Question

_______________ is the square root of the arithmetic mean of the squares of all deviations, the deviations being measured from the arithmetic mean of the distribution.

• A. Range
• B. Mean Deviation
• C. Standard Deviation
• D. Coefficient of Variation

Hint:

Standard Deviation is also referred to as the Root Mean Square Deviation (RMSD). This description directly translates to standard deviation.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
Measures of dispersion indicate how spread out or scattered the data values are around a central value like the arithmetic mean.
Key Formula or Approach:
The mathematical definition for a population standard deviation (\(\sigma\)) is:
\[ \sigma = \sqrt{\frac{\sum (x - \bar{x})^2}{N}} \]
where \(x\) represents individual values, \(\bar{x}\) is the arithmetic mean, and \(N\) is the total number of observations.

Step 2: Detailed Explanation:
Let us break down the definition given in the question:
1. "deviations being measured from the arithmetic mean of the distribution": This is represented by \((x - \bar{x})\).
2. "squares of all deviations": This is represented by \((x - \bar{x})^2\).
3. "arithmetic mean of the squares of all deviations": This is represented by \(\frac{\sum (x - \bar{x})^2}{N}\).
4. "square root of the arithmetic mean of the squares": This is represented by \(\sqrt{\frac{\sum (x - \bar{x})^2}{N}}\).
This mathematical formulation is the precise definition of the Standard Deviation.

Step 3: Final Answer
The standard deviation is the root-mean-square deviation from the arithmetic mean.