Question

if your calculator finds both kinds of standard deviation, the sample standard deviation and the population standard deviation, which of the two will be a larger number for a given set of data? (hint: recall the difference between how the two standard deviations are calculated.) fill in the blank below. the dropdown will be a larger number for a given set of data. population standard deviation sample standard deviation

Explanation:

Step1: Recall standard - deviation formulas

The sample standard deviation formula is $s=\sqrt{\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}{n - 1}}$, and the population standard deviation formula is $\sigma=\sqrt{\frac{\sum_{i = 1}^{N}(x_{i}-\mu)^{2}}{N}}$, where $n$ is the sample size and $N$ is the population size. For a given set of data, when calculating the sample standard deviation, we divide by $n - 1$ (a smaller divisor compared to $N$ in the population formula when $n=N$).

Step2: Analyze the impact on the result

Since dividing by a smaller number results in a larger quotient when the numerator is non - zero, $\sqrt{\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}{n - 1}}>\sqrt{\frac{\sum_{i = 1}^{N}(x_{i}-\mu)^{2}}{N}}$ for the same set of data values. So the sample standard deviation is larger.

Answer:

sample standard deviation