To calculate the standard deviation and range of the given sample data $x = [3, 8, 26, 27, 5]$, follow these steps:

1. Calculate the Mean

$\text{Mean} = \frac{3 + 8 + 26 + 27 + 5}{5} = \frac{69}{5} = 13.8$

2. Calculate the Variance

$\text{Variance} = \frac{\sum (x_i - \text{Mean})^2}{n-1}$ $= \frac{(3 - 13.8)^2 + (8 - 13.8)^2 + (26 - 13.8)^2 + (27 - 13.8)^2 + (5 - 13.8)^2}{4}$ $= \frac{( -10.8)^2 + ( -5.8)^2 + (12.2)^2 + (13.2)^2 + ( -8.8)^2}{4}$ $= \frac{116.64 + 33.64 + 148.84 + 174.24 + 77.44}{4}$ $= \frac{550.8}{4} = 137.7$

3. Calculate the Standard Deviation

$\text{Standard Deviation} = \sqrt{\text{Variance}} = \sqrt{137.7} \approx 11.7$

4. Calculate the Range

$\text{Range} = \text{Maximum} - \text{Minimum} = 27 - 3 = 24$

Final Results

• Standard Deviation: 11.7
• Range: 24

If you have any questions or need further details, feel free to ask! Here are some related questions to consider:

1. What is the difference between population and sample standard deviation?
2. How does changing one value in the dataset affect the standard deviation?
3. What are the implications of a large standard deviation in a dataset?
4. How do you interpret the range in terms of data variability?
5. Can you provide examples of when to use standard deviation versus variance?

Tip: Always check if your data set is a sample or a population, as this affects the formulas used for standard deviation and variance.