To calculate the mean and standard deviation of the final grades based on the given frequency distribution, we first need to make some assumptions about the data. In this case, we will assume the midpoint of each grade interval represents the grade for that range. Let's go through the steps:

Step 1: Calculate the midpoints of the intervals

The midpoint of each interval is calculated by averaging the lower and upper bounds of the intervals:

• For $0-6$: $\frac{0+6}{2} = 3$
• For $7-13$: $\frac{7+13}{2} = 10$
• For $14-20$: $\frac{14+20}{2} = 17$

So the midpoints are 3, 10, and 17.

Step 2: Organize the data

Final Grade Interval	Midpoint $x_i$	Frequency $f_i$	$f_i \times x_i$	$f_i \times x_i^2$
$0-6$	3	10	$10 \times 3 = 30$	$10 \times 3^2 = 90$
$7-13$	10	14	$14 \times 10 = 140$	$14 \times 10^2 = 1400$
$14-20$	17	2	$2 \times 17 = 34$	$2 \times 17^2 = 578$

Step 3: Calculate the total number of students and the sum of the products $f_i \times x_i$ and $f_i \times x_i^2$

• Total number of students $N = 10 + 14 + 2 = 26$
• Sum of $f_i \times x_i = 30 + 140 + 34 = 204$
• Sum of $f_i \times x_i^2 = 90 + 1400 + 578 = 2068$

Step 4: Calculate the mean

The mean $\mu$ is given by:

$\mu = \frac{\sum{f_i \times x_i}}{N} = \frac{204}{26} = 7.846$

So, the mean grade is approximately $7.85$.

Step 5: Calculate the variance

The variance $\sigma^2$ is calculated using the formula:

$\sigma^2 = \frac{\sum{f_i \times x_i^2}}{N} - \mu^2$

Substituting the values:

$\sigma^2 = \frac{2068}{26} - (7.846)^2 = 79.538 - 61.574 = 17.964$

Step 6: Calculate the standard deviation

The standard deviation $\sigma$ is the square root of the variance:

$\sigma = \sqrt{17.964} \approx 4.24$

Final Answers:

• Mean: $7.85$
• Standard Deviation: $4.24$

Would you like a breakdown of any of these steps?

Here are 5 related questions:

1. How is the variance related to the standard deviation?
2. Why are midpoints used in calculating the mean and standard deviation from grouped data?
3. How would the mean change if the frequencies were altered?
4. What is the significance of the range in each interval?
5. Can the same formula be applied for a different distribution?

Tip: When dealing with grouped data, always use the midpoint of intervals for approximations.