Question

the standard deviation of the data set { 3, 3, 4, 5, 7, 7, 8, 9, 11, 12 } is about _____. 3.015 3.178 6.90

Explanation:

Step1: Calculate the mean (μ) of the data set.

The data set is \( \{3, 3, 4, 5, 7, 7, 8, 9, 11, 12\} \). The number of data points \( n = 10 \).
The sum of the data points: \( 3 + 3 + 4 + 5 + 7 + 7 + 8 + 9 + 11 + 12 = 69 \)
Mean \( \mu=\frac{69}{10} = 6.9 \)

Step2: Calculate the squared differences from the mean.

For each data point \( x_i \), calculate \( (x_i - \mu)^2 \):

• \( (3 - 6.9)^2 = (-3.9)^2 = 15.21 \) (two times, since there are two 3s)
• \( (4 - 6.9)^2 = (-2.9)^2 = 8.41 \)
• \( (5 - 6.9)^2 = (-1.9)^2 = 3.61 \)
• \( (7 - 6.9)^2 = (0.1)^2 = 0.01 \) (two times, since there are two 7s)
• \( (8 - 6.9)^2 = (1.1)^2 = 1.21 \)
• \( (9 - 6.9)^2 = (2.1)^2 = 4.41 \)
• \( (11 - 6.9)^2 = (4.1)^2 = 16.81 \)
• \( (12 - 6.9)^2 = (5.1)^2 = 26.01 \)

Step3: Calculate the sum of the squared differences.

Sum of squared differences:
\( 2\times15.21 + 8.41 + 3.61 + 2\times0.01 + 1.21 + 4.41 + 16.81 + 26.01 \)
\( = 30.42 + 8.41 + 3.61 + 0.02 + 1.21 + 4.41 + 16.81 + 26.01 \)
\( = 90.9 \)

Step4: Calculate the variance (\( \sigma^2 \)).

Variance \( \sigma^2=\frac{\sum_{i = 1}^{n}(x_i - \mu)^2}{n}=\frac{90.9}{10}=9.09 \)

Step5: Calculate the standard deviation (\( \sigma \)).

Standard deviation \( \sigma=\sqrt{9.09}\approx3.015 \)

Answer:

3.015