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.
   = 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.