You might have learned about the standard deviation formula in your maths classes. Did you find it confusing? Do not worry. We will see in detail what a standard deviation formula is and the different properties related to it. A standard deviation is an absolute form of dispersion that is widely used to measure the scatteredness of any given distribution. It is considered the best measure of dispersion as it does not possess the demerits of other forms of absolute forms of dispersion like range and mean deviation.
Formula of Standard Deviation
The standard deviation for a given set of observations is defined as the root mean square deviation when the deviations are taken from the arithmetic mean of the observations. If a variable x assumes n values x1, x2, x3, …., xn then its standard deviation (s) is given by
S = √ (∑ (x – x̄) (x – x̄) / n)
This formula is for simple or discrete frequency distribution. For a grouped frequency distribution, the standard deviation is given by
S = √ (∑ f (x – x̄) (x – x̄) / N)
Sometimes the square of standard deviation, known as variance, is regarded as a measure of dispersion. Thus, variance = square of the standard deviation.
We can also calculate a relative measure of dispersion known as the coefficient of variation using the standard deviation. The coefficient of variation is defined as the ratio of standard deviation to the corresponding arithmetic mean, expressed as a percentage.
Coefficient of Variation ( CV ) = ( Standard deviation / Arithmetic mean ) / 100
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Example to Understand Calculation of Standard Deviation
Let us see an example and understand the calculation of standard deviation using the formula given above. We need to find the arithmetic mean, variance, and standard deviation for the following numbers: 5, 8, 9, 2, 6.
Solution: We know,
Arithmetic Mean = Sum of given observations / Total number of observations.
Thus, Arithmetic Mean = ( 5 + 8 + 9 + 2 + 6 ) / 5 = 6
Now, ∑x.x = 25 + 64 + 81 + 4 + 36 = 210
Variance = 210/5 – 6.6 = 42 – 36 = 6
We have learned before that variance is the square of standard deviation.
Thus, standard deviation = √6 = 2.45
Properties of Standard Deviation
1. If all the observations assumed by a variable are equal or constant, then the standard deviation is zero. This means that if all the values taken by a variable x is k, then the standard deviation = 0. This property is also applicable to range as well as the mean deviation.
2. Standard deviation is not affected due to a change of origin but is affected in the same ratio due to a change of scale which means that if there are two variables x and y related as y = a + b. x for any two constants a and b, then the standard deviation of y is given by
The standard deviation of y = | b | Standard deviation of x.
3. For any two numbers a and b, standard deviation is given by | a – b | / 2.
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