Standard Deviation Calculator

Standard Deviation Calculator

Enter Data:

x - x̄:

(x - x̄)2:



Rows: 1 Mean: 0 ∑(x - x̄)2: 0 Variance S2: 0 Standard Deviation: 0 Alert: None

The fast and accurate standard deviation calculator for any statistics problem, probability solution, and easily compute other essential mathematical numerical.

This free and online tool calculates the standard deviation and as well the Mean, ∑(x – x̄)2 and Variance for a given data set of real numbers. It generates two primary results, the 1st is single results that calculate x – x̄, (x – x̄)2 and Z-score for every separate data set. The 2nd is the final results that calculate Mean, ∑(x – x̄)2, Variance S2, and Standard Deviation.

How to Use STD Deviation Calculator

  1. Add rows according to your required data set by using the button.
  2. Use the “Delete” button to removing any extra or unnecessary row.
  3. The “Calculate” button is simply using for the calculation of your input data set.
  4. The final Results “Box” provides the overall result containing Mean, ∑(x – x̄)2, Variance S2, and standard deviation.
  5. Single results shows (x – x̄), (x – x̄)2, and Z-score value for every separate data input.

Features & Functions

  1. Provide 7 separate results.
  2. 100% tested and accurate results.
  3. Unlimited data set inputs.
  4. Adding and deleting function.
  5. A user-friendly interface.

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.

High and Low

The high standard deviation shows that data are more spread out, and low means data are clustered around the mean. Close to zero SD show data point are near to mean, while high or low SD indicates data points are above or below the mean.


The following formula is used for standard deviation.

Formula: S = 1/N-1 ni=1(xi-x̄)2

Where: S = Sample Deviation,

N = Numer of data points,

= Sum,

xi = Individual value,

= Sample mean


The equation of standard deviation is below.

Equation: σ = ∑(xi-μ)2/N

Where: σ = Population Standard Deviation,

N = The Size of the Population,

xi = Each Value From the Population,

μ = The Population Mean


To make it easier and compared to the variance, the SD always uses a similar unit of measurement as the variable in an observation or question.


  1. We have 3 observations of heights in inches 60, 64, 63.
  2. The mean of observed values is –
  3. mean = (60 + 64 + 63)/3 = 62.33 inches.
  4. The variance of observed values is –
  5. variance = (60 – 62.33)^2 + (64 – 62.33)^2 + (63 – 62.33)^2/3 = 2.88 square inches.
  6. Convert it into the same unit as the observed values, take the square root of it for Standard deviation.
  7. S.D = 2.88 = 1.69 inches.


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 estimated values about the return of investment.


Some examples with the different conditions help you to understand the value of data.
  • A teacher found the 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 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 (S2) as the answer of the population for your required problem.


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

Formula: σ = √1/N ni=1(xi-μ)2

Where: xi = Individual value,

μ = Mean value,

N = Total number,

= Sum


The symbol for Population SD is:

σ = ∑(xi-μ)2/N

Where: σ = Population Standard Deviation


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

Making 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. 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.


In mathematical texts and equations, it is commonly represented by the lower case Greek letter sigma σ and may be abbreviated SD.

Sample Size Calculator

In statistics, the sample size is the amount of error that will be tolerated. If the 10% result is NO and 90% resulted as YES shows that an enormous amount of error will be tolerated. The uncertain amount of tolerance is confidence level, and for a larger sample size, a higher confidence level required.


In 20,000 peoples, 50% of people drink milk in the morning. If we repeat this survey for 377 people, then 95% of the time, our survey would get YES answer for between 45% and 55% of people.


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.


The formula of the mean is given below.

Formula: x̄ = 1/N ni=1x

Where: = Mean,

N = Total number of values,

= Sum,

x = Observed valued,

n = Sample size

Combined Mean

A mean of two or more different and separate data set points is known as a combined mean. While combined SD also calculated for data set like means.


The formula for combined mean as:

Formula: xc = m.xa + n.xb / m+n

Where: xc = Combined Mean

m = Number of Items in the First Set

xa = Mean of the First Set

n = Number of Items in the Second Set

xb = Mean of the Second Set

Standard Error

The standard error of the estimate of the mean is the sampling distribution and the variance between the mean of different samples. These errors are strictly dependent on the sample size, and when the sample size increases, the standard error falls more. So it is clear that if a sample is bigger, the closer the sample means to the population’s mean and close to the actual value.


The equation or formula for standard error as follows.

Equation: SE = σ/√n

Where: σ = Standard Deviation

n = Sample Size

Probability Distribution Calculator

The mathematical function like probability distribution gives probabilities of different outcomes for a calculation or experiment. In terms of sample space and the probabilities of events, it is the mathematical description of a random phenomenon.


If X is used as the outcome of a coin toss, then the probability distribution of X would take the value of 0.5 for X=Heads, and 0.5 for X=Tails that assuming the coin is fair.


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 S2. 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 S2 values as the answer to your requires problem.


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

Formula: S2 = ∑(xi-x̄)2/n-1

Where: S2 = Variance,

= Sum,

xi = Data set term,

 = 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.


Below is the given formula for RSD.

RSD = S x 100 / x̄

Where: RSD = Relative SD,

S = STD,

= Mean

RSD Calculator

RSD calculatore let you quickly find out and solve the problem and give you an accurect result and answer. You can use the above free  online calculator for Relative SD.


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

Solution: RSD = 0.2 x 100 / 5.4

RSD = 3.70 (answer)


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.


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


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.

FAQs - Standard Deviation Calculator

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

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

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.

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.

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

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


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.

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.

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).

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