Question

find the range, mean, and standard deviation of the data set. {6, 10, 11, 13, 14, 18} the range is 12. (type a whole number.) the mean is 12. (type an integer or decimal.) the standard deviation is \boxed{}. (type an integer or decimal rounded to two decimal places as needed.)

Explanation:

Step1: Recall the formula for standard deviation

The formula for the sample standard deviation \( s \) is \( s = \sqrt{\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}{n - 1}} \), where \( x_{i} \) are the data points, \( \bar{x} \) is the mean, and \( n \) is the number of data points. We know \( \bar{x}=12 \) and \( n = 6 \), and the data points are \( 6,10,11,13,14,18 \).

Step2: Calculate \( (x_{i}-\bar{x})^{2} \) for each data point

• For \( x_1 = 6 \): \( (6 - 12)^{2}=(- 6)^{2}=36 \)
• For \( x_2 = 10 \): \( (10 - 12)^{2}=(-2)^{2}=4 \)
• For \( x_3 = 11 \): \( (11 - 12)^{2}=(-1)^{2}=1 \)
• For \( x_4 = 13 \): \( (13 - 12)^{2}=(1)^{2}=1 \)
• For \( x_5 = 14 \): \( (14 - 12)^{2}=(2)^{2}=4 \)
• For \( x_6 = 18 \): \( (18 - 12)^{2}=(6)^{2}=36 \)

Step3: Sum the squared deviations

\( \sum_{i = 1}^{6}(x_{i}-\bar{x})^{2}=36 + 4+1 + 1+4 + 36=82 \)

Step4: Calculate the variance

The variance \( s^{2}=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}{n - 1}=\frac{82}{6 - 1}=\frac{82}{5} = 16.4 \)

Step5: Calculate the standard deviation

The standard deviation \( s=\sqrt{16.4}\approx4.05 \)

Answer:

\( 4.05 \)