Standard deviation measures how spread out data is from the mean. In a normal distribution, about 68% of data falls within one standard deviation of the mean. Statisticians calculate standard deviation by finding the mean, calculating each data point’s deviation from the mean, squaring the deviation, and finding the square root of the variance. Standard deviation can be used to represent a population from a sample and to determine investment risk and volatility.
The standard deviation is a statistical value used to determine how distributed the data is in a sample and how close individual data points are to the mean – or mean – value of the sample. A standard deviation of a data set of zero means that all values in the set are the same. A larger value implies that individual data points are further away from the mean value.
In a normal distribution of data, also known as a bell curve, most of the data in the distribution – about 68% – will fall within plus or minus one standard deviation of the mean. For example, if the standard deviation of a data set is 2, most of the data in the set will fall within 2 more or 2 less than the mean. About 95.5% of normally distributed data is within two standard deviations of the mean, and over 99% is within three.
To calculate the standard deviation, statisticians first calculate the mean value of all data points. The mean is equal to the sum of all values in the data set divided by the total number of data points. Next, each data point’s deviation from the mean is calculated by subtracting its value from the mean value. The deviation of each data point is squared, and the individual squared deviations are calculated together. The resulting value is known as the variance. The standard deviation is the square root of the variance.
Typically, statisticians find the standard deviation of a sample from a population and use it to represent the entire population. Finding the exact data for a large population is impractical, if not impossible, so using a representative sample is often the best method. For example, if someone wanted to find the number of adult men in the state of California who weighed between 180 and 200 pounds, they could measure the weights of a small number of men and calculate their mean, variance, and standard deviation, and assume they were the same values hold for the population as a whole.
In addition to the uses of statistical analysis, standard deviation can also be used to determine the amount of risk and volatility associated with a particular investment. Investors can calculate the annual standard deviation of an investment’s returns and use that number to determine the investment’s volatility. A larger standard deviation would imply a riskier investment, assuming stability was the desired outcome.
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