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The median of the following data is ...

The median of the following data is
`{:("Marks obtained ","No. of students "),("less than 20",0),("less than 30",4),("less than 40",16),("less than 50",30),("less than 60",46),("less than 70",66),("less than 80",82),("less than 90",92),("less than 100",100):}`

A

1) 62

B

2) 64

C

3) 63

D

4) 52

Text Solution

AI Generated Solution

The correct Answer is:
To find the median of the given cumulative frequency distribution, we will follow these steps: ### Step 1: Construct a Frequency Table From the cumulative frequency data, we will create a frequency table. The cumulative frequencies given are: - Less than 20: 0 - Less than 30: 4 - Less than 40: 16 - Less than 50: 30 - Less than 60: 46 - Less than 70: 66 - Less than 80: 82 - Less than 90: 92 - Less than 100: 100 Now, we will find the frequency for each class interval by subtracting the cumulative frequency of the previous interval from the current one. | Class Interval | Cumulative Frequency | Frequency (f) | |----------------|---------------------|----------------| | Less than 20 | 0 | 0 | | 20 to 30 | 4 | 4 - 0 = 4 | | 30 to 40 | 16 | 16 - 4 = 12 | | 40 to 50 | 30 | 30 - 16 = 14 | | 50 to 60 | 46 | 46 - 30 = 16 | | 60 to 70 | 66 | 66 - 46 = 20 | | 70 to 80 | 82 | 82 - 66 = 16 | | 80 to 90 | 92 | 92 - 82 = 10 | | 90 to 100 | 100 | 100 - 92 = 8 | ### Step 2: Calculate Total Frequency (n) Now, we sum up the frequencies to find \( n \): \[ n = 0 + 4 + 12 + 14 + 16 + 20 + 16 + 10 + 8 = 100 \] ### Step 3: Find \( n/2 \) Next, we calculate \( n/2 \): \[ n/2 = 100/2 = 50 \] ### Step 4: Identify the Median Class We need to find the cumulative frequency just greater than \( n/2 \) (which is 50). From the cumulative frequency table: - The cumulative frequency just less than 50 is 46 (for the interval 50 to 60). - The cumulative frequency just greater than 50 is 66 (for the interval 60 to 70). Thus, the median class is **60 to 70**. ### Step 5: Identify Values for the Median Formula For the median class (60 to 70): - \( l = 60 \) (lower boundary of the median class) - \( f = 20 \) (frequency of the median class) - \( F = 46 \) (cumulative frequency of the class before the median class) - \( h = 10 \) (class interval width, which is 70 - 60) ### Step 6: Apply the Median Formula The formula for the median is: \[ \text{Median} = l + \frac{n/2 - F}{f} \times h \] Substituting the values we have: \[ \text{Median} = 60 + \frac{50 - 46}{20} \times 10 \] Calculating inside the brackets: \[ = 60 + \frac{4}{20} \times 10 \] \[ = 60 + 0.2 \times 10 \] \[ = 60 + 2 \] \[ = 62 \] ### Final Answer Thus, the median of the given data is **62**. ---
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