Which must be true about the standard deviation?

Question

BlackTom AI · Accepted Answer

The prompt asks about what must be true about the standard deviation, but the provided list of answer choices is missing, so we cannot evaluate each option in the usual way. Nevertheless, we can discuss the key concept implied by the included answer snippet: the idea that the standard deviation can equal 0.

Consider the general definition: the standard deviation measures the spread of data around the mean. If every data point in a dataset is exactly the same value, there is no variability at all, so all deviations from the mean are zero, and the standard deviation equals 0.

Why this can be true in practice: when a dataset consists of identical observations, or when sample data has zero variability, the squared deviations are all zero, leading to a zero variance and thus a zero standard deviation. This is a legitimate and sometimes encountered scenario in statistics.

Now, about the absence of answer options: since no list of choices is provided, I can't break down each alternative and explain their validity or invalidity. However, given the statement that the standard deviation can equal 0, the reasoning above shows a valid situation where that occurs.

In typical exams, other common statements might include: (a) the standard deviation is always nonnegative, (b) it can be negative, (c) it depends only on the mean, (d) it is unaffected by the scale of the data. These are frequent concepts used to test understanding of the standard deviation, and each would be evaluated accordingly from the properties of standard deviation.

If you can supply the actual list of answer choices, I can provide a complete, option-by-option analysis with tailored explanations for why each choice is correct or incorrect, following the same detailed, varied reasoning style.

STAT 1450.02 SP2025 (28132) Checkpoint 2.7 & 2.8

Which must be true about the standard deviation?

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