why is standard deviation a key risk measure 
standard deviation is an especially useful tool in investing and trading strategies as it helps measure market and security volatility and predict performance trends as it relates to investing for example an index fund is likely to have a low standard deviation versus its benchmark index as the fund s goal is to replicate the index on the other hand one can expect aggressive growth funds to have a high standard deviation from relative stock indices as their portfolio managers make aggressive bets to generate higher than average returns a lower standard deviation isn t necessarily preferable it all depends on the investments and the investor s willingness to assume risk when dealing with the amount of deviation in their portfolios investors should consider their tolerance for volatility and their overall investment objectives more aggressive investors may be comfortable with an investment strategy that opts for vehicles with higher than average volatility while more conservative investors may not standard deviation is one of the key fundamental risk measures that analysts portfolio managers and advisors use investment firms report the standard deviation of their mutual funds and other products a large dispersion shows how much the return on the fund is deviating from the expected normal returns because it is easy to understand this statistic is regularly reported to the end clients and investors key properties of standard deviationone key property of standard deviation is additivity which means that the standard deviation of a sum of random variables this property allows researchers to accurately quantify the variability of aggregated data and make meaningful comparisons between different groups or populations as opposed to only analyzing single points of data another property of standard deviation is scale invariance this is particularly useful in comparing the variability of datasets with different units of measurement for example if one dataset is measured in inches and another in centimeters their standard deviations can still be compared directly without needing to convert units last standard deviation has properties of symmetry and non negativity this means a standard deviation is always positive and symmetrically distributed around the mean this symmetry property implies that deviations above the mean are balanced by deviations below the mean resulting in a total balance of the entire data set the property of always being positive means a standard deviation has a higher degree of comparability when looking at standard deviations across data sets standard deviation vs variancevariance is derived by taking the mean of the data points subtracting the mean from each data point individually squaring each of these results and then taking another mean of these squares standard deviation is the square root of the variance all these calculations can be accomplished quickly using software like excel the variance helps determine the data s spread size when compared to the mean value as the variance gets bigger more variation in data values occurs and there may be a larger gap between one data value and another if the data values are all close together the variance will be smaller however this is more difficult to grasp than the standard deviation because variances represent a squared result that may not be meaningfully expressed on the same graph as the original dataset standard deviations are usually easier to picture and apply the standard deviation is expressed in the same unit of measurement as the data which isn t necessarily the case with the variance using the standard deviation statisticians may determine if the data has a normal curve or other mathematical relationship if the data behaves in a normal curve then 68 of the data points will fall within one standard deviation of the average or mean data point larger variances cause more data points to fall outside the standard deviation smaller variances result in more data that is close to average the standard deviation is graphically depicted as a bell curve s width around the mean of a data set the wider the curve the larger a data set s standard deviation from the mean business use cases of standard deviationsthere s a very diverse range of use cases in business where analysts or companies can use standard deviations some of the more common business uses may include strengths of standard deviationstandard deviation is a commonly used measure of dispersion many analysts are probably more familiar with standard deviation than compared to other statistical calculations of data deviation for this reason the standard deviation is often used in a variety of situations from investing to actuaries standard deviation is all inclusive of observations each data point is included in the analysis other measurements of deviation such as range only measure the most dispersed points without consideration for the points in between therefore standard deviation is often considered a more robust accurate measurement compared to other observations the standard deviation of two data sets can be combined using a specific combined standard deviation formula there are no similar formulas for other dispersion observation measurements in statistics in addition and unlike other means of observation the standard deviation can be used in further algebraic computations meaning there s some versatility to standard deviation limitations of standard deviationthere are some downsides to consider when using standard deviation the standard deviation does not actually measure how far a data point is from the mean instead it compares the square of the differences a subtle but notable difference from actual dispersion from the mean outliers have a heavier impact on standard deviation this is especially true considering the difference from the mean is squared resulting in an even larger quantity compared to other data points therefore be mindful that standard observation naturally gives more weight to extreme values last standard deviation can be difficult to calculate manually as opposed to other measurements of dispersion such as range the highest value minus the lowest value standard deviation requires several cumbersome steps and is more likely to incur computational errors compared to easier measurements this hurdle can be circumnavigated through the use of a bloomberg terminal consider leveraging excel when calculating standard deviation after entering your data use the stdev s formula if your data set is numeric or the stdeva when you want to include text or logical values there are also several specific formulas to calculate the standard deviation for an entire population example of standard deviationsay we have the data points 5 7 3 and 7 which total 22 you would then divide 22 by the number of data points in this case four resulting in a mean of 5 5 this leads to the following determinations x 5 5 and n 4 the variance is determined by subtracting the mean value from each data point resulting in 0 5 1 5 2 5 and 1 5 each of those values is then squared resulting in 0 25 2 25 6 25 and 2 25 the square values are then added together giving a total of 11 which is then divided by the value of n minus 1 which is 3 resulting in a variance of approximately 3 67 the square root of the variance is then calculated which results in a standard deviation measure of approximately 1 915 or consider shares of apple aapl for a period of five years historical returns for apple s stock were 88 97 for 2019 82 31 for 2020 34 65 for 2021 26 41 for 2022 and as of mid april 28 32 for 2023 the average return over the five years was thus 41 57 1the value of each year s return less the mean were then 47 40 40 74 6 92 67 98 and 15 57 respectively all those values are then squared to yield 22 47 16 60 0 48 46 21 and 2 42 the sum of these values is 0 882 divide that value by 4 n minus 1 to get the variance 0 882 4 0 220 the square root of the variance is taken to obtain the standard deviation of 0 4690 or 46 90 