Question

consider the following sample data values.
15 9 19 12 10 1 8 14 18
a) calculate the range.
b) calculate the sample variance.
c) calculate the sample standard deviation.

Explanation:

Part (a): Calculate the range

Step 1: Identify max and min

Find the maximum and minimum values in the data set: \(15, 9, 19, 12, 10, 1, 8, 14, 18\).
Max = \(19\), Min = \(1\).

Step 2: Compute range

Range = Max - Min = \(19 - 1 = 18\).

Step 1: Find the mean (\(\bar{x}\))

Sum of data: \(15 + 9 + 19 + 12 + 10 + 1 + 8 + 14 + 18 = 106\).
Number of data points (\(n\)) = \(9\).
Mean (\(\bar{x}\)) = \(\frac{106}{9} \approx 11.7778\).

Step 2: Compute squared deviations

For each data point \(x_i\), calculate \((x_i - \bar{x})^2\):

• \( (15 - 11.7778)^2 \approx 10.4444 \)
• \( (9 - 11.7778)^2 \approx 7.7160 \)
• \( (19 - 11.7778)^2 \approx 52.1111 \)
• \( (12 - 11.7778)^2 \approx 0.0494 \)
• \( (10 - 11.7778)^2 \approx 3.1503 \)
• \( (1 - 11.7778)^2 \approx 116.1111 \)
• \( (8 - 11.7778)^2 \approx 14.2691 \)
• \( (14 - 11.7778)^2 \approx 4.9444 \)
• \( (18 - 11.7778)^2 \approx 38.7160 \)

Step 3: Sum of squared deviations

Sum = \(10.4444 + 7.7160 + 52.1111 + 0.0494 + 3.1503 + 116.1111 + 14.2691 + 4.9444 + 38.7160 \approx 247.5556\).

Step 4: Sample variance (\(s^2\))

Sample variance = \(\frac{\text{Sum of squared deviations}}{n - 1} = \frac{247.5556}{9 - 1} \approx 30.9444\).

Step 1: Take square root of sample variance

Sample standard deviation (\(s\)) = \(\sqrt{\text{Sample variance}} = \sqrt{30.9444} \approx 5.56\).

Answer:

The range is \(\boldsymbol{18}\).

Part (b): Calculate the sample variance