객관식
Which of the following best describes the relationship between the and the of a procedure?
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3. Describing Data Numerically
Standard Deviation
객관식
Which formula should you use to calculate the variance and which formula should you use to calculate the standard deviation of a sample of observations , , ..., ?
A
Variance: ; Standard deviation:
B
Variance: ; Standard deviation:
C
Variance: ; Standard deviation:
D
Variance: ; Standard deviation:
검증된 단계별 안내1
Understand that the variance of a sample measures the average squared deviation of each observation from the sample mean, and the standard deviation is the square root of the variance, providing a measure of spread in the same units as the data.
Recall the formula for the sample variance, which uses the sum of squared differences between each observation \(x_i\) and the sample mean \(\bar{x}\), divided by the degrees of freedom \((n - 1)\), where \(n\) is the sample size:
\[
\text{Variance} = \frac{\sum_{i=1}^n (x_i - \bar{x})^2}{n - 1}
\]
Recognize that the sample standard deviation is the positive square root of the sample variance, so its formula is:
\[
\text{Standard Deviation} = \sqrt{\frac{\sum_{i=1}^n (x_i - \bar{x})^2}{n - 1}}
\]
Note why we divide by \(n - 1\) instead of \(n\): this adjustment, called Bessel's correction, corrects the bias in the estimation of the population variance and standard deviation from a sample.
Summarize that to calculate the sample variance and standard deviation, first compute the squared deviations from the mean, sum them, divide by \(n - 1\) for variance, and then take the square root of that result for the standard deviation.
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