# Standard Deviation Calculator

Our online standard deviation calculator determines the standard deviation, mean, variance, sum, coefficient of variance, statistical population, and error margin of the given set of data. Our sd calculator makes your calculations fast and gives accurate results.

## What Is the Standard Deviation?

“Standard deviation is the measurement of the variation between the given group or set of values.”

• The lower standard deviation shows the values are closer to the mean of the set.
• A high standard deviation indicates that values are spread in the wider range.

## Formula of Standard Deviation:

Our mean and standard deviation calculator online makes standard deviation calculations fast and accurate by using teh following equation:

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

Population standard deviation formula is:

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

Where

• σ is the population standard deviation
• xi is each value in the set of the data
• µ is the average mean value
• N is the total number of values

## How To Calculate Standard Deviation?

In order to understand how to find standard deviation of the mean, take a look at the example below.

### Example:

The math test scores of different students are 51, 49, 45, 91, and 97. How to calculate standard deviation?

#### Solution:

To find the standard deviation of the given class, we will find this with the help of the Sample standard deviation calculator formula:

$$SD = σ = \sqrt\frac{\sum(x-µ)^2}{N}$$

µ = 51+49+45+91+94/5

µ = 330/5

µ = 66

xi xi-µ (xi-µ)^2
51 51-66 = -16 (-16)^2 = 256
49 49-66 = -17 (-17)^2 = 289
45 45-66 = -21 (-21)^2 = 441
91 91-66 = 25 (25)^2 = 625
94 94-66 = 28 (28)^2 = 784
2395

Finding standard deviation from mean:

SD = σ = 2395/5

SD = σ = 48.94

## Working of Standard Deviation Calculator:

This standard deviation solver provides complete work with incredible speed. To use it, follow the steps below!

Input:

• Select the data set nature (Population or Sample)
• Enter the values of the data set in designated field
• Tap Calculate

Output:

Our variance and standard deviation calculator online will give you the following results:

• Standard deviation, variance, and mean of the data set
• Sum of the variance values
• Coefficient of the variance
• Standard error of the mean
• Sum of the square of the numbers
• Table for a given database
• Step-by-step calculations

## FAQs:

### What Does The Standard Deviation Depend On?

The standard deviation of the data is independent of any change in origin, but it depends on the change in the scale.

### What Reduces The Standard Deviation?

The standard deviation is inversely proportional to the mean of the sampling distribution. The standard deviation decreases as the sample size used to calculate the mean for sampling distribution increases.

## References:

From the source Wikipedia: Standard deviation, Basic examples, Definition of population values, Estimation, Identities, and mathematical properties.

From the source Khan Academy: sample standard deviation, Population, calculate sample standard deviation, sample variance.