How to Calculate Standard Deviation

The standard deviation indicates the dispersion of the values in the data set. ^{[1] X Research Source} To determine the value of the standard deviation, you first need to calculate a few other parameters such as the mean and the variance of the dataset. The variance represents the distribution of the data relative to the mean. ^{[2] X Research Source} The standard deviation is calculated by taking the square root of the variance. Here are the steps to help you find the mean, variance, and standard deviation of a set of data.

Table of Contents

Find the Average

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Review the data. Even if the unknowns you’re looking for are simple values like the mean or the center value, looking at the numbers is still an important step in the statistical calculations you need to perform. ^{[3] X Research Sources}

Specifies the number of values (or sizes) of the dataset.
Do these values vary widely? Or is there just a small difference between the values, like a few percents or thousandths.
Identify the type of metric you’re looking at. What properties of the dataset do these figures represent, e.g. heart rate, height, weight, score, etc.
Example: There is a set of test scores as follows: 10, 8, 10, 8, 8, and 4.

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Gather all the data. To determine the mean, you need all the values in the dataset. ^{[4] X Research Sources}

The mean is the average of all the data in the set.
The mean is calculated by adding all the numbers in the data set, dividing the result by the total number of values in the data set (usually denoted by n.)
There are 6 metrics in the set of test scores (10, 8, 10, 8, 8, 4), so n = 6.

Calculating the Variance of the Dataset

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Calculate variance. Variance is a value that represents the dispersion of the data relative to the mean. ^{[7] X Research Sources}

The variance provides information about the dispersion of the values in the dataset.
Small variance data sets are data sets with values close to the mean.
In contrast, large variance characterizes a set of data whose values are much larger or smaller than the mean.
Variance is often used to compare the dispersion of two sets of data.

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Subtract each value from the data set from the average value calculated in the previous step. The obtained results show the distance between each value compared to the mean value. ^{[8] X Research Sources}

For example, for the set of test scores (10, 8, 10, 8, 8, and 4), the mean is 8.
10 – 8 = 2; 8 – 8 = 0, 10 – 8 = 2, 8 – 8 = 0.8 – 8 = 0, and 4 – 8 = -4.
Repeat the above calculations to confirm the result. These results will be used for the next step, so you need to do these calculations correctly to be able to determine the exact value of the standard deviation.

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Square all the values obtained from the above subtractions. ^{[9] X Research Source}

We have taken the mean (8) and subtracted each value of the tuple (10, 8, 10, 8, 8, and 4), we get the values 2, 0, 2, 0, 0 and -4.
To calculate the variance, square the values in the previous step, we have 2 ² , 0 ² , 2 ² , 0 ² , 0 ² , and (-4) ² = 4, 0, 4, 0, 0, and 16.
Check the results again.

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Adding the sum of the above values after squaring, this value is also called the sum of squares. ^{[10] X Research Source}

For the data set we’re taking as an example, the squared error is: 4, 0, 4, 0, 0, and 16.
In this example, we started by subtracting each single value from the mean and squaring the result: (10-8)^2 + (8-8)^2 + (10- 2)^2 + (8-8)^2 + (8-8)^2 + (4-8)^2
4 + 0 + 4 + 0 + 0 + 16 = 24.
The sum of squares is 24.

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Divide the sum of squares by (n-1) where n is the size of the dataset. By dividing the sum of squares by (n-1), we get the variance. The use of (n-1) is to get the unbiased variance of the dataset as well as that of the population. ^{[11] X Research Source}

In the example with the set of test scores (10, 8, 10, 8, 8 and 4), there are 6 metrics, so n = 6.
n – 1 = 5.
The sum of the squares of this dataset is 24.
24 / 5 = 4.8
So, the variance of this dataset is 4.8.

Calculate Standard Deviation

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Determine the variance of the dataset. The variance of a dataset is the value required to calculate the standard deviation. ^{[12] X Research Source}

Variance is a measure of the dispersion of the data relative to the mean.
The standard deviation is also a value that represents the dispersion of the data.
The variance of the dataset in the test score example is 4.8.

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Taking the square root of the variance, we get the value of the standard deviation. ^{[13] X Research Source}

Typically, at least 68% of the values in the data set are within the equivalent of one standard deviation from the mean.
For the example given in this article, the variance has a value of 4.8.
√4.8 = 2.19. Therefore, the standard deviation of the dataset under consideration is 2.19.
5 out of 6 (83%) of the data in the set of test points (10, 8, 10, 8, 8, and 4) are within one standard deviation (2.19) of the value. average (8).

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Recalculate the mean, variance, and standard deviation to make sure you don’t make a mistake in the calculation. ^{[14] X Research Source}

While doing the calculation, you should write down all the steps to find the final answer, whether you calculate it by hand or by computer, it is quite necessary to rewrite the calculation procedure. necessary and important.
If you notice a difference between the values in the first and second calculation, check again.
Finally, if you cannot find the cause of the difference between the two calculations, repeat the steps above and compare the result with the results obtained from the previous two calculations.

Steps

Find the Average

Calculating the Variance of the Dataset

Calculate Standard Deviation

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