## Formula to Calculate Sample Standard Deviation

Sample standard deviation refers to the statistical metric that is used to measure the extent by which a random variable diverges from the mean of the sample and it is calculated by adding the squares of the deviation of each variable from the mean, then divide the result by a number of variables minus and then computing the square root in excel of the result.

Mathematically, it is represented as,

**σ = √ ∑**

_{i}^{n}(xi – X)^{2}/ (n-1)where

- x
_{i}= i^{th}random variable - X = Mean of the sample
- n = number of variables in the sample

### Calculation of Sample Standard Deviation (Step by Step)

**Step 1:**Firstly, gather random variables from a population of a large number of variables. These variables will form a sample. The variables are denoted by x_{i}.**Step 2:**Next, determine the number of variables in the sample, and it is denoted by n.**Step 3:**Next, determine the mean of the sample by adding all the random variables and dividing the result by the number of variables in the sample. The sample mean is denoted by x.

**Step 4:**Next, compute the difference between each variable of the sample and the sample mean, i.e., x_{i}– x.**Step 5:**Next, calculate the square of all the deviations, i.e. (x_{i}– x)^{2}.**Step 6:**Next, add all the of the squared deviations, i.e. ∑ (x_{i}– x)^{2}.**Step 7:**Next, divide the summation of all the squared deviations by the number of variables in the sample minus one, i.e. (n – 1).**Step 8:**Finally, the formula for sample standard deviation is calculated by computing the square root of the above-mentioned result, as shown below.

### Examples

#### Example #1

**Let us take the example of a sample of 5 students who were surveyed to see how many pencils they were using every week. Calculate the sample standard deviation of based on their given responses: 3, 2, 5, 6, 4**

Given,

- Sample size (n) = 5

Below is given data for the calculation of sample standard deviation.

**Sample Mean**

Calculation of Sample mean

Sample mean = (3 + 2 + 5 + 6 + 4) / 5

**Sample Mean = 4**

The squares of the deviations of each variable can be calculated as below,

- (3 – 4)
^{2}= 1 - (2 – 4)
^{2}= 4 - (5 – 4)
^{2}= 1 - (6 – 4)
^{2}= 4 - (4 – 4)
^{2}= 0

Now, the sample standard deviation can be calculated by using the above formula as,

- ơ = √ {(1 + 4 + 1 + 4 + 0) / (5 – 1)}

**Deviation will be –**

**ơ = 1.58**

Therefore, the sample standard deviation is 1.58.

#### Example #2

**Let us take the example of an office in New York where around 5,000 people work and a survey has been carried out on a sample of 10 people to determine the average age of the working population. Determine the sample standard deviation based on the ages of the 10 people given: 23, 27, 33, 28, 21, 24, 36, 32, 29, 25**

Given,

- Sample size(n) = 10

By using the above data we will first calculate the sample mean

**Sample Mean**

Calculation of Sample Mean

= (23 + 27 + 33 + 28 + 21 + 24 + 36 + 32 + 29 + 25) / 10

**Sample Mean = 27.8**

The squares of the deviations of each variable can be calculated as below,

- (23 – 27.8)
^{2}= 23.04 - (27 – 27.8)
^{2}= 0.64 - (33 – 27.8)
^{2}= 27.04 - (28 – 27.8)
^{2}= 0.04 - (21 – 27.8)
^{2}= 46.24 - (24 – 27.8)
^{2}= 14.44 - (36 – 27.8)
^{2}= 67.24 - (32 – 27.8)
^{2}= 17.64 - (29 – 27.8)
^{2}= 1.44 - (25 – 27.8)
^{2}= 7.84

**Deviation**

Now, the deviation can be calculated by using the above formula as,

- ơ = √ {(23.04 + 0.64 + 27.04 + 0.04 + 46.24 +14.44 +67.24 + 17.64 + 1.44 + 7.84) / (10 – 1)}

**Deviation will be –**

**ơ = 4.78**

You can refer to the given excel sheet above to understand the detailed calculation.

### Relevance and Uses

The concept of sample standard deviation is very important from the perspective of a statistician because usually, a sample of data is taken from a pool of large variables (population) from which the statistician is expected to estimate or generalize the results for the entire population. The measure of standard deviation is no exception to this, and hence, the statistician has to make an assessment of the population standard deviation on the basis of the sample drawn, and that is where such deviation comes into play.

### Recommended Articles

This has been a guide to Sample Standard Deviation Formula. Here we discuss the calculation of sample standard deviation along with examples and a downloadable excel template. You can learn more about excel modeling from the following articles –

- Standard Deviation Formula in Excel
- Formula of Relative Standard Deviation
- Portfolio Standard Deviation
- Compare Variance vs Standard Deviation

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