Consider the following for the next two (02) items that follow :
A frequency distribution is as follows:
Marks 18-26 27-35 36-44 45-53 54-62 63-71 72-80 Number
of
students5 7 10 15 8 3 2
What is the median of the distribution ?
Median Calculation for Frequency Distribution
The question asks us to find the median value for a given frequency distribution of marks obtained by students. The median represents the middle value of a dataset when it's ordered. For grouped data like this, we use a specific formula.
Steps to Calculate the Median
- Prepare the Data: First, we need to ensure the class intervals are continuous. The given intervals (e.g., 18-26, 27-35) have gaps. We convert them into class boundaries by subtracting 0.5 from the lower limit and adding 0.5 to the upper limit of each class. We also need the cumulative frequencies.
- Calculate Total Frequency (N): Sum up the number of students in all classes.
- Find the Median Class: Determine the position of the median, which is at \(\frac{N}{2}\). The median class is the class interval where the cumulative frequency first equals or exceeds this value.
- Apply the Median Formula: Use the formula for the median of a grouped frequency distribution.
Data Preparation and Calculations
Let's organize the data, calculate class boundaries, and find the cumulative frequencies (CF).
| Marks (Class Interval) | Number of Students (Frequency, f) | Class Boundaries | Cumulative Frequency (CF) |
| 18-26 | 5 | 17.5 - 26.5 | 5 |
| 27-35 | 7 | 26.5 - 35.5 | 12 (5 + 7) |
| 36-44 | 10 | 35.5 - 44.5 | 22 (12 + 10) |
| 45-53 | 15 | 44.5 - 53.5 | 37 (22 + 15) |
| 54-62 | 8 | 53.5 - 62.5 | 45 (37 + 8) |
| 63-71 | 3 | 62.5 - 71.5 | 48 (45 + 3) |
| 72-80 | 2 | 71.5 - 80.5 | 50 (48 + 2) |
Determining Median Class Parameters
- Total number of students, \(N = 50\).
- The position of the median is \(\frac{N}{2} = \frac{50}{2} = 25\).
- Looking at the cumulative frequencies (CF): 5, 12, 22, 37, 45, 48, 50. The cumulative frequency 37 is the first value greater than or equal to 25.
- Therefore, the median class is the interval corresponding to CF = 37, which is 45-53. Using class boundaries, this is the 44.5 - 53.5 interval.
- From the median class (44.5 - 53.5):
- The lower boundary (\(L\)) is 44.5.
- The frequency (\(f\)) of the median class is 15.
- The cumulative frequency (\(CF\)) of the class *preceding* the median class is 22.
- The class width (\(w\)) is \(53.5 - 44.5 = 9\).
Applying the Median Formula
The formula for the median of a grouped frequency distribution is:
Median = \(L + \frac{\frac{N}{2} - CF}{f} \times w\)
Substitute the values we found:
Median = \(44.5 + \frac{25 - 22}{15} \times 9\)
Median = \(44.5 + \frac{3}{15} \times 9\)
Median = \(44.5 + \frac{1}{5} \times 9\)
Median = \(44.5 + 0.2 \times 9\)
Median = \(44.5 + 1.8\)
Median = \(46.3\)
Conclusion
The median marks for the given distribution of students is 46.3.
Similar Questions
- 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 \(u\)?
- What is the value of \(v\)?
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 :
