Questions: Calculate the standard deviation of the sample quantitative data shown, to two decimal places. x 11.3 26.4 22.7 12.2 10.3 7 13.4 28 Standard deviation: Submit Question

Solution Steps

The mean \(\mu\) of the dataset is calculated as follows:

\[ \mu = \frac{\sum x_i}{n} = \frac{11.3 + 26.4 + 22.7 + 12.2 + 10.3 + 7 + 13.4 + 28}{8} = \frac{131.3}{8} = 16.41 \]

The variance \(\sigma^2\) for a sample is calculated using:

\[ \sigma^2 = \frac{\sum (x_i - \mu)^2}{n-1} \]

Calculating each \((x_i - \mu)^2\):

• \((11.3 - 16.41)^2 = 26.01\)
• \((26.4 - 16.41)^2 = 99.92\)
• \((22.7 - 16.41)^2 = 39.68\)
• \((12.2 - 16.41)^2 = 17.78\)
• \((10.3 - 16.41)^2 = 37.37\)
• \((7 - 16.41)^2 = 88.68\)
• \((13.4 - 16.41)^2 = 9.61\)
• \((28 - 16.41)^2 = 134.11\)

Sum of squared differences:

\[ \sum (x_i - \mu)^2 = 26.01 + 99.92 + 39.68 + 17.78 + 37.37 + 88.68 + 9.61 + 134.11 = 452.16 \]

Variance:

\[ \sigma^2 = \frac{452.16}{7} = 64.59 \]

The standard deviation \(\sigma\) is the square root of the variance:

\[ \sigma = \sqrt{64.59} = 8.04 \]

Final Answer