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To solve the problem, let’s calculate the range and standard deviation based on the data provided in the table.

### Data Interpretation:
- **Values:** 2, 3, 4, 5, 6
- **Frequencies:** 2, 5, 4, 0, 2

This means:
- The value 2 appears 2 times
- The value 3 appears 5 times
- The value 4 appears 4 times
- The value 5 appears 0 times
- The value 6 appears 2 times

### Step-by-Step Solution

#### (a) Range
The range of a dataset is the difference between the maximum and minimum values.
- **Minimum value:** 2
- **Maximum value:** 6

\[
\text{Range} = \text{Maximum value} - \text{Minimum value} = 6 - 2 = 4
\]

So, the range is **4.0**.

#### (b) Standard Deviation
To calculate the standard deviation, we’ll follow these steps for a sample standard deviation:

1. **Find the mean (average)**:
   - First, calculate the sum of all values multiplied by their frequencies.
   \[
   \text{Sum} = (2 \times 2) + (3 \times 5) + (4 \times 4) + (5 \times 0) + (6 \times 2)
   = 4 + 15 + 16 + 0 + 12 = 47
   \]
   - The total number of data points (N) is the sum of frequencies:
   \[
   N = 2 + 5 + 4 + 0 + 2 = 13
   \]
   - Calculate the mean:
   \[
   \text{Mean} = \frac{\text{Sum}}{N} = \frac{47}{13} \approx 3.6154
   \]

2. **Calculate the squared deviations** for each value and multiply by the frequency:
   \[
   (2 - 3.6154)^2 \times 2 \approx 5.2268
   \]
   \[
   (3 - 3.6154)^2 \times 5 \approx 1.8925
   \]
   \[
   (4 - 3.6154)^2 \times 4 \approx 2.2308
   \]
   \[
   (5 - 3.6154)^2 \times 0 = 0
   \]
   \[
   (6 - 3.6154)^2 \times 2 \approx 11.3516
   \]

   - Total sum of squared deviations:
   \[
   \text{Sum of squared deviations} \approx 5.2268 + 1.8925 + 2.2308 + 0 + 11.3516 = 20.7017
   \]

3. **Calculate the variance** (for a sample, divide by \(N - 1\)):
   \[
   \text{Variance} = \frac{20.7017}{13 - 1} \approx 1.7251
   \]

4. **Find the standard deviation** by taking the square root of the variance:
   \[
   \text{Standard Deviation} \approx \sqrt{1.7251} \approx 1.3
   \]

### Final Answers:
- **Range:** 4.0
- **Standard Deviation:** 1.3

Would you like further clarification on any of the steps?

---

#### Additional Questions
1. How would the range change if the frequency of the maximum or minimum value increased?
2. Why do we divide by \(N - 1\) instead of \(N\) for the sample standard deviation?
3. How would the standard deviation change if a value with a frequency of 0 was given a positive frequency?
4. What would the mean and standard deviation be if all values were increased by a fixed amount?
5. How can we verify if the calculated standard deviation accurately reflects the data spread?

#### Tip
When calculating sample variance or standard deviation, remember to use \(N - 1\) in the denominator if the data represents a sample.

Find (a) the range and (b) the standard deviation for the sample. (Value and frequency data provided in a table)

Learn how to calculate the range and standard deviation for a dataset using frequency values. Follow a step-by-step solution covering mean, variance, and standard deviation calculations for high school students in grades 10-12.

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