Question

The formula for the standard deviation of a sample is: s equals square root of fraction numerator 1 over denominator n minus 1 end fraction sum from i equals 1 to n of left parenthesis x subscript i space minus x with bar on top right parenthesis squared end root select the true statement for the following data set that has a mean of 6.75: 4, 6, 7, 10 answer choices are rounded to the hundredths place

Which of the following statements is true for the given data set with a mean of 6.75 (rounded to the hundredths place) and values 4, 6, 7, 10, based on the formula for the standard deviation of a sample?

a) The standard deviation is 2.22.
b) The standard deviation is 1.48.
c) The standard deviation is 2.50.
d) The standard deviation is 3.20.

Answer 1

b) The standard deviation is 1.48.

Step-by-step explanation:

The formula for the standard deviation of a sample is given by:

`\[ s = \sqrt{\frac{\sum_(i=1)^(n) (x_i - \bar{x})^2}{n-1}} \]`

For the data set with values 4, 6, 7, 10 and a mean
`(\(\bar{x}\))`of 6.75, the calculation involves finding the squared differences between each data point and the mean, summing these squared differences, dividing by
`\(n-1\) (where \(n\)` is the number of data points), and finally, taking the square root of the result.

`\[ s = \sqrt{((4-6.75)^2 + (6-6.75)^2 + (7-6.75)^2 + (10-6.75)^2)/(4-1)} \]`

After performing the calculations, the standard deviation (s) is approximately 1.48 (rounded to the hundredths place). This value represents the measure of the dispersion or spread of the data points around the mean.

The correct answer is option (b), stating that the standard deviation is 1.48, as per the calculations. It is crucial for understanding the variability within the data set and assessing how closely individual data points cluster around the mean.