An introduction to standard deviation along with its definition, types, and calculations

In statistics, the term standard deviation is a method that is widely used in descriptive statistics, hypothesis, and other relevant branches to evaluate problems easily. The variation in the collection of the tax on a daily, monthly, and yearly basis can be determined by drawing the data graph with the help of the standard deviation.

There is another term related and relevant to the standard deviation which is variance. It is used for taking the squared results of the data set. In this article, we will briefly introduce the term standard deviation along with its definition, types, and examples with calculations.

What is the standard deviation?

In statistics, a method that is helpful for calculating the measure of the variability of sample and population data set is known as the standard deviation. It is the average of the relation of data observations from the expected value (mean).

The square root (√) of the variance also gives the result of the standard deviation. It is measured in linear units. From the variance and the standard deviation, the term variance is less accurate as compared to the standard deviation.

The closeness and fairness of data observations from the mean are totally dependent on the smaller and larger output of the standard deviation. Such as the smaller output of the standard deviation tells the data observations are closer to the mean. While the larger value of the standard deviation indicates that the data values are far away from the mean.

Types of Standard Deviation

There are two kinds of data such as sample and population data so the term standard deviation must be divided into two categories depending upon the sample and population type of data values.

1. 1. Sample Standard Deviation

2. 2.Population Standard Deviation

Sample and population standard deviations are frequently used to calculate the different types of data sets to measure the variability in the data.

Population Standard Deviation

The term that deals with the population data observations to measure the estimated variability of data values from the mean is known as population standard deviation. Because the population is taken for the whole.

The term population standard deviation is denoted by “σ”.

The general formula to evaluate the population standard deviation is:

σ = √ [∑ (x_i – µ)²/ n]

You can follow the below steps to evaluate the population standard deviation.

1. 1. First of all, take the population data and evaluate the mean of the given population data.

2. 2. Subtract the population mean from each individual data value of the given population set.

3. 3. Take the square of each subtracted term in order to make them positive.

4. 4. Add all the squared terms and take the sum of squares.

5. 5. Divide the sum of squares by the total number of observations (n). It will give you the result of the variance.

6. 6. Take the square root of the variance in order to get the result of the standard deviation.

Let us take an example of this type of standard deviation to understand how to calculate it manually.

Example: for population data

Calculate the population standard deviation of the given population observations.

Population data = 5, 7, 9, 12, 14, 15, 21, 13

Solution

Step 1:First of all, take the given population data then add all the observations and divide the result by the total number of observations to get the population mean.

Population data = 5, 7, 9, 12, 14, 15, 21, 13

Sum of Population data = 5 + 7 + 9 + 12 + 14 + 15 + 21 + 13

Sum = 96

Total number of observation = n = 8

Population Mean = µ = 96/8

Population Mean = µ = 48/4 = 24/2 = 12

Step 2:Now subtract the mean from each data value and then take the square to make all the observations positive.

Data values	xi -µ	(xi -µ)2
5	5 – 12 = -7	(-7)2 = 49
7	7 – 12 = -5	(-5)2 = 25
9	9 – 12 = -3	(-3)2 = 9
12	12 – 12 = 0	(0)2 = 0
14	14 – 12 = 2	(2)2 = 4
15	15 – 12 = 3	(3)2 = 9
21	21 – 12 = 9	(9)2 = 81
13	13 – 12 = 1	(1)2 = 1

Step 3:Now add squared terms to get the sum of squares. 

∑ (x_i - µ)² = 49 + 25 + 9 + 0 + 4 + 9 + 81 + 1

∑ (x_i - µ)² = 178

Step 4:Now take the quotient of the sum of squares and the total number of observations (n).

∑ (x_i - µ)² / N = 178 / 8

∑ (x_i - µ)² / N = 89 / 4

∑ (x_i - µ)² / N = 22.25

Step 5: Now take the square root of the result of the variance to determine the sample standard deviation.

√[∑ (x_i - µ)² / N] = √22.25

√[∑ (x_i - µ)² / N] = 4.717

A standarddeviationcalculator.io is a helpful resource to evaluate the problems of standard deviation according to the formulas with steps.

Sample Standard Deviation

The term that deals with the sample data observations to measure the estimated variability of data values from the mean is known as sample standard deviation. Because the sample is taken from the whole just to estimate the results to avoid a larger calculation.

The term sample standard deviation is denoted by “s”.

The general formula to evaluate the sample standard deviation is:

s = √ [∑ (x_i – x̄)²/ n – 1]

You can follow the below steps to evaluate the sample standard deviation.

1. 1. First of all, take the sample data and evaluate the mean of the given sample data.

2. 2. Subtract the sample mean from each individual data value of the given sample set.

3. 3. Take the square of each subtracted term in order to make them positive.

4. 4. Add all the squared terms and take the sum of squares.

5. 5. Divide the sum of squares by the degree of freedom (n-1). It will give you the result of the variance.

6. 6. Take the square root of the variance in order to get the result of the standard deviation.

Let us take an example of this type of standard deviation to understand it more accurately.

Example: for sample data

Evaluate the measure of the variability of the given sample data.

Sample data = 2, 1, 8, 5, 7, 9, 11, 14, 15, 12, 4

Solution

Step 1:First of all, take the given sample data then add all the observations and divide the result by the total number of observations to get the sample mean. 

Sample data = 2, 1, 8, 5, 7, 9, 11, 14, 15, 12, 4

Sum of Sample data = 2 +1 + 8 + 5 + 7 + 9 + 11 + 14 + 15 + 12 + 4

Sum = 88

Total number of observation = N = 11

Sample Mean = x̄ = 88/11

Sample Mean = x̄ = 8

Step 2:Now subtract the mean from each data value and then take the square to make all the observations positive.

Data values	xi - x̄	(xi - x̄)2
2	2 – 8= -6	(-6)2 = 36
1	1 – 8 = -7	(-7)2 = 49
8	8 – 8 = 0	(0)2 = 0
5	5 – 8 = -3	(-3)2 = 9
7	7 – 8 = -1	(-1)2 = 1
9	9 – 8 = 1	(1)2 = 1
11	11 – 8 = 3	(3)2 = 9
14	14 – 8 = 6	(6)2 = 36
15	15 – 8 = 7	(7)2 = 49
12	12 – 8 =4	(4)2 = 16
4	4 – 8 = -4	(-4)2 = 16

Step 3:Now add squared terms to get the sum of squares.

∑ (x_i - x̄)² = 36 + 49 + 0 + 9 + 1 + 1 + 9 + 36 + 49 + 16 + 16

∑ (x_i - x̄)² = 222

Step 4:Now take the quotient of the sum of squares and the degree of freedom (n-1).

∑ (x_i - x̄)² /(n – 1) = 222 / 11 – 1

∑ (x_i - x̄)² / (n – 1) = 222 / 10

∑ (x_i - x̄)² / (n – 1) = 111/5 = 22.2

Step 5: Now take the square root of the result of the variance to determine the sample standard deviation.

√[∑ (x_i - x̄)² / (n – 1)] = √22.2

√[∑ (x_i - x̄)² / (n – 1)] = 4.71

Final words

Now you can grab all the basics of the measure of the variability through this post. We have discussed all the basics of sample and population standard deviation along with their formulas and solved examples.