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Q2:
If the median of the distribution given below is 28.5, find the values of $x$ and $y$.
If the median of the distribution given below is 28.5, find the values of $x$ and $y$.
Solution :
Given: A frequency distribution table with a total frequency $N = 60$ and a median value of $28.5$.
| Class Interval | Frequency ($f$) |
|---|---|
| 0-10 | 5 |
| 10-20 | $x$ |
| 20-30 | 20 |
| 30-40 | 15 |
| 40-50 | $y$ |
| 50-60 | 5 |
| Total | 60 |
To find: The values of $x$ and $y$.
Step 1: Constructing the Cumulative Frequency Table
| Class Interval | Frequency ($f$) | Cumulative Frequency ($cf$) |
|---|---|---|
| 0-10 | 5 | 5 |
| 10-20 | $x$ | $5 + x$ |
| 20-30 | 20 | $25 + x$ |
| 30-40 | 15 | $40 + x$ |
| 40-50 | $y$ | $40 + x + y$ |
| 50-60 | 5 | $45 + x + y$ |
Step 2: Establishing the first linear equation
Since the total frequency is given as $60$, we have:
$45 + x + y = 60$$x + y = 60 - 45$
$x + y = 15$ --- (Equation 1)
Step 3: Identifying the Median Class
The median is given as $28.5$. Since $28.5$ lies between $20$ and $30$, the median class is $20-30$.
From the median class $20-30$:
- Lower limit ($l$) = $20$
- Frequency of median class ($f$) = $20$
- Cumulative frequency of the class preceding the median class ($cf$) = $5 + x$
- Class size ($h$) = $10$
- $N/2 = 60/2 = 30$
Step 4: Applying the Median Formula
The formula for the median is: $\text{Median} = l + \left( \frac{\frac{N}{2} - cf}{f} \right) \times h$
Substituting the known values:
$28.5 = 20 + \left( \frac{30 - (5 + x)}{20} \right) \times 10$$28.5 - 20 = \left( \frac{30 - 5 - x}{20} \right) \times 10$
$8.5 = \frac{25 - x}{2}$
$8.5 \times 2 = 25 - x$
$17 = 25 - x$
$x = 25 - 17$
$x = 8$
Step 5: Solving for $y$
Substitute $x = 8$ into Equation 1:
$8 + y = 15$$y = 15 - 8$
$y = 7$
Final Answer: The values are $x = 8$ and $y = 7$.
More Questions from Class 10 Mathematics Statistics 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.
- 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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