For a distribution, the empirical relation is, Mode = 3×Median – 2×Mean. If the Mean and Median satisfy, \(20 \le \text{Mean} \le 25\) and \(20 \le \text{Median} \le 25\), what is the possible range of the Mode?
\(10 \le \text{Mode} \le 35\)
Mode = 3 × Median – 2 × Mean increases with the Median and decreases with the Mean, so the extremes come from the end values of each interval.
Least Mode: take the smallest Median (20) and the largest Mean (25): 3 × 20 – 2 × 25 = 60 – 50 = 10.
Greatest Mode: take the largest Median (25) and the smallest Mean (20): 3 × 25 – 2 × 20 = 75 – 40 = 35.
So the Mode can take values from 10 to 35, i.e. \(10 \le \text{Mode} \le 35\).
Check: with Mean = Median = 20 the Mode is 60 – 40 = 20, and with Mean = Median = 25 it is 75 – 50 = 25; both lie inside 10 to 35, as they must.
Hence, \(10 \le \text{Mode} \le 35\).
Similar Questions
In an exam, marks scored by 16 students of section A (out of 50) are 33, 41, 56, 51, 53, 46, 49, 41, 45, 56, 51, 49, 46, 41, 53, 33 and marks scored by 13 students of section B (out of 50) are 47, 54, 42, 50, 47, 39, 42, 52, 56, 49, 54, 39, 47. Calculate the difference between the mode of section B and section A.
Find the weighted mean of 15, 20 and 25 with weights 2, 3 and 5, respectively.
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A data set has a mean of 15 and a mode of 20. Each value is changed by multiplying by 3 and then adding 5. Assuming the empirical relation between the mean, median, and mode holds, what is the median of the new data set?
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Important Questions from Measures of Central Tendency
A random sample of 20 people is classified in the following table according to their ages:
Age
Frequency
15 – 25
2
25 – 35
4
35 – 45
6
45 – 55
5
55 - 65
3
What is the mean age of this group of people?
The median of the following observations 46, 64, 87, 41, 58, 77, 35, 90, 55, 92, 33 is 58. If 92 is replaced by 99 and 41 by 43 in the above data. The new median is:
If the difference of mode and median is 36, then the difference of median and mean is:
In a Mathematics test 15 students scored 80 marks, 20 students scored 75 marks, 28 students scored 65 marks and 25 students scored 60 marks, mode of the score is:
If the mode of the scores 10, 12, 13, 15, 15, 13, 12, 10, x is 15, then what is the value of x?
