Statistics | Exercise 13.3

Question 2

If the median of the distribution given below is 28.5, find the values of xx and yy.

Question diagram 1
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Solution
Understand the Question
  • The sum of all frequencies is given as N=60N = 60, which allows us to set up a linear equation in terms of xx and yy: x+y=15x + y = 15.
  • The median is given as 28.528.5, which lies in the class interval 20−3020 - 30. Therefore, the median class is 20−3020 - 30.
  • Using the median formula for grouped data, Median=l+(N2−cff)×h\text{Median} = l + \left(\dfrac{\frac{N}{2} - cf}{f}\right) \times h, we can solve for xx and subsequently find yy.

Step 1 · Form equation using total frequency

Given total frequency N=60N = 60.Diagram 1

Sum of frequencies:

5+x+20+15+y+5=6045+x+y=60x+y=60−45x+y=15…(1)\begin{aligned} 5 + x + 20 + 15 + y + 5 &= 60 \\[0.6em] 45 + x + y &= 60 \\[0.6em] x + y &= 60 - 45 \\[0.6em] x + y &= 15 \quad \dots (1) \end{aligned}

Step 2 · Identify median class and solve for xx

Since Median =28.5= 28.5, it lies in the class interval 20−3020 - 30. Therefore, the median class is 20−3020 - 30.

From the distribution:

  • Lower limit of median class, L=20L = 20
  • Frequency of median class, f=20f = 20
  • Cumulative frequency of preceding class, cf=5+xcf = 5 + x
  • Class size, h=30−20=10h = 30 - 20 = 10
  • N2=602=30\dfrac{N}{2} = \dfrac{60}{2} = 30

Using the median formula: Median=L+(N2−cff)×h\text{Median} = L + \left(\dfrac{\dfrac{N}{2} - cf}{f}\right) \times h

28.5=20+(30−(5+x)20)×1028.5−20=(30−5−x20)×108.5=(25−x20)×108.5=25−x28.5×2=25−x17=25−xx=25−17x=8\begin{aligned} 28.5 &= 20 + \left(\dfrac{30 - (5 + x)}{20}\right) \times 10 \\[0.6em] 28.5 - 20 &= \left(\dfrac{30 - 5 - x}{20}\right) \times 10 \\[0.6em] 8.5 &= \left(\dfrac{25 - x}{20}\right) \times 10 \\[0.6em] 8.5 &= \dfrac{25 - x}{2} \\[0.6em] 8.5 \times 2 &= 25 - x \\[0.6em] 17 &= 25 - x \\[0.6em] x &= 25 - 17 \\[0.6em] x &= 8 \end{aligned}

Step 3 · Calculate the value of yy

Substitute x=8x = 8 into equation (1)(1):

8+y=15y=15−8y=7\begin{aligned} 8 + y &= 15 \\[0.6em] y &= 15 - 8 \\[0.6em] y &= 7 \end{aligned}
Answer

x=8,y=7x = 8, \quad y = 7

Common Mistakes
  • Sign Error in cfcf: Forgetting parentheses when subtracting cumulative frequency: 30−(5+x)=30−5−x=25−x30 - (5 + x) = 30 - 5 - x = 25 - x. Omitting brackets often leads to the incorrect expression 30−5+x=25+x30 - 5 + x = 25 + x.
  • Wrong Median Class Selection: Attempting to find the median class using N2=30\frac{N}{2} = 30 in the cumulative frequency column (which contains unknowns xx) instead of locating the given median value (28.528.5) directly within the class intervals (20−3020 - 30).

More questions in Exercise 13.3

Q1

The following frequency distribution gives the monthly consumption of electricity of 68 consumers of a locality. Find the median, mean and mode of the data and compare them.

Q2

If the median of the distribution given below is 28.5, find the values of xx and yy.

Q3

A life insurance agent found the following data for distribution of ages of 100 policy holders. Calculate the median age, if policies are given only to persons having age 18 years onwards but less than 60 year.

Q4

The lengths of 40 leaves of a plant are measured correct to the nearest millimetre, and the data obtained is represented in the following table :

Find the median length of the leaves.

(Hint : The data needs to be converted to continuous classes for finding the median, since the formula assumes continuous classes. The classes then change to 117.5 - 126.5, 126.5 - 135.5, . . . , 171.5 - 180.5.)

Q5

The following table gives the distribution of the life time of 400 neon lamps :

Find the median life time of a lamp.

Q6

100 surnames were randomly picked up from a local telephone directory and the frequency distribution of the number of letters in the English alphabets in the surnames was obtained as follows:

Determine the median number of letters in the surnames. Find the mean number of letters in the surnames? Also, find the modal size of the surnames.

Q7

The distribution below gives the weights of 30 students of a class. Find the median weight of the students.

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