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# Standard Deviation Calculator

Enter the data set values to calculate the standard deviation (σ).

## Standard Deviation Calculator:

Use this standard deviation calculator to calculate the standard deviation either for a sample or population data set. You can use this calculator to find:

• Standard Deviation (s): How the given data is spread according to sample or population, with the steps shown
• Count (n): The total number of values in the given data set
• Sum (Σx): The sum of total values within the data set
• Mean (x̄): Average of the values contained in the data set
• Variance (s²): The average squared deviation of data values from the mean
• Coefficient of Variance: The relative ratio of the standard deviation to the mean
• Standard Error of The Mean: The variability of sample means around the population mean
• Frequency Table: The frequency of each value in a dataset

Limitation: The calculator is not able to handle complex data structures such as matrices or time series data etc, it only focuses on general numerical data sets.

## What Is Standard Deviation?

Standard deviation (σ) is a statistical measurement that informs how spread out the data points are in a set of data relative to the mean (average). A higher standard deviation points towards a higher variability in the data. A Greek symbol sigma “σ” is used to represent standard deviation. The following image shows the graphical representation of a normal distribution having a width of 1 band.

It can be used for:

• Understanding Data Spread: It helps to quantify how much the data points deviate from the mean.
• Identifying Outliers: If the values fall more than 2-3 standard deviations away from the mean, then they are considered outliers.
• Analyzing Data Consistency: Standard deviation helps to measure the dispersion of different data sets, even if they are not normally distributed

## Standard Deviation Formula:

### 1. Sample Standard Deviation:

The given formula is used for finding the standard deviation of a sample (subset of data drawn from the population):

$$s = \sqrt{\dfrac{1}{N – 1} \sum_{i=1}^N\left(x_{i} – \bar{x}\right)^2}$$

Where

• S = Sample standard deviation
• $$x_{i}$$ = Each single value in the data set
• x = Sample mean
• N = Total sample size

### 2. Population Standard Deviation:

When all the members of the population can be sampled, then the following standard deviation formula is used:

$$σ = \sqrt{\dfrac{1}{N} \sum_{i=1}^N\left(x_{i} – μ\right)^2}$$

Where

• σ = Population standard deviation
• $$x_{i}$$ = Individual value
• μ = Average mean value/expected value
• N = Total number of values

## Other Statistical Formulas Used By Our Calculator:

The standard deviation calculator uses the following formulas to make statistical calculations of standard deviation:

### Sum:

$$\text{Sum} = \sum_{i=1}^{n}x_i$$

### Size, Count:

$$\text{Size} = n = \text{count}(x_i)_{i=1}^{n}$$

### Mean:

For Sample:

$$\overline{x} = \dfrac{\sum_{i=1}^{n}x_i}{n}$$

For Population:

$$\mu = \dfrac{\sum_{i=1}^{n}x_i}{n}$$

### Sum of Squares:

For Sample:

$$\ SS = \sum_{i=1}^{n}(x_i - \overline{x})^{2}$$

For Population:

$$\ SS = \sum_{i=1}^{n}(x_i - \mu)^{2}$$

### Variance:

For Sample:

$$\ s^{2} = \dfrac{\sum_{i=1}^{n}(x_i - \overline{x})^{2}}{n - 1}$$

For Population:

$$\sigma^{2} = \dfrac{\sum_{i=1}^{n}(x_i - \mu)^{2}}{n}$$

## How To Calculate Standard Deviation?

These are the steps:

1. Calculate the mean
2. Measure the distance from the mean for all the values in the given data set
3. Calculate the square of the distance for each value
4. Sum all the squared values
5. Now divide the sum by “n” to obtain the population variance $$\ ?^2$$ for population standard deviation or by the “n – 1” to obtain the variance $$\ ?^{2}$$ for sample standard deviation
6. Find the square root of the answer, and that’s it. Now, you will have the standard deviation for the given set of data

## Example:

Suppose you have a data set (3, 4, 9, 7, 2, 5 ), find its standard deviation.

Solution:

Step #1(Calculate Mean Value):

$$\bar{x} = {\dfrac{3 + 4 + 9 + 7 + 2 + 5}{6}}$$

$$\bar{x} = \dfrac{30}{6}$$ $$\bar{x} = 5$$

Step #2(Calculate The Value Of $$\left(x_{i} – \bar{x}\right)$$:

 Data Values (xi) xi - x̅ (xi - x̅ )2 3 3 - 5 = -2 (-2)2 = 4 4 4 - 5 = -1 (-1)2 = 1 9 9 - 5 = 4 (4)2 = 16 7 7 - 5 = 2 (2)2 = 4 2 2 - 5= -3 (-3)2 = 9 5 5 - 5 = 0 (0)2 = 0

Step # 3 (Calculate Sample Standard Deviation):

Now we know that:

$$\ s = \sqrt{\dfrac{1}{N – 1} \sum_{i=1}^N\left(x_{i} – \bar{x}\right)^2}$$

$$\ s = \sqrt {\dfrac { 4+1+16+4+9+0}{ 6-1}}$$

$$\ s = \sqrt {\dfrac { 34 }{5}}$$

$$\ s = \sqrt {6.8}$$

Standard Deviation =$$\ s = 2.6076...$$

## FAQ’s:

### Why Is Standard Deviation Important?

• Understanding Data Spread: Standard deviation measures how spread out data points are in a dataset. A high standard deviation means that the data points are more scattered. On the other hand, a low standard deviation means that the data points are clustered near to the mean.
• Financial Risk Assessment: Standard deviation helps to measure the volatility associated with investments. A high standard deviation indicates fluctuations are more (riskier)
• Identifying Errors in Surveys: It helps identify unexpected variations in the survey reports. For instance, if a question has a high standard deviation, then it means the question was poorly worded or possibly confusing.
• Quality Control in Manufacturing: It helps to assess the changes in the manufactured product and its deviation from the range in which it should fall. If the value falls outside the range, then it may be crucial for you to make changes in the production line to have the expected quality of the final product

### What Is The Standard Deviation of 5 5 9 9 9 10 5 10 10?

The standard deviation of data set {5 5 9 9 9 10 5 10 10} is approximately 2.2913. To check your answers, use the above standard deviation calculator.

### Is A High or Low Standard Deviation Better?

It depends on what you are trying to understand about your data:

• High Standard Deviation: A high standard deviation value indicates that the data is widespread from the mean. It is beneficial if you want to understand the full range of values in your set of data and how much variation exists
• Low Standard Deviation: It indicates that the data is tightly clustered to the mean. A low standard deviation is useful, if you want to focus on a single value to represent your data, assuming the data is normally distributed

References:

From the source of Wikipedia: General understanding and basic examples

From the site of scribbr.com: Formulas for population and sample standard deviation