The standard deviation is frequently used in statistics to determine the spread of observations of the given data set. The process of finding the standard deviation is similar to the variance as the standard deviation is the square root of the variance.

The result of the standard deviation is in the same units of the measurement while the variance has the squared units of observations. According to this fact, the standard deviation is more accurate than the variance.

In this article, we’ll study the definition and formulas of the standard deviation along with solved examples.

**What is the Standard Deviation?**

In probability distribution, the standard deviation is similar to the variance and is used to measure the deviation of the given observation from the mean. This is the process of finding the difference between all the terms from the expected value.

The data of the standard deviation can be taken in two ways:

- Population
- Sample

The standard deviation uses population observation if the measure of the spread of the value is to be found from the whole population. While the term sample is used to measure the approximated values by taking some observations from the whole.

**Types of the Standard Deviation**

In statistics, there are two types of standard deviation.

- Population standard deviation
- Sample standard deviation

Let’s discuss the types of the standard deviation briefly.

**Population Standard Deviation **

In probability, the population standard deviation is a square root of the population variance. While the population variance (σ^{2}) is the average of the differences from each observation of the population from the mean.

The population standard deviation formula must be divided by the total number of observations available in the population data. The population is the whole data e.g., the number of boys in a city.

The formula of the population standard deviation is:

**σ = √ [∑ (x _{i} – μ)^{2}/n]**

**Sample Standard Deviation **

In probability distribution, the sample standard deviation measures scattering data from the mean. The sample data of the standard deviation is statistics and used to measure some of the data from the whole.

The sample standard deviation is the square root of the sample variance. All the working and calculation of the sample standard deviation are similar to sample variance just a difference of square root is present between these terms.

The formula of the sample standard deviation is:

**s = √ [∑ (x _{i} – x̅)^{2}/n – 1]**

**How to calculate the standard deviation?**

The standard deviation can be calculated easily either by using its formulas or a standard deviation calculator. Follow the steps below to calculate standard deviation problems with sample and population data.

- First of all, find the mean of given sample or population data values (x̅ or μ) by finding the sum of the given values.
- After finding the mean of the sample or population observation, take the difference of each data value from the mean of sample or population data values. The difference between the mean and data values is said to be the deviation and the deviation can either be positive or negative.
- After that take the square of each deviation to make all the terms positive.
- Determine the summation of the squared deviations.
- Divide the summation of the squared deviation by “n” for population standard deviation or by n – 1 for sample standard deviation.
- In the end, take the square root of the quotient of the summation of squared deviation and the total number of observations to get the population or sample standard deviation.

**Example-I: For sample standard deviation**

Calculate the sample standard deviation of the given sample data.

3, 6, 8, 9, 14, 15, 21, 24, 26

**Solution **

**Step-I:** First of all, find the mean of the given observations of sample data.

Sum of sample values = 3 + 6 + 8 + 9 + 14 + 15 + 21 + 24 + 26

= 126

Total number of observations = n = 9

Mean of sample data set = x̅ = 126/9 = 42/3

= 14

**Step-II:** Now find the deviation of each data value from the mean.

x_{1} – x̅ = 3 – 14 = -11

x_{2} – x̅ = 6 – 14 = -8

x_{3} – x̅ = 8 – 14 = -6

x_{4} – x̅ = 9 – 14 = -5

x_{5} – x̅ = 14 – 14 = 0

x_{6} – x̅ = 15 – 14 = 1

x_{7} – x̅ = 21 – 14 = 7

x_{8} – x̅ = 24 – 14 = 10

x_{9} – x̅ = 26 – 14 = 12

**Step-III:** Find the square of deviations.

(x_{1} – x̅)^{2} = (-11)^{2} = 121

(x_{2} – x̅)^{2} = (-8)^{2} = 64

(x_{3} – x̅)^{2} = (-6)^{2} = 36

(x_{4} – x̅)^{2} = (-5)^{2} = 25

(x_{5} – x̅)^{2} = (0)^{2} = 0

(x_{6} – x̅)^{2} = (1)^{2} = 1

(x_{7} – x̅)^{2} = (7)^{2} = 49

(x_{8} – x̅)^{2} = (10)^{2} = 100

(x_{9} – x̅)^{2} = (12)^{2} = 144

**Step IV:** Now find the summation of the squared deviations.

∑ (x_{i} – x̅)^{2} = 121 + 64 + 36 + 25 + 0 + 1 + 49 + 100 + 144

∑ (x_{i} – x̅)^{2} = 540

**Step-V:** Now divide the summation of the squared deviations by n – 1.

∑ (x_{i} – x̅)^{2} / n – 1 = 540 / 9 – 1

= 540 / 8

= 67.5

**Step VI:** Take the square root of the summation of the squared deviations by n – 1 to get the sample standard deviation.

√ [∑ (x_{i} – x̅)^{2} / n – 1] = √67.5

= 8.216

**Example-II: For population standard deviation**

Calculate the population standard deviation of the given population data.

3, 5, 7, 11, 19, 22, 23, 25, 29

**Solution **

**Step-I:** First of all, find the mean of the given observations of population data.

Sum of population values = 3 + 5 + 7 + 11 + 19 + 22 + 23 + 25 + 29

= 144

Total number of observations = n = 9

Mean of population data set = μ = 144/9 = 48/3

= 16

**Step-II:** Now find the deviation of each data value from the mean.

x_{1} – μ = 3 – 16 = -13

x_{2} – μ = 5 – 16 = -11

x_{3} – μ = 7 – 16 = -9

x_{4} – μ = 11 – 16 = -5

x_{5} – μ = 19 – 16 = 3

x_{6} – μ = 22 – 16 = 6

x_{7} – μ = 23 – 16 = 7

x_{8} – μ = 25 – 16 = 9

x_{9} – μ = 29 – 16 = 13

**Step-III:** Find the square of deviations.

(x_{1} – μ)^{2} = (-13)^{2} = 169

(x_{2} – μ)^{2} = (-11)^{2} = 121

(x_{3} – μ)^{2} = (-9)^{2} = 81

(x_{4} – μ)^{2} = (-5)^{2} = 25

(x_{5} – μ)^{2} = (3)^{2} = 9

(x_{6} – μ)^{2} = (6)^{2} = 36

(x_{7} – μ)^{2} = (7)^{2} = 49

(x_{8} – μ)^{2} = (9)^{2} = 81

(x_{9} – μ)^{2} = (13)^{2} = 169

**Step IV:** Now find the summation of the squared deviations.

∑ (x_{i} – μ)^{2} = 169 + 121 + 81 + 25 + 9 + 36 + 49 + 81 + 169

∑ (x_{i} – μ)^{2} = 740

**Step-V:** Now divide the summation of the squared deviations by n.

∑ (x_{i} – μ)^{2} / n = 740 / 9

= 740 / 9

= 82.2222

**Step VI:** Take the square root of the summation of the squared deviations by n to get the population standard deviation

√ [∑ (x_{i} – μ)^{2} / n] = √82.2222

= 9.068

**Calculating Standard Deviation – Summary**

In this article, we’ve discussed the definition, types, and formulas of the standard deviation. Now you can grab all the basics of the standard deviation from this post. The standard deviation is frequently used and helpful for hypothesis testing and z score calculations.

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