Question

which formula gives the standard deviation of a data set?
a.
\\(\sqrt{\frac{\sum \left(x - \overline{x}\
ight)^2}{(n + 1)}}\\)
b.
\\(\sqrt{\frac{\sum \left(x - \overline{x}\
ight)^2}{(n - 1)}}\\)
c.
\\(\sqrt{\frac{\sum \left(x - \overline{x}\
ight)^2}{n}}\\)
d.
\\(\sqrt{\frac{\sum \left(x + \overline{x}\
ight)^2}{(n - 1)}}\\)

Explanation:

Step1: Recall standard deviation formulas

For a population standard deviation, the formula is $\sqrt{\frac{\sum (X-\bar{X})^2}{n}}$, where $n$ is the total number of data points. For a sample standard deviation, the formula uses $n-1$ (Bessel's correction) in the denominator: $\sqrt{\frac{\sum (X-\bar{X})^2}{n-1}}$. Both are valid standard deviation formulas, but option C represents the population standard deviation, which is a core standard deviation formula for a full data set.

Step2: Eliminate incorrect options

• Option A uses $n+1$ in the denominator, which is not a standard deviation formula.
• Option D uses $(X+\bar{X})^3$, which is mathematically irrelevant to standard deviation (we use squared deviations from the mean, not summed values cubed).
• While option B is sample standard deviation, option C is the standard formula for the full population data set, which is the most general answer for "a data set" when not specified as a sample.

Answer:

C. $\sqrt{\frac{\sum (X-\bar{X})^2}{n}}$