For the following two (02) items: Consider the following distribution having median value 24:Marks Number of Students Less than 10 5 Less than 20 30 Less than 30 $30+k$ Less than 40 $48+k$ Less than 50 $55+k$
Understanding the Problem: Median in a Cumulative Frequency Distribution
The question provides a cumulative frequency distribution table showing the number of students who scored less than certain marks. We are given that the median value for this distribution is 24. Our goal is to find the value of the unknown '\(k\)'.
Key Concepts: Median and Cumulative Frequency
- Median: The median is the middle value in a dataset that separates the higher half from the lower half. For grouped data, it's calculated using a specific formula.
- Cumulative Frequency Distribution (Less than type): This type of distribution shows the total number of observations up to a certain value. The table given is in this format.
- Median Value: The median value (24) tells us that half of the students scored less than 24 and the other half scored more than 24. Since 24 falls between 20 and 30, the "median class" is the interval 20-30.
Setting up the Data
Let's organize the data and calculate the frequencies for each class interval. The total number of students (\(N\)) is the final cumulative frequency.
| Marks Range | Cumulative Frequency (CF) | Frequency (f) |
|---|---|---|
| Less than 10 | 5 | 5 |
| Less than 20 | 30 | \(30 - 5 = 25\) |
| Less than 30 | \(30+k\) | \((30+k) - 30 = k\) |
| Less than 40 | \(48+k\) | \((48+k) - (30+k) = 18\) |
| Less than 50 | \(55+k\) | \((55+k) - (48+k) = 7\) |
From the table:
- The median value is 24.
- This means the median class is 20-30.
- The lower boundary of the median class (\(L\)) is 20.
- The cumulative frequency of the class preceding the median class (\(CF\)) is 30.
- The frequency of the median class (\(f\)) is \(k\).
- The width of the median class (\(w\)) is \(30 - 20 = 10\).
- The total number of students (\(N\)) is \(55+k\).
- The position of the median is \(N/2 = (55+k)/2\).
Applying the Median Formula
The formula for calculating the median for grouped data is:
\(\text{Median} = L + \left(\frac{N/2 - CF}{f}\right) \times w\)
Step-by-Step Calculation for k
- Substitute the known values into the median formula: \(24 = 20 + \left(\frac{(55+k)/2 - 30}{k}\right) \times 10\)
- Isolate the fraction term: Subtract 20 from both sides. \(24 - 20 = \left(\frac{(55+k)/2 - 30}{k}\right) \times 10\) \(4 = \left(\frac{(55+k)/2 - 30}{k}\right) \times 10\)
- Simplify the equation: Divide both sides by 10. \(\frac{4}{10} = \frac{(55+k)/2 - 30}{k}\) \(0.4 = \frac{(55+k)/2 - 30}{k}\)
- Multiply both sides by \(k\): \(0.4k = \frac{55+k}{2} - 30\)
- Simplify the right side by finding a common denominator: \(0.4k = \frac{55+k - (30 \times 2)}{2}\) \(0.4k = \frac{55+k - 60}{2}\) \(0.4k = \frac{k - 5}{2}\)
- Multiply both sides by 2: \(2 \times 0.4k = k - 5\) \(0.8k = k - 5\)
- Solve for \(k\): Rearrange the terms to group \(k\) together. \(5 = k - 0.8k\) \(5 = 0.2k\) \(k = \frac{5}{0.2}\) \(k = \frac{5}{1/5}\) \(k = 5 \times 5\) \(k = 25\)
Verification
Let's check if \(k=25\) yields a median of 24.
- If \(k=25\), the total number of students (\(N\)) is \(55+25 = 80\).
- The median position is \(N/2 = 80/2 = 40\).
- The median class is 20-30.
- \(L=20\), \(CF=30\), \(f=k=25\), \(w=10\).
- Median = \(20 + \left(\frac{40 - 30}{25}\right) \times 10\)
- Median = \(20 + \left(\frac{10}{25}\right) \times 10\)
- Median = \(20 + (0.4) \times 10\)
- Median = \(20 + 4 = 24\).
The calculated median matches the given median value. Therefore, the value of \(k\) is 25.
Similar Questions
What is the median of the distribution ?
- What is the mode of the distribution?
- Consider the following data :
110, 41, 43, 95, 127, 99, 61, 92, 71, 93, 110, 36.
If 93 is replaced by 94, then consider the following statements:
1. The difference between new median and old median is 1.
2. The difference between new mean and old mean is less than 0·1.
3. The difference between new mode and old mode is zero.
Which of the statements given above are correct? - Which measure of central tendency is least affected by the presence of extreme observations in the data?
- To find the average ratio like price/unit, work done/hour, kilometre/hour under certain conditions, the suitable measure of central tendency applicable is
- The frequency distribution of marks of 100 candidates in a particular examination is as follows:What are the average marks of the candidates?
Marks Number of Candidates More than 10 100 More than 20 75 More than 30 60 More than 40 40 - The arithmetic mean of 200 observations is 60. If 5 is multiplied to each observation, then what will be the new arithmetic mean?
- A distribution consists of 3 components with frequencies 45, 40 and 55 having their means 2, 2.5 and 2 respectively. What is the mean of the combined distribution?
- Which one of the following is a positional average?
- What is the value of \(u\)?
Important Questions from Elementary Statistics
In a colony 5 families have 1 child, 7 families have 2 children, 8 families have 3 children and 3 families have 4 children.What is the mode of the number of children.
For which set of numbers do the mean, median and mode all have the same value?
The median of 5, 8, 25, 22, 34, 18 is
Find the mean of first 5 two-digit multiples of 4.
The difference between the upper limit and the lower limit of a class is called :
