Standard Deviation: An introduction to its types and Calculations

Standard deviation is frequently used in statistics for different purposes in descriptive statistics, inferential statistics, and quality control. It has also wide uses in hypothesis testing, a measure of variability, normal distribution, and risk assessment.

In this article, we’ll learn the definition, types, and solved examples of the standard deviation.

In descriptive statistics, a measure that describes the variation and dispersion in the data set is said to be the standard deviation. It is a technique for measuring the data deviation from the expected value to measure the dispersion in a data set.

Types of standard deviation

There are two main types of standard deviation:

• Sample standard deviation

• Population standard deviation

Here is a brief introduction to the types of standard deviation.

Sample Standard Deviation

In standard deviation, sample data is used to measure the variation in data observations and is said to be the sample standard deviation. The measure of variability is frequently used in this type of standard deviation. The sample standard deviation is denoted by “s”.

When the subset of a larger population is available, then this type of standard deviation is used. It is generally used when you want to estimate the standard deviation of the entire population based on the sample.

• A larger sample standard deviation indicates greater variability in the data

• A smaller sample standard deviation indicates that the data is clustered more closely around the sample mean.

The formula for calculating the sample standard deviation is:

 $s=\sqrt{(\sum\left(x-\overline{x}\right)^{2}/\left(n-1\right)})$ s is the sample STD x is each data point, x̄ is the sample average (expected term) Σ is the sum of all the data points n is the total number in the sample subset of entire population.

Population Standard Deviation

In standard deviation, populationdata is used to measure the variation in data observations and is said to be the population standard deviation. The measure of variability is frequently used in this type of standard deviation. The populationstandard deviation is denoted by “σ”.

When entire observations of a larger population are available, then this type of standard deviation is used. It is generally used when you want to estimate the standard deviation of the entire population.

• A larger population standard deviation indicates greater variability in the data

• A smaller population standard deviation indicates that the data is clustered more closely around the population mean.

The formula for calculating the population standard deviation is:

 $\sigma=\sqrt{(\sum\left(X-\mu\right)^{2}/N})$ σ is the population standard deviation x is each data point μ is the population mean Σ is the sum of all the data points N is the total number in the set of entire population.

A population standard deviation calculator is a useful resource to evaluate the problems of entire population data according to the formula.

Guidelines for working with a standard deviation

Here are some guidelines to keep in mind when working with standard deviation:

• A lower standard deviation indicates that the data is clustered more closely around the mean.

• The standard deviation cannot be negative since it represents a measure of distance from the mean.

• Standard deviation can be influenced by extreme values.

What are the sample and population mean?

The sample mean and population mean are the average values of the sample and population data values that are helpful for the calculations of sample and population standard deviations.

 Sample mean Population mean The sample mean is a statistical term that refers to the average of a subset of data taken from a larger population. It is also known as the "sample average." Population mean is a statistical term that refers to the average of all the values in a specific population. It is also known as the "population average." The sample mean “x̄” is evaluated by adding up all the values “Σx” in the sample and dividing the sum “Σx” by the number of values in the sample. Population mean “μ” is evaluated by adding up all the values in the population and dividing the sum “Σx” by the number of values in the population. It is denoted by “x̄” (x bar). It is denoted by the symbol "μ" (mu). The formula for calculating the sample mean is: x̄ = (Σx) / n The formula for calculating the sample mean is: μ = (Σx) / n Where x̄ is the sample mean Σx is the sum of all the values in the sample n is the number of values in the sample. Where μ is the sample mean Σx is the sum of all the values in the sample n is the number of values in the sample.

Examples of sample and population standard deviation

Example-1: for population data

Find population standard deviation if:

42, 44, 48, 55, 60, 69, 71, 74, 75, 82

Solution

Step 1:First of all, calculate the population mean “μ”.

 Given data x = 42, 44, 48, 55, 60, 69, 71, 74, 75, 82 Sum Σx = 42 + 44 + 48 + 55 + 60 + 69 + 71 + 74 + 75 + 82 = 620 population mean μ = (Σx) ÷ n μ = 620 ÷ 10 = 62

Step 2:Now take the difference of each population observation from the population mean and square them.

 Data values xi – μ (xi - μ)2 42 42 – 62 = -20 (-20)2 = 400 44 44 – 62 = -18 (-18)2 = 324 48 48 – 62 = -14 (-14)2 = 196 55 55 – 62 = -7 (-7)2 = 49 60 60 – 62 = -2 (-2)2 = 4 69 69 – 62 = 7 (7)2 = 49 71 71 – 62 = 9 (9)2 = 81 74 74 – 62 = 12 (12)2 = 144 75 75 – 62 = 13 (13)2 = 169 82 82 – 62 = 20 (20)2 = 400

Step 3:Now add the values of the third column.

∑ (xi - μ)2 = 400 + 324 + 196 + 49 + 4 + 49 + 81 + 144 + 169 + 400

∑ (xi - μ)2 = 1816

Step 4:Now divide the above sum by n.

∑ (xi - μ)2 ÷ (n) = 1816 ÷ 10

∑ (xi - μ)2 ÷ (n) = 908 ÷ 5

∑ (xi - μ)2 ÷ (n) = 181.6

Step 5: Now take the square root.

√[∑ (xi - μ)2 ÷ (n)] = √181.6

√[∑ (xi - μ)2 ÷ (n)] = 13.48

Example-2: for sample data

Find sample standard deviation if:

12, 21, 8, 15, 20, 19, 31, 24, 15, 52, 36

Solution

Step 1:First of all, calculate the sample mean “x̄”.

 Given data x = 12, 21, 8, 15, 20, 19, 31, 24, 15, 52, 36 Sum Σx = 12 + 21 + 8 + 15 + 20 + 19 + 31 + 24 + 15 + 52 + 36 = 253 Sample mean x̄ = (Σx) ÷ n x̄ = 253 ÷ 11 = 23

Step 2:Now take the difference of each sample observation from the sample mean and square them.

 Data values xi - x̄ (xi - x̄)2 12 12 – 23 = -11 (-11)2 = 121 21 21 – 23 = -2 (-2)2 = 4 8 8 – 23 = -15 (-15)2 = 225 15 15 – 23 = -8 (-8)2 = 64 20 20 – 23 = -3 (-3)2 = 9 19 19 – 23 = -4 (-4)2 = 16 31 31 – 23 = 8 (8)2 = 64 24 24 – 23 = 1 (1)2 = 1 15 15 – 23 = 8 (8)2 = 64 52 52 – 23 = 29 (29)2 = 841 36 36 – 23 = 13 (13)2 = 169

Step 3:Now add the values of the third column.

∑ (xi - x̄)2 = 121 + 4 + 225 + 64 + 9 + 16 + 64 + 1 + 64 + 841 + 169

∑ (xi - x̄)2 = 1578

Step 4:Now divide the above sum by n – 1.

∑ (Xi - x̄)2 ÷ (N – 1) = 1578 ÷ 11 – 1

∑ (Xi - x̄)2 ÷ (N – 1) = 1578 ÷ 10

∑ (Xi - x̄)2 ÷ (N – 1) = 789 ÷ 5 = 157.8

Step 5: Now take the square root.

√ [∑ (Xi - x̄)2 ÷ (N – 1)] = √157.8

√ [∑(Xi - x̄)2 ÷ (N – 1)] = 12.56

Conclusion

The standard deviation is explained with its definition, types, and solved examples in this post. Now you can grab all the basics of the standard deviation from this article for the concept to calculations.