How to Calculate the Standard Deviation From a Frequency Distribution

High standard deviation implies higher volatility.
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Analysts and researchers can use frequency distributions to evaluate historical investment returns and prices. Investment types include stocks, bonds, mutual funds and broad market indexes. A frequency distribution shows the number of occurrences for different data classes, which could be single data points or data ranges. The standard deviation is one of the ways to examine the spread or distribution of a data sample -- this helps predict rates of return, volatility and risk.


Step 1

Format the data table. Use a software spreadsheet tool, such as Microsoft Excel, to simplify the calculations and eliminate math errors. Label the columns data class, frequency, midpoint, the square of the difference between the midpoint and the mean, and the product of the frequency and the square of the difference between the midpoint and the mean. Use symbols to label the columns and include an explanatory note with the table.


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Step 2

Populate the first three columns of the data table. For example, a stock price table could consist of the following price ranges in the data class column -- $10 to $12, $13 to $15 and $16 to $18 -- and 10, 20 and 30 for the corresponding frequencies. The midpoints are $11, $14 and $17 for the three data classes. The sample size is 60 (10 plus 20 plus 30).


Step 3

Approximate the mean by assuming that all distributions are at the midpoint of the respective ranges. The formula for the arithmetic mean of a frequency distribution is the sum of the product of the midpoint and the frequency for each data range divided by the sample size. Continuing with the example, the mean is equal to the sum of the following midpoint and frequency multiplications -- $11 multiplied by 10, $14 multiplied by 20 and $17 multiplied by 30 -- divided by 60. Therefore, the mean is equal to $900 ($110 plus $280 plus $510) divided by 60, or $15.


Step 4

Fill the other columns. For each data class, compute the square of the difference between the midpoint and the mean, and then multiply the result by the frequency. Continuing with the example, the differences between the midpoint and the mean for the three data ranges are -$4 ($11 minus $15), -$1 ($14 minus $15) and $2 ($17 minus $15), and the squares of the differences are 16, 1 and 4, respectively. Multiply the results by the corresponding frequencies to get 160 (16 multiplied by 10), 20 (1 multiplied by 20) and 120 (4 multiplied by 30).


Step 5

Calculate the standard deviation. First, sum the products from the previous step. Second, divide the sum by the sample size minus 1, and finally calculate the square root of the result to get the standard deviation. To conclude the example, the standard deviation is equal to the square root of 300 (160 plus 20 plus 120) divided by 59 (60 minus 1), or about 2.25.



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