3. There were 26 students in the course STATS 1234: Statistics for students who love
statistics. Unfortunately, the original database containing the final grades was lost and
all that remains is the frequency table below. Use the info in the table to calculate the
mean and standard deviation of the final grades of all the students STATS 1234.
Final Grade frequency
0-6 10
7-13 14
14-20 2

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.

There were 26 students in the course STATS 1234: Statistics for students who love statistics. Unfortunately, the original database containing the final grades was lost and all that remains is the frequency table below. Use the info in the table to calculate the mean and standard deviation of the final grades of all the students STATS 1234. Final Grade frequency: 0-6 (10), 7-13 (14), 14-20 (2).

Learn how to calculate the mean and standard deviation from a grouped frequency table in STATS 1234: Statistics for students who love statistics. The problem involves three grade intervals (0-6, 7-13, and 14-20) and covers essential concepts in frequency distribution and grouped data analysis. Suitable for college-level introductory statistics.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation