Certainly! Let's analyze the given data and determine the number of students below and above the mean class mark step-by-step.
The table of marks and the number of students is:
[tex]\[
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
Marks & 10-20 & 20-30 & 30-40 & 40-50 & 50-60 & 60-70 \\
\hline
No. of students & 2 & 5 & 7 & 6 & 3 & 2 \\
\hline
\end{tabular}
\][/tex]
### Step 1: Determine the total number of students
First, we sum the number of students in each mark interval:
[tex]\[
\text{Total students} = 2 + 5 + 7 + 6 + 3 + 2 = 25
\][/tex]
### Step 2: Calculate the mean class mark
To calculate the mean mark, we first find the midpoints of each mark interval. The midpoints are calculated as follows:
- For 10-20, midpoint [tex]\( = \frac{10 + 20}{2} = 15 \)[/tex]
- For 20-30, midpoint [tex]\( = \frac{20 + 30}{2} = 25 \)[/tex]
- For 30-40, midpoint [tex]\( = \frac{30 + 40}{2} = 35 \)[/tex]
- For 40-50, midpoint [tex]\( = \frac{40 + 50}{2} = 45 \)[/tex]
- For 50-60, midpoint [tex]\( = \frac{50 + 60}{2} = 55 \)[/tex]
- For 60-70, midpoint [tex]\( = \frac{60 + 70}{2} = 65 \)[/tex]
Next, we multiply each midpoint by the number of students in that interval and sum the results to find the total marks sum:
[tex]\[
\text{Total marks sum} = (15 \times 2) + (25 \times 5) + (35 \times 7) + (45 \times 6) + (55 \times 3) + (65 \times 2) = 30 + 125 + 245 + 270 + 165 + 130 = 965
\][/tex]
The mean mark is then calculated by dividing the total marks sum by the total number of students:
[tex]\[
\text{Mean mark} = \frac{965}{25} = 38.6
\][/tex]
### Step 3: Count the number of students below and above the mean mark
Now, we compare the midpoint of each mark interval with the mean mark (38.6) and count the number of students accordingly.
#### Below the Mean Mark:
- For 10-20 (midpoint 15, students 2)
- For 20-30 (midpoint 25, students 5)
- For 30-40 (midpoint 35, students 7)
Total students below the mean mark:
[tex]\[
2 + 5 + 7 = 14
\][/tex]
#### Above the Mean Mark:
- For 40-50 (midpoint 45, students 6)
- For 50-60 (midpoint 55, students 3)
- For 60-70 (midpoint 65, students 2)
Total students above the mean mark:
[tex]\[
6 + 3 + 2 = 11
\][/tex]
### Step 4: Comparison of students below and above the mean mark
We have:
[tex]\[
\text{Students below mean} = 14
\][/tex]
[tex]\[
\text{Students above mean} = 11
\][/tex]
Comparison:
- There are 3 more students below the mean mark than above it (14 - 11 = 3).
In conclusion:
- The number of students below the mean mark (38.6) is 14.
- The number of students above the mean mark (38.6) is 11.
- There are more students below the mean class mark than above it by 3 students.
