150]
- The variable x and y are given by y = 2x + 11. If the median of x is 3, find the median of y. [Ans. 17]
5.2 QUARTILE DEVIATION
Quartiles are such values which divide the total number of observations into 4 equal parts. Obviously, there
are 3 quartiles—
(i) First quartile (or Lower quartile): Q 1
(ii) Second quartile, (or Middle quartile) : Q 2
(iii) Third quartile (or Upper quartile): Q 3
The number of observations smaller than Q 1 , is the same as the number lying between Q 1 and Q 2 , or
between Q 2 and Q 3 , or larger than Q 3. For data of continuous type, one-quarter of the observations is
smaller than Q 1 , two-quarters are smaller than Q 2 , and three-quarters are smaller than Q 3. This means that
Q 1 , Q 2 , Q 3 are values of the variable corresponding to ‘less-than’ cumulative frequencies N/4, 2N/4, 3N/
4 respectively. Since, 2N/4 = N/2, it is evident that the second quartile Q 2 is the same as median.
Q 1 < Q 2 < Q 3 ; Q 2 = Median.
Quartiles are used for measuring central tendency, dispersion and skewness. For instance, the second quartile
Q 2 is itself taken as a measure of central tendency, where it is known as Median.
Quartile deviation is defined as half the difference between the upper and the lower quartiles.
Quartile Deviation = Q^32 −Q^1
The difference Q3- Q1 being the distance between the quartiles can also be called inter quartile range;
half of this Semi- inter quartile Range. Thus the name ‘Semi - inter quartile Range’ itself gives the definition
of Quartile Deviation.