To determine which is larger, the range or the standard deviation, let's calculate both for the given data set:

Data Set: $11, 11, 12, 12, 12, 12, 13, 15, 17, 22, 22, 22$

1. Range Calculation:

The range is the difference between the maximum and minimum values in the data set.

$\text{Range} = \text{Max value} - \text{Min value}$ $\text{Range} = 22 - 11 = 11$

2. Standard Deviation Calculation:

The standard deviation is a measure of the amount of variation or dispersion of a set of values.

Step-by-step calculation:

• Mean (μ): $\mu = \frac{\sum x_i}{n} = \frac{11 + 11 + 12 + 12 + 12 + 12 + 13 + 15 + 17 + 22 + 22 + 22}{12} = \frac{181}{12} \approx 15.08$

• Variance (σ²): $\sigma^2 = \frac{\sum (x_i - \mu)^2}{n}$ $\sigma^2 \approx \frac{(11 - 15.08)^2 + (11 - 15.08)^2 + \cdots + (22 - 15.08)^2}{12} = \frac{247.92}{12} \approx 20.66$

• Standard Deviation (σ): $\sigma = \sqrt{\sigma^2} \approx \sqrt{20.66} \approx 4.55$

3. Comparison:

• Range = 11
• Standard Deviation ≈ 4.55

Conclusion:
The range (11) is larger than the standard deviation (4.55).

Would you like more details or have any questions?

Here are 5 related questions:

1. What is the significance of the standard deviation in data analysis?
2. How does the range compare to the interquartile range?
3. Can the range and standard deviation ever be equal?
4. What is the effect of outliers on the standard deviation and range?
5. How do variance and standard deviation differ?

Tip: The range is a simple measure of dispersion but can be misleading if the data set has outliers, while the standard deviation provides a more comprehensive view of data spread.