https://file.aiquickdraw.com/mathbot/file/1728458679757tmhod40n.jpg

Let's address both questions from the image you've uploaded:

### 1. Question 3 (Scores of students in Sections A, B, and C)

You are asked to compute the following for each section: **range**, **variance**, and **standard deviation**. Then, you need to determine which section has the most consistent score.

#### a) **Range**:
- **Range** is the difference between the maximum and minimum values in a data set.

For **Section A**:
- Max = 25, Min = 7, Range = 25 - 7 = **18**

For **Section B**:
- Max = 25, Min = 7, Range = 25 - 7 = **18**

For **Section C**:
- Max = 25, Min = 18, Range = 25 - 18 = **7**

#### b) **Variance** and **Standard Deviation**:
Variance (\( \sigma^2 \)) measures how much the data points deviate from the mean. The **standard deviation** (\( \sigma \)) is the square root of the variance.

I'll calculate the variance and standard deviation for each section:

**Formula for Variance**:
\[
\sigma^2 = \frac{1}{N}\sum_{i=1}^N (x_i - \mu)^2
\]
Where:
- \( N \) = Number of values
- \( x_i \) = Each individual value
- \( \mu \) = Mean of the values

I'll work on these detailed calculations next.

### 2. Question 4 (Coefficient of Variation)
Here, you need to compute the **coefficient of variation (CV)** for Set A and Set B. CV is defined as:
\[
\text{CV} = \frac{\sigma}{\mu} \times 100\%
\]
Where \( \sigma \) is the standard deviation and \( \mu \) is the mean of the data set. After calculating the CV, you'll compare which set has more variation and which is more consistent (the lower the CV, the more consistent the data set).

I'll start by calculating the means and standard deviations for both sets.

I will now calculate both the standard deviation and variance for Sections A, B, C, and the coefficient of variation for Set A and Set B.Here are the results of the calculations:

### 1. Question 3 (Sections A, B, and C)

#### a) **Range**:
- Section A: **18**
- Section B: **18**
- Section C: **7**

#### b) **Variance**:
- Section A: **35.76**
- Section B: **39.25**
- Section C: **6.09**

#### c) **Standard Deviation**:
- Section A: **5.98**
- Section B: **6.26**
- Section C: **2.47**

#### Most Consistent Section:
- **Section C** has the **lowest standard deviation (2.47)**, meaning it has the most consistent scores.

---

### 2. Question 4 (Coefficient of Variation for Set A and Set B)

#### a) **Coefficient of Variation (CV)**:
- Set A: **36.31%**
- Set B: **75.12%**

#### b) Interpretation:
- Set A has a lower CV (36.31%) compared to Set B (75.12%). This indicates that **Set A is more consistent** and **Set B has more variation**.

---

Let me know if you'd like further details on any step, or if you have any questions!

Here are five additional related questions:
1. How is the mean affected by extreme values in a data set?
2. Why is the coefficient of variation useful in comparing data sets?
3. What is the difference between population variance and sample variance?
4. Can the range alone give a clear picture of data variability?
5. How does standard deviation reflect the consistency of a data set?

**Tip:** When comparing the consistency of data sets, the coefficient of variation is a powerful tool as it normalizes the standard deviation by the mean, allowing for a fair comparison.

1. Compute for the range, variance, and standard deviation of scores from three sections (A, B, C). Which section has the most consistent score? 2. Compute the coefficient of variation for two data sets (Set A and Set B). Which data set is more consistent?

Learn how to calculate the range, variance, and standard deviation for scores of students in three sections, and determine the most consistent section. Additionally, compute the coefficient of variation for two data sets and identify which set is more consistent. Suitable for Grades 10-12.

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