Question

Which formula is used to determine the value of the standard deviation of ungrouped data?
1. \(\sqrt {\frac{{\sum {D^2}}}{N}} \)
2. \( {\frac{{\sum {D^2}}}{N}} \)
3. \(\sqrt {\frac{{\sum {N}}}{D^2}} \)
4. \( {\frac{{\sum {D}}}{N}} \)

Answer 1

Correct Answer - Option 1 : \(\sqrt {\frac{{\sum {D^2}}}{N}} \)

Standard Deviation (S. D.): One of the most stable measure of variability, it is the most important and commonly used measure of dispersion. It measures the absolute dispersion or variability of a distribution. Standard deviation is the positive square root of the mean of the squared deviations of observations from their mean. It is denoted by S.D. or σx.

Actual Mean Method: Let X variable takes on N values i.e. X₁, X₂, ... X_N. The standard deviation of these N observations using the actual mean method can be computed as follows:

• Obtain the arithmetic mean (x̄) of the given data
• Obtain the deviation of each ith observation from X i.e. (x_i – x̄). (Note that Σ(x_i – x ) = 0)
•  Square each deviation i.e (xi – x̄)²
•  Obtain the sum in step 3
•  Obtain the square root of the mean of these squared deviations as follows:

Standard deviation (σ_x) = \(\sqrt {\frac{{∑ {D^2}}}{N}} \), where ∑D² = (xi – x̄)² and N = Total No. of observation

Hence, we conclude that the formula used to determine the value of the standard deviation of ungrouped data is \(\sqrt {\frac{{∑ {D^2}}}{N}} \)