Questions: For the following set of data, find the sample standard deviation, to the nearest hundredth. 19,20,30,27,17,22,25,30

Solution Steps

To find the mean \( \mu \) of the dataset, we use the formula: \[ \mu = \frac{\sum x_i}{n} \] where \( \sum x_i \) is the sum of all data points and \( n \) is the number of data points. For our dataset: \[ \sum x_i = 19 + 20 + 30 + 27 + 17 + 22 + 25 + 30 = 190 \] Thus, the mean is: \[ \mu = \frac{190}{8} = 23.75 \]

The variance \( \sigma^2 \) is calculated using the formula: \[ \sigma^2 = \frac{\sum (x_i - \mu)^2}{n-1} \] First, we find \( (x_i - \mu)^2 \) for each data point:

• \( (19 - 23.75)^2 = 22.5625 \)
• \( (20 - 23.75)^2 = 14.0625 \)
• \( (30 - 23.75)^2 = 38.0625 \)
• \( (27 - 23.75)^2 = 10.5625 \)
• \( (17 - 23.75)^2 = 45.5625 \)
• \( (22 - 23.75)^2 = 3.0625 \)
• \( (25 - 23.75)^2 = 1.5625 \)
• \( (30 - 23.75)^2 = 38.0625 \)

Now, summing these values: \[ \sum (x_i - \mu)^2 = 22.5625 + 14.0625 + 38.0625 + 10.5625 + 45.5625 + 3.0625 + 1.5625 + 38.0625 = 190.5 \] Now, we can calculate the variance: \[ \sigma^2 = \frac{190.5}{8-1} = \frac{190.5}{7} \approx 25.07 \]

The standard deviation \( \sigma \) is the square root of the variance: \[ \sigma = \sqrt{\sigma^2} = \sqrt{25.07} \approx 5.01 \]

Thus, the sample standard deviation is \( 5.01 \).

Final Answer