Mean and standard deviation calculator with steps

'Mean and Standard Deviation Calculator' is an online tool that helps to calculate the mean and standard deviation for the given numbers. 

What is Mean and Standard Deviation Calculator?

Online Mean and Standard Deviation Calculator helps you to calculate the mean and standard deviation for the given numbers in a few seconds.

Mean and Standard Deviation Calculator

NOTE: Enter values inside the bracket, separated by a comma.

How to Use Mean and Standard Deviation Calculator?

Please follow the steps below to find the mean and standard deviation for the given numbers:

• Step 1: Enter the numbers separated by a comma in the given input box.
• Step 2: Click on the "Calculate" button to find the mean and standard deviation for the given numbers.
• Step 3: Click on the "Reset" button to clear the fields and find the mean and standard deviation for the different numbers.

How to Find Mean and Standard Deviation Calculator?

The mean or average of a given data is defined as the sum of all observations divided by the number of observations. The mean is calculated using the formula:

Mean or Average(x) = (x1 + x2 + x3...+ xn) / n , where n = total number of terms, x1, x2, x3, . . . , xn = Different n terms

Standard deviation is commonly denoted as SD, and it tells about the value that how much it has deviated from the mean value.

Standard deviation = √(∑(xi - x)2 / (N - 1)),

where xi is individual values in the sample, and x is the mean or an average of the sample, N is the number of terms in the sample.

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Solved Examples on Mean and Standard Deviation Calculator

Example 1:

Find the mean and standard deviation for the following set of data: {51,38,79,46,57}

Solution:

Given N = 5

Standard deviation = √(∑(xi - x)2 / (N - 1))

Mean(x) = 51 + 38 + 79 + 46 + 57 / 5 = 54.2

Standard deviation = √(51 − 54.2)2 + (38 − 54.2)2 + (79 − 54.2)2 + (46 − 54.2)2 + (57 − 54.2)2 / (5 - 1)

= 15.5

Therefore, mean = 54.2, and standard deviation = 15.5

Example 2:

Find the mean and standard deviation for the following set of data: {1, 6, 7, 2, 9}

Solution:

Given N = 5

Standard deviation = √(∑(xi - x)2 / (N - 1))

Mean(x) = 1 + 6 + 7 + 2 + 9 / 5 = 5

Standard deviation = √(1 - 5)2 + (6 - 5)2 + (7 - 5)2 + (2 - 5)2 + (9 - 5)2 / (5 - 1)

= 3.39

Therefore, mean = 5, and standard deviation = 3.39

Example 3:

Find the mean and standard deviation for the following set of data: {4, 8, 11, 19}

Solution:

Given N = 4

Standard deviation = √(∑(xi - x)2 / (N - 1))

Mean(x) = 4 + 8 + 11 + 19 / 4 = 42/4 = 10.5

Standard deviation = √(4 - 10.5)2 + (8 - 10.5)2 + (11 - 10.5)2 + (19 - 10.5)2 / (4 - 1)

= 6.35

Therefore, mean = 10.5, and standard deviation = 6.35

Similarly, you can try the calculator to find the mean and standard deviation for the following: 

a)  21,14,16,8,2,4,15,8

b)  25,1,7,15,6,14,14,25,7

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• Mean
• Standard deviation

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Find the standard deviation for your data set by entering the numbers in the calculator below. Keep reading to learn how to calculate standard deviation and the formula.

How to Find the Standard Deviation

In statistics, the standard deviation is a measure of the dispersion or variability between observations in a data set. A smaller standard deviation value indicates that the data are relatively close to the mean, while a higher value suggests that the data are more widely spread out.

Standard deviation is sometimes shortened to SD but is often represented using the symbol σ (the Greek letter sigma) or the letter s for sample data.

A low standard deviation means that the data points in the set are close to the mean, while a higher standard deviation means the data is highly dispersed.

Standard Deviation Formula

The standard deviation is equal to the square root of the variance. However, the formula to calculate the variance is different for a population versus a sample, so there are actually different formulas to calculate the standard deviation for population and sample data sets.

Population Standard Deviation Formula

You can calculate the population standard deviation using the following formula:[1]

\sigma = \sqrt{\frac{\sum \left ( x_{i}-\mu \right )^{2}}{N}}

Thus, the standard deviation for a population σ is equal to the square root of the sum of squares ∑(xi – μ)² divided by the population size N.

Sample Standard Deviation Formula

You can calculate a sample standard deviation using the following formula:[2]

s= \sqrt{\frac{\sum \left ( x_{i}-\bar{x} \right )^{2}}{n-1}}

Thus, the standard deviation for a sample s is equal to the square root of the sum of squares ∑(xi – x̄)² divided by the sample size n minus 1.

One thing you might notice that’s different in these formulas is that the standard deviation for a sample divides the sum of squares by n – 1 rather than just n. The reason for this is that when working with a sample, the estimation for the variance and standard deviation includes some amount of bias.

In the formula for a sample, the denominator n – 1 is referred to as degrees of freedom.

Since the sum of squares for a sample is lower than the sum of squares for a population, subtracting one from the sample size artificially increases the SD and variance to account for this bias. This is known as Bessel’s Correction.[3] Bessel’s Correction is more useful for smaller sample sizes, but for large sample sizes that approach the population size, it’s less necessary.

Steps to Calculate the Standard Deviation

We can break down the formulas above to find the standard deviation into six easy steps.[4]

Step One: Calculate the Mean

The first step to finding the standard deviation is to find the mean for the sample or population.

To calculate the mean, sum each observation in the dataset, then divide the result by the sample size or population size.

\text{mean}=\frac{\text{sum of observations}}{\text{count of observations}}

More formally, the mean formula looks more like this:

\mu=\frac{\sum_{i=1}^{N}x_{i}}{N}

\mu=\frac{x_{1}+x_{2}+…+x_{N}}{N}

The mean is equal to the sum of each observation xi divided by the total number of observations N.

Step Two: Calculate the Deviations From the Mean

The next step is to find the deviation from the mean for each value in the data set. The formula below illustrates this:

\text{deviation} = x_{i}-\mu

Step Three: Square Each Deviation

Then, for each deviation from the mean, calculate its square.

\text{squared deviation} = \left ( x_{i}-\mu \right )^{2}

Step Four: Calculate the Sum of Squares

The next step is to find the sum of squares. You can use the sum of squares formula to calculate it.

SS = \sum \left ( x_{i}-\mu \right )^{2}

The sum of squares SS is equal to the sum of the deviations of each value from the mean, squared.

Step Five: Calculate the Variance

Next, using the sum of squares, it’s time to calculate the variance in the data. The formula to calculate the variance for a population and sample is slightly different:

Variance for a Population

The variance for a population is equal to the sum of squares divided by the population size.

\sigma^{2} = \frac{SS}{N}

Variance for a Sample

The variance for a sample is equal to the sum of squares divided by the sample size minus one, or rather, the sum of squares divided by the degrees of freedom.

s^{2} = \frac{SS}{n-1}

Step Six: Calculate the Standard Deviation

Finally, using the variance, you can calculate the standard deviation.

Standard Deviation for a Population

\sigma = \sqrt{\sigma^{2}}

Standard Deviation for a Sample

s = \sqrt{s^{2}}

Thus, the standard deviation is equal to the square root of the variance.

For example, let’s calculate the standard deviation for the sample data [1,3,4,7,8,9].

We’ll use the six steps for the SD calculation shown above.

Step One: Find the mean by adding the values together, then dividing by the sample size.

\bar{x}=\frac{1+3+4+7+8+9}{6}

\bar{x}=\frac{32}{6}=5.33

Step Two: Find the deviations from the mean by subtracting the mean from each sample observation.

Step Three: Square the deviations from the mean by raising each deviation to the power of 2.

Step Four: Find the sum of squares by adding the squared deviations together.

SS = 18.78+5.44+1.78+2.78+7.11+13.44

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SS = 49.33

Step Five: Find the variance by dividing the sum of squares by the sample size minus 1.

s^{2} = \frac{49.33}{6-1}

s^{2} = 8.22

Step Six: Find the SD by taking the square root of the variance.

s = \sqrt{8.22}

s = 2.87

So, for the data points in this sample, the standard deviation is equal to 2.87.

If you know the standard deviation of the population, then you can also use the central limit theorem to find the sample standard deviation.

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