To solve these questions, let's go through each step:

a) Calculate the Mean Score of the Students

1. Determine the Midpoint (or Class Mark) for Each Score Range:

   • 50–59: midpoint = $\frac{50 + 59}{2} = 54.5$
   • 60–69: midpoint = $\frac{60 + 69}{2} = 64.5$
   • 70–79: midpoint = $\frac{70 + 79}{2} = 74.5$
   • 80–89: midpoint = $\frac{80 + 89}{2} = 84.5$
   • 90–99: midpoint = $\frac{90 + 99}{2} = 94.5$

2. Multiply Each Midpoint by the Number of Students to get the total score contribution for each range:

   • 54.5 × 5 = 272.5
   • 64.5 × 8 = 516
   • 74.5 × 12 = 894
   • 84.5 × 10 = 845
   • 94.5 × 5 = 472.5

3. Calculate the Sum of These Products: $272.5 + 516 + 894 + 845 + 472.5 = 3000$

4. Calculate the Total Number of Students: $5 + 8 + 12 + 10 + 5 = 40$

5. Mean Score: $\text{Mean} = \frac{\text{Total Score}}{\text{Total Number of Students}} = \frac{3000}{40} = 75$

The mean score is 75.

b) Determine the Median of the Score

1. Find the Cumulative Frequency:

   • 50–59: 5
   • 60–69: $5 + 8 = 13$
   • 70–79: $13 + 12 = 25$
   • 80–89: $25 + 10 = 35$
   • 90–99: $35 + 5 = 40$

2. Determine the Position of the Median: Since there are 40 students, the median will be at the $\frac{40}{2} = 20^{\text{th}}$ position.

3. Identify the Interval Containing the Median: The cumulative frequency up to the 70–79 range is 25, so the 20th student lies in the 70–79 range.

4. Use the Median Formula for a Grouped Data (optional here, as we only need the interval): Since the 20th student falls in the 70–79 interval, we take the midpoint of this interval as the median score:

   • Midpoint of 70–79 is 74.5

The median score is 74.5.

Would you like more details on any part of this solution?

Here are some related questions:

1. How would the mean change if the highest score range had more students?
2. What is the mode of the scores in this distribution?
3. How would you find the range of the scores?
4. If you added another score range, how would it affect the median?
5. Can you calculate the standard deviation for these scores?

Tip: When finding the median in grouped data, always check the cumulative frequency to locate the correct interval.