Sample Standard Deviation Formula How To Calculate

Formula to Calculate Sample Standard Deviation

Mathematically, it represents as,

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where

xi = ith random variableX = Mean of the samplen = number of variables in the sample

Calculation of Sample Standard Deviation (Step by Step)

Examples

Example #1

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

Firstly, gather random variables from a population of a large number of variables. These variables will form a sample. The variables are denoted by xi. Next, determine the number of variables in the sample, and denote by n. Next, determine the sample’s mean 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. Next, compute the difference between each variable of the sample and the sample mean, i.e., xi – x. Next, calculate the square of all the deviations, i.e. (xi – x)2. Next, add all the squared deviations, i.e. ∑ (xi – x)2. Next, divide the summation of all the squared deviations by the number of variables in the sample minus one, i.e. (n – 1). Finally, the formula for sample standard deviation is calculated by computing the result’s square root, as shown below.

Given,

Sample size (n) = 5

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

Sample Mean

The calculation of the sample mean:

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

Sample Mean = 4

One can calculate the squares of the deviations of each variable as below,

(3 – 4)2 = 1(2 – 4)2 = 4(5 – 4)2 = 1(6 – 4)2 = 4(4 – 4)2 = 0

Now, one can calculate the sample standard deviation by using the above formula,

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. But, first, determine the sample standard deviation based on the ages of the 10 people: 23, 27, 33, 28, 21, 24, 36, 32, 29, and 25.

Sample size(n) = 10

The calculation of sample mean:

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

Sample Mean = 27.8

One can calculate the squares of the deviations of each variable 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)}
ơ = 4.78

You can refer to the Excel sheet above to understand the detailed calculation.

Relevance and Uses

The concept of sample standard deviation is very important from a statistician’s perspective because a sample of data usually takes from a pool of large variables (population) from which the statistician expects to estimate or generalize the results for the entire population. The measure of standard deviation is no exception to this. Hence, the statistician has to assess the population standard deviation based on the sample drawn, and that is where such deviation comes into play.

Formula to Calculate Sample Standard Deviation#

Calculation of Sample Standard Deviation (Step by Step)#

Examples#

Example #1#

Example #2#

Relevance and Uses#

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