Question

find the range, mean, and standard deviation of the data set {7, 8, 11, 13, 15, 18} the range is (type a whole number.) the mean is (type an integer or decimal.) the standard deviation is (type an integer or decimal rounded to two decimal places as needed.)

Explanation:

Step1: Calculate the range

The range is the difference between the maximum and minimum values in the data set.
For the data set \(\{7,8,11,13,15,18\}\), the maximum value \(x_{max}=18\) and the minimum value \(x_{min} = 7\).

$$Range=x_{max}-x_{min}=18 - 7=11$$

Step2: Calculate the mean

The mean \(\bar{x}\) of a data set \(x_1,x_2,\cdots,x_n\) is given by \(\bar{x}=\frac{\sum_{i = 1}^{n}x_i}{n}\).
Here \(n = 6\), \(x_1=7\), \(x_2 = 8\), \(x_3=11\), \(x_4=13\), \(x_5=15\), \(x_6=18\).
\(\sum_{i=1}^{6}x_i=7 + 8+11+13+15+18=72\)
\(\bar{x}=\frac{72}{6}=12\)

Step3: Calculate the standard deviation

The formula for the standard deviation \(s\) of a sample is \(s=\sqrt{\frac{\sum_{i = 1}^{n}(x_i-\bar{x})^2}{n - 1}}\).
First, calculate \((x_i-\bar{x})^2\) for each \(i\):

• When \(x_1 = 7\), \((7 - 12)^2=(-5)^2 = 25\)
• When \(x_2=8\), \((8 - 12)^2=(-4)^2 = 16\)
• When \(x_3=11\), \((11 - 12)^2=(-1)^2 = 1\)
• When \(x_4=13\), \((13 - 12)^2=(1)^2 = 1\)
• When \(x_5=15\), \((15 - 12)^2=(3)^2 = 9\)
• When \(x_6=18\), \((18 - 12)^2=(6)^2 = 36\)

\(\sum_{i = 1}^{6}(x_i-\bar{x})^2=25+16 + 1+1+9+36=88\)

Since \(n = 6\), then \(n-1=5\)

\(s=\sqrt{\frac{88}{5}}=\sqrt{17.6}\approx4.20\)

Answer:

The range is \(11\), the mean is \(12\), and the standard deviation is \(4.20\)