Part 1: Find the Range

The range is the difference between the largest and smallest values in the dataset.

Data: $12, 10, 6, 12, 20$

1. Smallest value: $6$
2. Largest value: $20$

$\text{Range} = 20 - 6 = 14$

The range is $14$, which has already been provided.

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Part 2: Find the Standard Deviation

The standard deviation is calculated using the formula: $\text{Standard Deviation} = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n}}$ Where:

• $x_i$: Each data point
• $\bar{x}$: Mean of the data
• $n$: Number of data points

Step 1: Calculate the Mean ($\bar{x}$)

$\bar{x} = \frac{\text{Sum of all data points}}{\text{Number of data points}} = \frac{12 + 10 + 6 + 12 + 20}{5} = \frac{60}{5} = 12$

Step 2: Calculate the Variance

Variance is the average of the squared differences from the mean.

1. Subtract the mean from each data point and square the result: $(x_i - \bar{x})^2 = \{(12-12)^2, (10-12)^2, (6-12)^2, (12-12)^2, (20-12)^2\}$ $(x_i - \bar{x})^2 = \{0^2, (-2)^2, (-6)^2, 0^2, 8^2\} = \{0, 4, 36, 0, 64\}$

2. Sum the squared differences: $\text{Sum of squares} = 0 + 4 + 36 + 0 + 64 = 104$

3. Divide by the number of data points ($n = 5$): $\text{Variance} = \frac{104}{5} = 20.8$

Step 3: Calculate the Standard Deviation

Take the square root of the variance: $\text{Standard Deviation} = \sqrt{20.8} \approx 4.56$

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Final Answers:

1. Range: $14$
2. Standard Deviation: $4.56$ (rounded to the nearest hundredth)

Would you like further clarification or to explore more related problems?

Related Questions:

1. How does standard deviation differ from variance?
2. Can we use the same data to calculate the coefficient of variation?
3. What is the effect of adding a constant to all data points on the range and standard deviation?
4. How would removing the outlier (20) affect the standard deviation?
5. Can you compute the z-scores for each data point in this dataset?

Tip:

Always round intermediate results only at the final step to maintain accuracy in your calculations.