Calculating Means and Standard Deviations

STANDARD DEVIATIONSThe standard deviation is conventionally calculated by finding the difference betweeneach value and the mean value, squaring each of these differences, summing up these squareddifferences, dividing this number by the number of observations minus one, and taking thesquare root of this result:S.D. =n( Â (Xi - X) )/(n -1)i=12However, this formula leads to substantial rounding error because the subtraction steptakes place before squaring the differences between the values and the mean. A more robustformula is the following: †S.D. =2nn[ Â (Xi) - (( Â Xi) / n)]/ (n - 1)i=1i=12Using the symbols (butterfly, whale & octopus) below to represent the quantitiesdefined above, the formula may be expressed as:=n2Â ( Xi)= ( Â Xi)= ni=1ni=1Arrange the data as follows:Xi (Xi) 22 43 95 258 6410 10028 202