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, ... XN. 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. (xi – x̄). (Note that Σ(xi – 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 (σx) = \(\sqrt {\frac{{∑ {D^2}}}{N}} \), 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 \(\sqrt {\frac{{∑ {D^2}}}{N}} \)