Question

a teacher weighed the backpacks of 4 students. here are the weights (in kilograms):
5,7,9,11
the mean is (\bar{x} = 8) kg.
which of these formulas gives the standard deviation?
choose 1 answer:
(\boldsymbol{\text{a}}) (s_x = \frac{5 + 7 + 9 + 11}{4}) ...

Explanation:

Step1: Recall Standard Deviation Formula

The formula for the population standard deviation (since we have all 4 students' weights, a population) is $\sigma = \sqrt{\frac{\sum_{i = 1}^{n}(x_{i}-\mu)^{2}}{n}}$, where $\mu$ is the mean, $x_{i}$ are the data points, and $n$ is the number of data points. For sample standard deviation, it's $s = \sqrt{\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}{n - 1}}$, but here $n=4$ (small, but let's check the options).

Step2: Analyze Option A

Option A: $s_{x}=\frac{5 + 7+9 + 11}{4}$ is the formula for the mean (since $\bar{x}=\frac{\sum x_{i}}{n}$), not standard deviation. So A is incorrect.

(Note: Since the full options aren't shown, but based on the start, the correct formula for standard deviation should involve squaring the differences from the mean, summing them, dividing by $n$ (or $n - 1$), and taking the square root. For example, if there was an option like $\sqrt{\frac{(5 - 8)^{2}+(7 - 8)^{2}+(9 - 8)^{2}+(11 - 8)^{2}}{4}}$, that would be the population standard deviation formula. Assuming the other options include this, we can deduce the correct one by eliminating A (which is mean) and identifying the one with squared differences, sum, division by $n$, and square root.)

Answer:

(Assuming the correct option is the one with $\sqrt{\frac{(5 - 8)^{2}+(7 - 8)^{2}+(9 - 8)^{2}+(11 - 8)^{2}}{4}}$ (or with $n - 1$ if sample, but since it's all 4 students, population). Since the original problem's option A is mean, the correct option would be the one following the standard deviation formula structure. For example, if the options are as typical, the correct formula is the one with squared deviations from the mean, summed, divided by $n$, square - rooted. So if the options include that, that's the answer. Since the user's image shows A as mean, the correct answer would be the option with the standard deviation formula, e.g., if option B (not fully shown) is $\sqrt{\frac{(5 - 8)^{2}+(7 - 8)^{2}+(9 - 8)^{2}+(11 - 8)^{2}}{4}}$, then the answer is that option. But based on the given info, we know A is wrong (it's mean), so the correct option is the one with the standard deviation formula structure. )

(If we assume the options are:
A. $\frac{5 + 7+9 + 11}{4}$ (mean)
B. $\sqrt{\frac{(5 - 8)^{2}+(7 - 8)^{2}+(9 - 8)^{2}+(11 - 8)^{2}}{4}}$ (population standard deviation)
C. Other incorrect ones

Then the answer is B. $\sqrt{\frac{(5 - 8)^{2}+(7 - 8)^{2}+(9 - 8)^{2}+(11 - 8)^{2}}{4}}$)