# Standard Deviation Calculator

**FINAL RESULTS**

This online calculator calculates the standard deviation and as well the Mean, ∑(x – x̄)^{2} and Variance for a data set of real numbers. It generates two primary results, the **1 ^{st}** is single results that calculate x – x̄, (x – x̄)

^{2}and Z-score for every separate data. The

**2**is the final results that calculate Mean, ∑(x – x̄)

^{nd}^{2}, Variance S

^{2}and Standard Deviation.

## Use of Calculator

- Add rows according to your required data set by using the button.
- Use the delete button to removing any extra or unnecessary row.
- The calculate button is simply using for the calculation of your input data set.
- Final Results box provides the overall result containing Mean, ∑(x – x̄)
^{2}, Variance S^{2}and standard deviation. - Single results shows (x – x̄), (x – x̄)
^{2}and Z score value for every separate data input.

## Features & Functions

- Gives 7 separate results.
- 100% tested and accurate results.
- Fast calculation.
- Unlimited data set inputs.
- Adding and deleting function.
- A user-friendly interface.
- 1.0 Updated version.

## Standard Deviation

In statistical terminology, it is the measurement of the dispersion of the data set. It is donated by Greek letter **σ**. The higher standard deviation tells that the values of a data set are spread out very much while the lower indicates that the data set is very near to the mean. Many times it is used for the measurement of statistical results such as error margins and due to this, it is also known as the standard means error.

### Formula

The following formula is used for standard deviation.

*Formula:* **S** = **√1/N-1 ^{n}∑_{i=1}(x_{i}-x̄)^{2}**

**Where:** **S** = sample deviation, **N **= Numer of data points, **∑ **= Sum, **x _{i}** = Individual value,

**x̄**= Sample mean

### Applications

The standard deviation is mainly used for the industrial configuration to test models of real-world data like to take control of the quality of the products.

- It is used to find out the maximum and minimum time percentage values of products.
- It is also used for the climatic and weather prediction of different regions.
- Another important application is in the finance field, it provides the estimate values about the return of investment.

### Examples

Some examples with the different conditions that help you to understand the value of data.

- A teacher found mean score was 85% after the class test, after calculating the standard deviation of other tests the result values found very small which indicate that most students take marks very close to 85%.
- A financial researcher takes a survey to find out how a large of people might answer the same question. So after the survey, the lower standard deviation will indicate the answer is very efficient for peoples.
- A reporter is analyzing the high temperature for different regions for different dates versus the real high temperature that recorded on every date. so the lower standard deviation will show the forecast reliability.

## Population Standard Deviation

In statistics, the population standard deviation is the square root of the variance of a data set and also the definition of **σ**. It is used when we need to measure the standard deviation of the entire population. So after the calculation just note down the value of Variance (**S ^{2}**) as the answer of the population for your required problem.

### Formula

The below formula is using for the manual calculation of the entire population of standard deviation.

*Formula:* **σ = √1/N ^{n}∑_{i=1}(x_{i}-μ)^{2 }**

**Where:** **x _{i}** = Individual value,

**μ**= Mean value,

**N**= Total number,

**∑**= Sum

### Example

The following problem is the best example of an entire population.

*Example:* **μ** = (1+2+4+6+8)/4 = 5.25

*Solution:* **σ** = [(1-5.25)^{2} + (2-5.25)^{2}+…+(8-5.25)^{2})]/5

**σ** = (18.06+10.56+1.56+0.56+7.56)/5

**σ** = 7.66 *(answer)*

## Sample Standard Deviation

To make every member sample in the population is not possible. So it is used to determine the large population of the sample data set, such as **x1….xN***. ***x̄** shows the mean of the sample data set, and **N** shows the size of the sample data point. The sample standard deviation is the common estimator for **σ** and denoted by **S**. Just like the sample mean, the sample standard deviation also has no single estimator and that is unbiased, enough and has maximum similarity.

## Mean

In standard deviation, the sample mean is the average and the sum of all observed outcomes and by dividing the total number of events. In mathematical terms, the sample mean is denoted by x̄ and used for many purposes.

### Formula

The formula of the mean is given below.

*Formula:* **x̄ = 1/N ^{n}∑_{i=1}x**

**Where:** **x̄** = Mean, **N** = Total number of values, **∑ **= Sum, **x** = Observed valued, **n** = Sample size

## Variance

The variance and standard deviation are the mathematics basic concept and are mostly used for the measurement of spread while the variance is denoted by **S ^{2}**. The variance is the measure that how a data set is spread out. It is considered as the average squared deviation of a data set from the mean of each value. Write down the

**S**values as the answer to your requires problem.

^{2}### Formula

This is the formula of variance that will solve your problem through manual calculation by putting your given values in it.

*Formula:*** S ^{2}** =

**∑(x**

_{i}-x̄)^{2}/n-1**Where:** **S ^{2}** = Variance,

**∑**= Sum,

**x**= Data set term,

_{i}**x̄**= Sample mean,

**n**= Sample size

### Range Variance

The range of standard deviation is the difference between the highest and the smallest values of the data set. In other words, we can say that it is the representation of a single number of a data set. It is very easy to find out the range by this calculator because it shows a separate and single result for every input data.

## Relative Standard Deviation

It is the special form of standard deviation and its shorts form is (RSD). By comparing to the mean of a specific data set, it indicates whether the regular standard deviation is higher or smaller from the mean, It also tells how the data are closely rounded from the mean.

### Formula

Below is the given formula for RSD.

**RSD = S x 100 / x̄**

**Where:** **RSD** = Relative SD, **S** = S**TD**, **x̄** = Mean

### Example

*Problem:* If STD = 0.2, Mean = 5.4, Find the RSD

*Solution:* **RSD** = 0.2 x 100 / 5.4

**RSD** = 3.70 *(answer)*

## Average

The average deviation is the measurement of variability but its calculation is exactly the same as the standard deviation. Also, it is using positive values instead of negative values. In statistics, we can say that it is the absolute value difference between the data point and their means. Follow the steps to calculate the average deviation.

**Steps**

- Do the operation of subtraction between all data points and each data values.
- Add them and average the positive differences values.

### Difference

The STD is mostly used for creating different marketing, financial and investment strategies because it predicts the performance trends. While the average deviation is used in later phases means that in the more complex and inner calculation.

# Frequently Ask Questions

## Why is the Standard Deviation is Preferred Over the Variance?

There are two main reasons that standard deviation is preferred over the variance that is below.

- It has the same units as the original statics.
- It can estimate the percentage of items in any section of the population.

## What Does a Negative Standard Deviation Mean?

This is a tricky question but standard deviation can not be negative as it is the measurement of dispersion that how much your data distanced from the mean so in this way the distance can never be negative. Its calculation is based on the square of the difference which makes it positive not what difference is.

## What is a Standard Deviation of 1?

A normal distribution with a standard deviation of 1 and a mean of 0 is called the standard normal distribution. In normal cases, the STD of 1 would be 1 standard deviation from the mean. For instance, a Z of -2.5 represents a value 2.5 standard deviations below the mean.

## How Do You Manually Calculate Standard Deviation?

The standard deviation formula may look confusing, but it will make sense after we break it down.

- Find the mean.
- For each data point, find the square of its distance to the mean.
- Sum the values from Step 2.
- Divide by the number of data points.
- Take the square root.

Done

## How Can I Calculate Standard Deviation in Excel?

It is very simple just use the Excel Formula =STDEV( ) and select the range of values that contain the data and you will get your answer.

## Is Standard Deviation a Percentage?

Because of its reliable scientific properties, 68 percent of the qualities in any informational collection exist in one standard deviation of the mean, and 95 percent exist in two standard deviations of the mean.

## Is It Better to Have a Higher or Lower Standard Deviation?

A high standard deviation shows that the data is spread (not better), and a low standard deviation indicates that the data are closely around from the mean (yes better).