# Standard deviation: Definition, Formulas, Uses, and Examples

In statistics, a standard deviation is a measure of the variability of a set of data points. It is also used to measure the spread of data from the average value. In data analysis and interpretation, the study of standard deviation is very essential.

The term standard deviation is very helpful to identify unusual data points, comparing data sets, evaluate the reliability of data, and make predictions based on statistical models. In this article, you will get the comprehensive guide and basics concepts of the term standard deviation such as definition, formulas, uses, and solved examples.

## What is the standard deviation?

Standard deviation is a statistical measure that is used to measure how much the points of the given data set differ from the expected value of the data set. The amount of dispersion in a data set is measured with this statistical measure.

In other words, the standard deviation is a statistical technique that deals with the measure of data variation from the average value of the data set. It is also used to make a comparison between various data sets with the help of quantitative measures of the level of variability.

 Low standard deviation High standard deviation A low STD shows that the data points are more collected around the average value A high STD shows that the data points are more spread out.

## Formulas for calculating standard deviation

### The formula for calculating sample standard deviation.

The formula for calculating the sample standard deviation of a set of data is as follows:

s = sqrt((Σ(xi – x̄)2) / (n – 1))

Where:

• s is the sample standard deviation
• Σ represents the sum of
• xi is each data point in the data set
• is the mean of the data set
• n is the number of data points in the data set
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### The formula for calculating population standard deviation.

The formula for calculating the population standard deviation of a set of data is as follows:

σ = sqrt((Σ(xi – µ)2) / (n))

Where:

• σ is the population standard deviation
• Σ represents the sum of
• xi is each data point in the data set
• µ is the mean of the data set
• n is the number of data points in the data set

## What is the use of STD in data analysis?

In data analysis, the term standard deviation is very essential for identifying the level of variability and comparing different data sets. Such as in finance and scientific experiments, the term standard deviation is used frequently.

 Finance Scientific Experiments In finance, the standard deviation is used to measure the risk associated with a particular investment.·         A higher standard deviation indicates a higher risk level.·         A lower standard deviation indicates a lower risk level. In scientific experiments, standard deviation helps to evaluate the significance of results by indicating how much variation there is in the data.

## How to calculate standard deviation?

Follow the below steps to calculate the standard deviation of a set of data

 Steps Symbol Calculate the sample or population average of the data set by adding all the data points together and dividing by the number of data points. x̄ = Σxi / n Subtract the mean “x̄” from the data point to get the deviation from the mean. xi – x̄ Square each deviation from the average value. (xi – x̄)2 Add up all the squared deviations from the average value. Σ (xi – x̄)2 Divide the sum of the squared deviations by the number of data points minus one (n – 1) or the number of data points. Σ (xi – x̄)2 / n – 1Σ (xi – µ)2 / n Take the square root of the result to get the standard deviation. Sqrt [Σ (xi – x̄)2 / n – 1]Sqrt [Σ (xi – µ)2 / n]

An SD calculator is an online source to evaluate the problems of STD without any difficulty according to the above mentioned steps. You can take help from the below solved examples of STD to evaluate it manually.

Example I: for population data

Find the variability in the given data set if it is a population set of data.

3, 12, 2, 6, 1, 14, 5, 9, 11

Solution

Step 1: Take the given population data and find the population average.

 Given data x = 3, 12, 2, 6, 1, 14, 5, 9, 11 Sum Σx = 3 + 12 + 2 + 6 + 1 + 14 + 5 + 9 + 11 = 63 population mean μ = (Σx) ÷ nμ = 63 ÷ 9 = 7

Step 2: Now subtract the average population from each observation of the population data set.

 Data values xi – μ (xi – μ)2 3 3 – 7 = -4 (-4)2 = 16 12 12 – 7 = 5 (5)2 = 25 2 2 – 7 = -5 (-5)2 = 25 6 6 – 7 = -1 (-1)2 = 1 1 1 – 7 = -6 (-6)2 = 36 14 14 – 7 = 7 (7)2 = 49 5 5 – 7 = -2 (-2)2 = 4 9 9 – 7 = 2 (2)2 = 4 11 11 – 7 = 4 (4)2 = 16

Step 3: Now Add up all the squared deviations from the average value.

∑ (xi – μ)2 = 16 + 25 + 25 + 1 + 36 + 49 + 4 + 4 + 16

∑ (xi – μ)2 = 176

Step 4: Now Divide the sum of the squared deviations by the number of data points.

∑ (xi – μ)2 ÷ (n) = 176 ÷ 9

∑ (xi – μ)2 ÷ (n) = 19.56

Step 5: Now Take the square root of the result to get the standard deviation.

√[∑ (xi – μ)2 ÷ (n)] = √19.56

√[∑ (xi – μ)2 ÷ (n)] = 4.422

Example-2: for sample data

Find the variability in the given data set if it is a sample set of data.

12, 14, 15, 17, 2, 4, 9, 21, 24, 32

Solution

Step 1: Take the given sample data and find the sample average.

 Given data x = 12, 14, 15, 17, 2, 4, 9, 21, 24, 32 Sum Σx = 12 + 14 + 15 + 17 + 2 + 4 + 9 + 21 + 24 + 32 Sample mean x̄ = (Σx) ÷ nx̄ = 150 ÷ 10 = 15

Step 2: Now subtract the average sample from each observation of the sample data set.

 Data values xi – x̄ (xi – x̄)2 12 12 – 15 = -3 (-3)2 = 9 14 14 – 15 = -1 (-1)2 = 1 15 15 – 15 = 0 (0)2 = 0 17 17 – 15 = 2 (2)2 = 4 2 2 – 15 = -13 (13)2 = 169 4 4 – 15 = -11 (-11)2 = 121 9 9 – 15 = -6 (-6)2 = 36 21 21 – 15 = 6 (6)2 = 36 24 24 – 15 = 9 (9)2 = 81 32 32 – 15 = 17 (17)2 = 289

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

∑ (xi – x̄)2 = 9 + 1 + 0 + 4 + 169 + 121 + 36 + 36 + 81 + 289

∑ (xi – x̄)2 = 746

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

∑ (Xi – x̄)2 ÷ (N – 1) = 746 ÷ 10 – 1

∑ (Xi – x̄)2 ÷ (N – 1) = 746 ÷ 9

∑ (Xi – x̄)2 ÷ (N – 1) = 82.89

Step 5: Now take the square root.

√ [∑ (Xi – x̄)2 ÷ (N – 1)] = √82.89

√ [∑(Xi – x̄)2 ÷ (N – 1)] = 9.104

## Conclusion

The term standard deviation is a very essential system of measurement in statistics that are used in data analysis for various purposes. The calculations of the standard deviation can be done with the help of sample and population formulas.

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