To compute the mean marks of all the students from the given data for 50 schools in a city, we follow these steps:
1. Determine the Mid-Point for Each Marks Range:
- More than 35: The mid-point is set at 37.5.
- 30-35: The mid-point is the average of 30 and 35, which is [tex]\(\frac{30 + 35}{2} = 32.5\)[/tex].
- 25-30: The mid-point is the average of 25 and 30, which is [tex]\(\frac{25 + 30}{2} = 27.5\)[/tex].
- 20-25: The mid-point is the average of 20 and 25, which is [tex]\(\frac{20 + 25}{2} = 22.5\)[/tex].
- 15-20: The mid-point is the average of 15 and 20, which is [tex]\(\frac{15 + 20}{2} = 17.5\)[/tex].
- Less than 15: The mid-point is set at the average of 0 and 15, which is [tex]\(\frac{0 + 15}{2} = 7.5\)[/tex].
So, the mid-points are:
[tex]\[
[37.5, 32.5, 27.5, 22.5, 17.5, 7.5]
\][/tex]
2. List the Frequencies (Number of Schools):
[tex]\[
[200, 250, 300, 200, 150]
\][/tex]
3. Calculate the Total Number of Schools (Frequencies Sum):
[tex]\[
200 + 250 + 300 + 200 + 150 = 1100
\][/tex]
4. Calculate the Sum of Frequency [tex]\(\times\)[/tex] Mid-Point for Each Mark Range:
[tex]\[
\sum (f_i \cdot x_i) = 200 \cdot 37.5 + 250 \cdot 32.5 + 300 \cdot 27.5 + 200 \cdot 22.5 + 150 \cdot 17.5
\][/tex]
Computing each term separately:
[tex]\[
200 \cdot 37.5 = 7500
\][/tex]
[tex]\[
250 \cdot 32.5 = 8125
\][/tex]
[tex]\[
300 \cdot 27.5 = 8250
\][/tex]
[tex]\[
200 \cdot 22.5 = 4500
\][/tex]
[tex]\[
150 \cdot 17.5 = 2625
\][/tex]
Summing these products:
[tex]\[
7500 + 8125 + 8250 + 4500 + 2625 = 31000
\][/tex]
5. Calculate the Mean Marks:
[tex]\[
\text{Mean} = \frac{\sum (f_i \cdot x_i)}{\text{Total Frequency}} = \frac{31000}{1100} \approx 28.18
\][/tex]
Thus, the mean marks of all the students from 50 schools in the city is approximately [tex]\( 28.18 \)[/tex].
