To calculate the standard deviation of the dataset $\{2, 1, 1, 1, 3, 6, 4\}$, follow these steps:

Step 1: Find the Mean

$\text{Mean} = \frac{\sum \text{data points}}{n} = \frac{2 + 1 + 1 + 1 + 3 + 6 + 4}{7} = \frac{18}{7} \approx 2.57$

Step 2: Calculate Each Data Point's Deviation from the Mean

$\text{Deviation} = \text{Data Point} - \text{Mean}$

• For 2: $2 - 2.57 = -0.57$
• For 1: $1 - 2.57 = -1.57$
• For 1: $1 - 2.57 = -1.57$
• For 1: $1 - 2.57 = -1.57$
• For 3: $3 - 2.57 = 0.43$
• For 6: $6 - 2.57 = 3.43$
• For 4: $4 - 2.57 = 1.43$

Step 3: Square Each Deviation

$\text{Squared Deviations:}$

• For -0.57: $(-0.57)^2 = 0.3249$
• For -1.57: $(-1.57)^2 = 2.4649$
• For -1.57: $(-1.57)^2 = 2.4649$
• For -1.57: $(-1.57)^2 = 2.4649$
• For 0.43: $(0.43)^2 = 0.1849$
• For 3.43: $(3.43)^2 = 11.7649$
• For 1.43: $(1.43)^2 = 2.0449$

Step 4: Find the Average of the Squared Deviations

$\text{Variance} = \frac{0.3249 + 2.4649 + 2.4649 + 2.4649 + 0.1849 + 11.7649 + 2.0449}{7} = \frac{21.7144}{7} \approx 3.10$

Step 5: Take the Square Root of the Variance

$\text{Standard Deviation} = \sqrt{3.10} \approx 1.76$

The standard deviation of the dataset is approximately 1.76.

Would you like more details or have any questions?

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Here are some follow-up questions related to this topic:

1. How does variance relate to standard deviation?
2. What is the difference between sample and population standard deviation?
3. How can standard deviation be used to measure data spread?
4. In what situations is a low standard deviation desirable?
5. How would the standard deviation change if we added another number to the dataset?

Tip: A higher standard deviation means the data points are more spread out from the mean, while a lower standard deviation indicates they are closer.