5.38 I FUNDAMENTALS OF BUSINESS MATHEMATICS AND STATISTICSMeasures of Central Tendency and Measures of Dispersion
Solution:
Table : Construction of frequency table and hence calculation of median and mode.
Marks f cf class boundaries
10–19 3 3 9.5 – 19.5
20–29 5 8 19.5 – 29.5
30 – 39 13 21 29.5 – 39.5
40 – 49 6 27 39.5 –49.5
50 – 59 3 30 49.5 – 59.5Median = value of N 2 th i.e.,^302 i.e. 15th term
So median class is (29.5 – 39.5)∴ median =29.5+39.5 29.5 13 − (15 8− ) =29.5+^1130 ×^7
= 29.5 + 5.38 = 34.88 marks
Highest frequency is 13 (= f 1 ), f 0 = 5, f 2 = 6
∴ Mode^10
1 0 2= l + f - f ×i
2f - f - f
29.5^1 3 5 10
2 13 5 6
= + − ×
× − −29.5^8 10 29.5 5.33
= + 15 × = + = 34.83 marks.
Advantages of mode :
(i) It can often be located by inspection.
(ii) It is not affected by extreme values. It is often a really typical value.
(iii) It is simple and precise. It is an actual item of the series except in a continuous series.
(iv) Mode can be determined graphically unlike Mean.
Disadvantages of mode :
(i) It is unsuitable for algebraic treatment.
(ii) When the number of observations is small, the Mode may not exist, while the Mean and Median can
be calculated.
(iii) The value of Mode is not based on each and every item of series.
(iv) It does not lead to the aggregate, if the Mode and the total number of items are given.
5.1.7. Empirical Relationship among Mean, Median and Mode
A distribution in which the values of Mean, Median and Mode coincide, is known symmetrical and if the
above values are not equal, then the distribution is said asymmetrical or skewed. In a moderately skewed
distribution, there is a relation amongst Mean, Median and Mode which is as follows :
Mean – Mode = 3 (Mean – Median)
If any two values are known, we can find the other.