Sample Standard Deviation Formula
Before learning the sample standard deviation formula, let us see when do we use it. In a practical situation, when the population size N is large it becomes difficult to obtain value x_{i} for every observation in the population and hence it becomes difficult to calculate the standard deviation (or variance) for the population. In such cases, we can estimate the standard deviation by calculating it on a sample of size n taken from the population of size N. This estimated variance is called the sample standard deviation(S). Let us explore the sample standard deviation formula below.
What Is the Sample Standard Deviation Formula?
There are two types of standard deviations, population standard deviation and sample standard deviation. Given a sample of data (observations) for the random variable x, its sample standard deviation formula is given by
\(S = \sqrt{\dfrac{1}{n1} \sum^{n}_{i=1}(x_i  \bar{x})^2} \)
Here,
 \(\overline{x}\) = sample average
 x = individual values in sample
 n = count of values in sample
Let us see the applications of sample standard deviation formula in the below solved examples.
Solved Examples using Sample Standard Deviation Formula

Example 1: There are 45 students in a class. 5 students were randomly selected from this class and their weights(in kg) were recorded as follows:
51 38 79 46 57 Calculate the sample standard deviation of their weights. (Use √962.73 = 31)
Solution:
N = 5
Mean (\(\bar{x}\))= \(\dfrac{51+38+79+46+57}{5}\) = 54.2 kg
Using the sample standard deviation formula,
S = \( \sqrt{\dfrac{\Sigma (x_i\bar{x})^2}{N1}} \) = \( \sqrt{\dfrac{(5154.2)^2 +(3854.2)^2 +(7954.2)^2 +(4654.2)^2 +(5754.2)^2}{4}} \) = 15.5
Answer: Sample Standard Deviation for this data is 15.5

Example 2: There are 4smallsized trees in a garden. Given below are the heights (in cm) of these trees.
Height (in cm)
812 836 982 769 Find the sample standard deviation of their heights using the sample standard deviation formula. (Take √23.1= 4.8)
Solution:
Sample size (N) = 4.
Let X_{i }be the height of the i^{th} tree.
Sample Mean (X̄) = \( \dfrac{812+836+982+769}{4} \) = 849.75 cm
Sample Variance = \( \dfrac{\sum^{N}_{i=1} (X_i  \bar{X})}{N1} \) = \( \dfrac{\sum^{4}_{i=1} (X_i  849.75)}{3} \) = \( \dfrac{(812  849.75)^2 + (836  849.75)^2 + (982  849.75)^2 + (769  849.75)^2}{3} \) = 92.4 cm^{2}
Sample Standard Deviation = \(\sqrt{92.4}\) = \(2 \sqrt{23.1}\) = 9.6 cm
Answer: Sample Standard Deviation for this data is 9.6 cm