Standard Deviation Calculator

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Standard Deviation Calculator

The Standard Deviation is the average of the variability given. On average, it gives how far a value lies and differs from the mean. In case of a high standard deviation, the calculator gives values that differ from the mean. At the same time, the low value depicts the values clustered.


In simple terms, the numbers are separated by commas to calculate the standard deviation. It could be done with the help of variance, mean, sum, and margin of error.

Similarly, a higher standard deviation implies a wide range of values. It is similar to other mathematical and statistical concepts. There are various situations in which standard deviation is used. It is used with different equations. Concerning expressing population variation, the standard variation is used in measuring statistical results like the margin.

The six steps to calculate the Standard Deviation

It could be calculated via software or the Standard Deviation Calculator. But the following are the steps involved.

Finding the mean value

To find the mean value. It is advised to add all the scores and divide them by several scores.

Find the standard deviation for each score

Subtracting mean from each score. It helps in getting the deviation of the mean.

Square each deviation

Multiplying each deviation with the mean. It gives positive results.

Find the sum of Squares

Adding all the square deviation. It is called the sum of squares.

Finding the variance

Dividing the sum of squares. Use n-1 for a sample and N for the population. The result of this is variance.

Finding the square root of variance

To find the standard deviation take the square root of it.

If used in the same process, the standard deviation is the standard error in the mean. It is also a standard error estimated concerning the mean.

What is a Population Standard Deviation?

A Population Standard Deviation and the standard definition is σ. With this, the entire population can be measured. It is the square root of variance for a given set. In some cases, every member of a population is sampled.

The following equation is used to find the standard deviation for the entire population.

The formula is


Here x= Individual Value.

μ=mean or expected value.

A total number of values.

Individuals who need to learn about this equation. It is daunting. Although when it's addressed with individual components. The summation is simple.

Here the i=1 in the formula. It indicates the starting index. So that means for data set 1, 3, 5,7, 8, i=1. So here i=2, and it could be three and more. So the summation notion means to operate (xi - μ)2

On every value of N, here it is five, so there are five values in the data set.

Now example

= 1+3+4+7+8) / 5 = 4.6

σ = √[(1 - 4.6)2 + (3 - 4.6)2 + and on.

σ = √(12.96 + 2.56 + 0.36 + 5.76 + 11.56)/5 = 2.577.

What is a sample Standard Deviation?

In most cases, it is only possible to sample some population members. So the above equation can be modified. The standard deviation is measured with a random sample for the population studied.

The sample standard deviation is a common way to estimate or the estimator

σ. It is typically denoted by s. It is a thing to notice that there are many different equations for calculating standard sample deviation.

Unlike the simple mean, the sample standard deviation has no single estimator. It is unbiased and efficient. So it comes with a maximum likelihood.

The equation given below is the Standard Deviation. It is a corrected new version of the equation obtained by modifying the population standard deviation.

So this corrected version of the equation by modifying the population standard deviation equation by sample size. Because it is the Size of the population, it removes all the bias of the equation.

Unbiased calculation of Standard deviation is highly involved and varies depending on the distribution value.

So the "Corrected sample deviation" is the most common way to calculate the population's standard deviation. It is a better way to calculate the unrevised version. It still has a significant bias for small sample sizes like (N<10)


Now here

xi=Sample value.

x̄=Sample mean.

n=Sample Size.

So referring to the Population Standard Deviation as an example of how to use summation.

The equation is mostly the same. Here the exception is the N-1 term. It is in the corrected sample deviation equation. These are sample values.

Let's understand the applications of Standard Deviation

So the Standard deviation is widely used in experiments and industries. It is used as a setting to test models against real-world data. One example of industrial settings is the control of quality for some products.

The Standard Deviation can also calculate the maximum and minimum values. Some aspects of the product should fall under a high percentage of that time.

In some cases, values fall outside the calculated category. So it is crucial to make necessary changes in the production process as it will ensure quality control.

The Standard Deviation is also used in weather to find regional climate change differences.

Suppose there are two cities. One is situated on the deep island, and one is on the coast.

So they have the same temperature, 75°F. It can support the belief that these two cities' temperature is virtually the same. The reality could only be understood if the mean is considered and the standard deviation is completely ignored.

Coastal regions tend to have a more stable temperature. It is because of the regulation of large bodies of water. Now water, as we know it, has a higher heating capacity than land. It makes water less susceptible to changes in weather. So coastal areas are mostly warm even in winter. At the same time, they remain cool in summer. Now it is because of the energy required to change the temperature of the water. So when the coastal city may have temperatures, it can vary from 60°F and 85°F for some time. To get the result in a mean of 75°F, the inland city can have a temperature range of 30°F to 110°F. So it will give the same meaning.

The other aspect in which standard deviation is largely used in finance. It is often used to measure the associated risk that comes with price fluctuations. The fluctuation occurs in assets of a portfolio of those assets. So when using the Standard Deviation in such cases, it estimates uncertainty. The uncertainty is in the case of future funds. It could also be a given investment.

So let's consider this example

When you compare stock A with an average return of 7℅, it has a standard deviation of 10℅ against stock B since it has the same average return with a standard deviation of 50℅.

The first stock, without a doubt, is the safe and best option. Now, the standard deviation of Stock B is larger than that of Stock A. It is for the same return.

It is safe to say Stock A is a better investment option. The Standard Deviation can skew the mean in any direction. Stock A has a high probability, with an average return close to 7%.

So Stock 8 can give a larger return.

These were some examples of how the standard deviation can be used. However, there are many ways to use it.

In general, cases calculating Standard Deviation is valuable at any given time. So to find the mean, a typical value from a distribution can be used.

Our Standard Deviation Calculator simplifies the process of calculating variances, finding the mean, and obtaining accurate statistical results with ease.


Q. What does a Standard Deviation mean?

A. A standard deviation tells or measures the dispersed data compared to the mean value.

Q. How to calculate Standard Deviation?

A. Standard Deviation cannot be calculated on hands. It is important to use the formula. σ= √∑ |x1 -µ| ^ 2)/ N)

Q. What does a Standard Error mean?

A. The standard deviation of distribution or mean is called the Standard Error. Whenever there is, research is conducted by a whole group. A small sample size from the population is used to collect data which can be relatively different and have different means.

Q. How to determine the Population Standard Deviation?

A. The Population Standard Deviation determines the variation amongst individual points in a population. It calculates the data, which is spread out from the mean.

Q. Why is a Sample Standard Deviation used?

A. The Sample Standard is the square root of its variance. It is also used to measure the spread out data.

Q. What are the applications of Standard Deviation?

A. Standard Deviation is widely used in experimental industrial sets, Financial sectors, the Climatic sector, and more. It is used to test the data against the real world's data.