A statistical measure of the variability of a distribution around its mean is referred to as __________.
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The statistical measure of the variability of a distribution around its mean is referred to as the standard deviation. The standard deviation is a commonly used measure of the variability or dispersion of a set of data points from the mean or average value. It is calculated by finding the square root of the sum of the squared deviations from the mean, divided by the number of data points minus one.
For example, let's say we have a set of 10 data points representing the annual returns on a particular stock investment: 10%, 5%, 8%, -2%, 12%, 7%, 9%, 11%, 6%, and 4%. The mean, or average, return is (10+5+8-2+12+7+9+11+6+4)/10 = 7%. To find the standard deviation, we first calculate the deviations of each data point from the mean:
(10-7) = 3 (5-7) = -2 (8-7) = 1 (-2-7) = -9 (12-7) = 5 (7-7) = 0 (9-7) = 2 (11-7) = 4 (6-7) = -1 (4-7) = -3
Then, we square each deviation:
3^2 = 9 (-2)^2 = 4 1^2 = 1 (-9)^2 = 81 5^2 = 25 0^2 = 0 2^2 = 4 4^2 = 16 (-1)^2 = 1 (-3)^2 = 9
Next, we find the sum of the squared deviations:
9+4+1+81+25+0+4+16+1+9 = 150
We divide this sum by the number of data points minus one (10-1 = 9), then take the square root:
sqrt(150/9) ≈ 4.082
Therefore, the standard deviation of the annual returns on this stock investment is approximately 4.082%.
The coefficient of variation is another measure of variability, which is calculated by dividing the standard deviation by the mean. This measure allows us to compare the variability of two or more sets of data with different means. However, it is not as commonly used as the standard deviation.