Standard deviation can be determined by the square root of the

Analyze the question: It asks which quantity you take the square root of to get the standard deviation.

Explain the concept: The standard deviation measures how spread out data values are around the mean. It is built in these steps:

1. Find the mean, x̄.
2. Find each deviation from the mean, (x − x̄).
3. Square each deviation, (x − x̄)². Squaring removes negative signs.
4. Average the squared deviations. This gives the variance.
5. Take the square root of the variance to return to the original units.

Formula:

• Variance: σ² = Σ(x − x̄)² / n
• Standard deviation: σ = √(Σ(x − x̄)² / n) = √variance

Check the options:

• A. Mean is the average. It is not the quantity under the root.
• B. Mean deviation uses absolute values and is a different measure of spread.
• C. Mode is the most frequent value.
• D. Variance. The standard deviation is its square root.
• E. Variance divided by the mean is not a standard measure here. (The coefficient of variation is SD divided by mean.)

Answer: The standard deviation is the square root of the variance.

Exam tip: Remember "SD = √Variance" and "Variance = SD²". Also remember that the variance is in squared units, so we take the square root to get back to the original units.