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Standard deviation or SD, which is also referred to as root mean square, is the 'mean of the mean'. It is used to measure variation or dispersion. In simple words, it is a measure about how the statistical data is spread. Standard deviation was first implemented by Karl Pearson in 1894. Standard deviation can be of two types, namely; low standard deviation and high standard deviation. The low standard deviation indicates that the data points are close to the mean point, while the high standard deviation indicates that the data points are spread out and away from the mean point. As we have different calculators like loan calculator, mortgage calculator, etc., there are free standard deviation calculators available online which can help you in calculating standard deviation.

Calculation of Standard Deviation

Standard deviation is represented with the symbol lower case sigma. In standard deviation (Sn), average is represented by x bar and sum is represented by upper case symbol of sigma. Let's take a look at the formula for standard deviation calculation.

S_n = √ 1 / (N - 1) ∑ ( x _i - x̄ ) ²
(where ∑ ranges from i = 1 to N)

Steps on Calculating Standard Deviation

Method 1: This is a manual method for calculating standard deviation.

Step 1: The first step is to calculate the mean or the average of the numbers.

Step 2: The next step is to get the deviations. In order to get the deviations, subtract the mean obtained from step 1 from all the numbers.

Step 3: Now, square the numbers (or multiply the number by itself), which are obtained as results from step 2.

Step 4: Now, add all the numbers.

Step 5: The next step is to divide this number by 1 less than the total numbers.

Step 6: The last step requires you to take the square root of this number.

Example

Consider the numbers 1, 2, 3, 4 and 5

Step 1
1 + 2 + 3 + 4 + 5 = 15

Step 2
1 – 15 = -14
2 – 15 = -13
3 – 15 = -12
4 – 15 = -11
5 – 15 = -10

Step 3
14 * 14 = 196
13 * 13 = 169
12 * 12 = 144
11 * 11 = 121
10 * 10 = 100

Step 4
196 + 169 + 144 + 121 + 100 = 730

Step 5
730 / 4 = 182.5

Step 6
Square root of 182.5 = 13.5092561

In order to calculate population standard deviation, follow the same procedure as mentioned above except for a bit of change in step 5. Instead of dividing the number by one less than the total numbers, divide the number by the number which is the same as the total numbers used. So, the formula in this case is:

S_n = √ 1 / N ∑ ( x _i - x̄ ) ²
(where ∑ ranges from i = 1 to N)

Method 2: In this method, you will have to calculate the standard deviation with the help of an excel spreadsheet.

Step 1
The first step is to open the excel spreadsheet and enter, say 10 numbers, in different cells. Make sure you start from the first cell and enter the numbers one below the other.

Step 2
Now, place the cursor on the 11th cell. From the menu bar, select the insert tab and then the function tab. After doing this, a function dialog box will appear on the screen.

Step 3
From this function category dialog box, click on statistical and then select average and press the enter button.

Step 4
After pressing the enter button, a dialog box will appear on the screen asking for the confirmation of the range. It will ask if you want to perform operation on the numbers present in the cells 1, 2,...10. For confirmation, press enter. After pressing enter, the average of the numbers will appear in the 11th cell.

Step 5
Now, place the cursor in the 12th cell. Again follow the same procedure as mentioned above and from menu bar select insert and then function.

Step 6
In the select function window, scroll down and select STDEV.

Step 7
Pressing the enter button, will open a dialog box. This dialog box will ask you to select the range. According to the calculations, you will have to select the range. After you select the range, press the enter tab. Pressing the enter button will give the standard deviation. This result will appear in the 12th cell.

Hope you must have understood how the methods mentioned above will help you in calculating standard deviation. These methods are simple to do, if you have proper inputs handy. With these formulas, you can calculate standard deviation from various data including population surveys.