Finding the Mean of Incomplete Distribution with Median Given

Given information:
- Total number of pupils = 100
- Median mark = 32

To find:
- Mean mark of the distribution

Method:
1. Determine the number of pupils below and above the median mark
2. Use the formula for finding the median of a distribution to determine the value of the missing frequency
3. Calculate the sum of all the marks in the distribution using the mean formula
4. Use the sum of marks and the total number of pupils to calculate the mean mark

Detailed Steps:

1. Determine the number of pupils below and above the median mark
- Since the total number of pupils is 100, the median pupil will be the 50th pupil
- If the median mark is 32, this means that the 50th pupil scored 32 marks
- We need to find out how many pupils scored marks below and above the median mark
- To do this, we need to consider the cumulative frequency of the distribution
- Let's assume that the frequency of marks below the median is x and the frequency of marks above the median is y
- Then, the cumulative frequency of the 50th pupil will be x + y + 1 (since the 50th pupil is included in the distribution)
- We know that the median mark is 32, so we can use this information to set up an equation:
x + 1/2 = 50
Solving for x, we get x = 49.5
- Since we cannot have a fractional frequency, we can assume that 49 pupils scored marks below the median and 51 pupils scored marks above the median

2. Use the formula for finding the median of a distribution to determine the value of the missing frequency
- The formula for finding the median of a distribution is:
Median = L + ((n/2 - F)/f) * w
where L is the lower limit of the median class, n is the total number of pupils, F is the cumulative frequency of the class before the median class, f is the frequency of the median class, and w is the width of the median class
- In this case, we don't know the frequency of the median class, but we know all the other values
- We can use the formula to find the missing frequency:
32 = L + ((50 - 49)/f) * w
where L = lower limit of the median class, w = width of the median class
- We don't know L or w, but we can assume that the median class is symmetrical around the median mark
- This means that the lower limit of the median class will be 31.5 (since the 49th pupil scored 31 marks) and the upper limit of the median class will be 32.5 (since the 51st pupil scored 33 marks)
- Therefore, the width of the median class is 1 mark (32.5 - 31.5)
- Substituting these values into the formula, we get:
32 = 31.5 + ((50 - 49)/f) * 1
Solving for f, we get f = 2

3. Calculate the sum of all the marks in the distribution using the mean formula
- The mean formula is:
Mean = (Sum of marks) / (Number of pupils)
- We don't

Mean =32