Standard Deviation Calculator


Population
Sample

Both give the same result, but the computational formula is simpler to calculate step-by-step.
Standard
Computational

Solution:

1. Start by writing the computational formula for the standard deviation of a sample: $$ {s}= \sqrt{\frac{{\sum}{x^2} - \frac{({\sum}{x})^2}{n}}{n-1}}$$

2. Create a table of 2 columns and 10 rows. There will be a header row and a row for each data value. The header row should be labeled with ${x}$ and $ x^2$. Enter the data values in the ${x}$ column, with each data value in its own row. In the second column, put the square of each of the data values, ${x^2}$.

$x$

$x^2$

13919321
19337249
14621316
18032400
21546225
25665536
21345369
14521025
27273984

3. Find the sum of all the values in the first column, ${\sum}{x}$.

$$ \sum{x} = 1759 $$

4. Square the answer from step 3, then divide that number by the size of the sample.

$$ \frac{({\sum}{x})^2}{n} = \frac{3094081}{9} = 343786.77777778 $$

5. Find the sum of all the values in the second column, ${\sum}{x^2}$.

$$ {\sum}{x^2} = 362425 $$

6. Subtract the answer in step 4 from the answer in step 5.

$$ {\sum}{x^2} - \frac{({\sum}{x})^2}{n} = 362425 - 343786.77777778 = 18638.222222222 $$

7. Divide the answer in step 6 by n - 1, one less than the size of the sample. This answer is the variance of the sample. $$ {s^2}= \frac{{\sum}{x^2} - \frac{({\sum}{x})^2}{n}}{n-1} = \frac{ 18638.222222222 }{8} = 2329.7777777778$$

8. Take the square root of the answer found in step 7 above. This number is the standard deviation of the sample. It is symbolized by ${s}$ . Here, we round the standard deviation to at most 4 decimal places.

$$ {s} = \sqrt{2329.7777777778} = 48.2678$$

Standard Deviation Calculator with an Easy Step by Step Solution
Standard Deviation Calculator with an Easy Step by Step Solution

Using the Standard Deviation Calculator

The standard deviation calculator above offers a simple way to both calculate and learn how to find the standard deviation of a set of numbers.  Better than any standard calculator, this calculator provides a step by step solution for how to find the answer on your own.  This standard deviation calculator is an excellent teaching tool to help guide you in getting the correct answers in your own work.  If you also need to find the range of a data set, see the page Measures of Variability Calculator.  That calculator will find all three measures of variability, the range, variance and standard deviation, and show you a step by step solution.

What’s the Standard Deviation?

The standard deviation definition is a measure of the “spread” of the data values within the data set. The “spread” refers to how close or far away the data values are as compared to the data set’s mean.  The variance is the square of the standard deviation.  Both the variance and standard deviation are measures of variability.

This standard deviation calculator not only gives you an answer to your problem, it also guides you through a step-by-step solution.

What Does a Large Standard Deviation Imply?

By the standard deviation definition, it measures the spread of data values from the mean.  If there is a large standard deviation, then there is a large spread of data values.  This means the values are more spread out far away from the mean.  This implies great variability in the data set. If the standard deviation is small, then the data values in a data set are less spread out from the mean.  This implies less variability and more consistency.

Suppose you take an exam and the standard deviation for the class grades is 5.0.  At this point, we can’t really say if your class performed consistently or not, because we have nothing to compare it to.  Now, your friend in a different class takes an exam and the standard deviation for those class grades is 15.0. When we compare the two standard deviations, there is more consistency and less variability in your class.  There is less consistency and more variability in your friend’s class.

If you use the standard deviation calculator to find the standard deviations of two different data sets, the standard deviation that is smaller is for the data set that is more consistent, and the standard deviation that is larger is for the data set that is more variable.

Income Example – Comparing Two Cities

Suppose you have two data sets consisting of family income.  The first data set consists of the population of incomes of families in city ‘A’, and the second data set consists of the population of incomes of families in city ‘B.’  City ‘A’ and city ‘B’ both have mean family incomes of $65,000.  So far, we have:

City A mean:
µ = 65,000

City B mean:
µ = 65,000

If the standard deviation for the data set of incomes from City A is $ \$ 5,500.00 $, and the standard deviation for the data set of incomes from City B is $ \$ 2,100.00 $, then we know that the incomes in City A are spread out further away from the mean, while the incomes in City B are closer, or clustered more tightly, around the mean.  The incomes in City A have greater variability than the incomes in City B.

Symbol For the Standard Deviation

The symbol for the standard deviation of data set that represents a sample is s.  The symbol for the standard deviation of a data set that represents  population is σ (lowercase Greek sigma).  We have the population information for both City ‘A’ and City ‘B’.  Therefore, the symbol for the standard deviation for both are:

City A standard deviation:
σ  = $5,500

City B standard deviation:
σ = $2,100

Standard Deviation for No Variability

A standard deviation is always a positive number, or possibly 0.  Suppose in City ‘C,’ every family has the same income, $ \$ 65,000 $.  While realistically this is not possible, mathematically this would mean that the mean for incomes in City ‘C’ is $ \$ 65,000 $, and the standard deviation is 0.  A standard deviation of 0 indicates that a data set has no variability at all, and every data value in the data set is exactly the same.

Try it! Using the standard deviation calculator, enter the following:

5, 5, 5, 5, 5, 5, 5, 5

You’ll see that the standard deviation will calculate to 0, and the steps for the solution will show you why it’s 0.

Units Used for the Standard Deviation

The units for the standard deviation are the same as the units for the data values in the data set.  In our example above, the data values are incomes in dollars, therefore the standard deviation is in dollars.

What is the Variance?

Related to the standard deviation of a data set is the variance of a data set.  The variance of a data set is the square of the standard deviation, and therefore the units for the variance are squared from that of the units of the standard deviation.  The symbol for the sample variance is s2, an the symbol for the population variance is σ2. In our example above, the variances for City A and City B are:

City A variance:
σ2  = 30,250,000 $

City B variance:
σ2 = 4,410,000 $2

Just as you would do manually, the standard deviation calculator finds the variance first, and then takes the square root to find the standard deviation.

Applying the Standard Deviation and Variance Formulas

Now that you know the standard deviation definition, do you want to learn how to calculate the standard deviation and variance?  You can either apply the standard deviation and variance formulas, or you can scroll up and use the standard deviation calculator online.  In the tutorial below I’ll show you how to find the standard deviation and variance by hand using formulas.

Do you want to know how to find the standard deviation or variance of a data set manually?  Then, you’ll need to use the variance and/or standard deviation formulas.  These formulas can look complex, but when taken in small steps, the process for calculating them is very manageable.  The formulas use different symbols, depending if the data set represents a population or a sample.

There are two versions of the variance and standard deviation formulas, the standard and computational formulas.  I’ll be using the computational formula in this article.  It’s simpler to calculate by hand and has less rounding errors.  If you want to see the standard formula solution, the Standard Deviation Calculator above can show you solutions using both formulas.

Population Variance Formula and Sample Variance Formula

Population Variance FormulaSample Variance Formula

$$ {\sigma^2}= \frac{{\sum}{x^2} – \frac{({\sum}{x})^2}{N}}{N}$$

Where $\sigma^2$ is the population variance symbol,
$x$ is each data value in the population,
and $ N $ is the size of the population.

$$ {s^2}= \frac{{\sum}{x^2} – \frac{({\sum}{x})^2}{n}}{n-1}$$

Where $s^2$ is the sample variance symbol,
$ x $ is each data value in the sample,
and $ n $ is the size of the sample.

There is  very simple step between getting the variance and then getting the standard deviation.  Once you have the variance, just take the square root to get the standard deviation.

Population Standard Deviation Formula and Sample Standard Deviation Formula

Population Standard Deviation FormulaSample Standard Deviation Formula

$$ {\sigma}= \sqrt{\frac{{\sum}{x^2} – \frac{({\sum}{x})^2}{N}}{N}} $$

Where $\sigma$ is the population standard deviation symbol,
$x$ is each data value in the population,
and $ N $ is the size of the population.

$$ {s}= \sqrt{\frac{{\sum}{x^2} – \frac{({\sum}{x})^2}{n}}{n – 1}} $$

Where $s$ is the sample standard deviation symbol,
$ x $ is each data value in the sample,
and $ n $ is the size of the sample.

Example on How to Find the Standard Deviation and Variance

Let’s walk through how to find the standard deviation and variance for a small data set, given that the data set represents a sample of children’s heights.  After we get the variance, we’ll then take one small step to get the standard deviation.  We’ll calculate our answers by completing a series of 8 steps.

The Problem: Find the variance and standard deviation for the following.  Suppose you have a sample of 5 children and their heights are:

56 in, 49 in, 61 in, 60 in, 63 in

Step 1 – Write the Sample Variance and Sample Standard Deviation Formulas

Because this problem states that the 5 values represent a sample, we’ll use the sample variance and sample standard deviation formulas.  First, start by writing the computational formulas for the sample variance and sample standard deviation:

$$ {s^2}= \frac{{\sum}{x^2} – \frac{({\sum}{x})^2}{n}}{n-1}$$

$$ {s}= \sqrt{\frac{{\sum}{x^2} – \frac{({\sum}{x})^2}{n}}{n – 1}} $$

Step 2 – Create a Table for All Values of $ x $ and $x^2$

Next, draw a table of 2 columns and 5 rows for each data value, and a header row.  Label the header row with $ x $ and $ x^2 $.  Now, put each of the data values in the $ x $ column.  Each data value has its own row.  Square each value of x in the first column, and put these values  in the second column.

$x$

$x^2$

563136
492401
613721
603600
633969

Step 3 – Add up All The Values in the First Column

After the table and columns are created, take the sum of all the values in the first column.  This is symbolized as $ \sum{x} $.

$$ \sum{x} = 56+49+61+60+63 $$

$$ \sum{x} = 289 $$

Step 4 – Square and Divide

Now, take the answer from Step 3, 289, and square it. Then, divide by the size of the sample.

$$ \frac{(\sum{x})^2}{n} = \frac{83521}{5} = 16704.2 $$

Step 5 – Add up All The Values in the Second Column

Next, take the sum of all the values in the second column.  This is symbolized as $ \sum{x^2} $.

$$ \sum{x^2} = 3136+2401+3721+3600+3969$$

$$ \sum{x^2} = 16827 $$

Step 6 – Subtract $ \sum{x^2} – \frac{(\sum{x})^2}{n} $

In this step, you’ll take the answer from step 5 and subtract the answer from step 4.

$$\sum{x^2} – \frac{(\sum{x})^2}{n}$$

$$ 16827 – 16704.2 = 122.8 $$

Step 7 – Divide and Get the Variance

Here, take the answer from step 6 and divide by $n – 1$, one less than the sample size.  That’s the variance!

$$ {s^2}= \frac{{\sum}{x^2} – \frac{({\sum}{x})^2}{n}}{n-1}
= \frac{ 122.8 }{4} = 30.7 $$

Step 8 – How to Find the Standard Deviation from the Variance

Lastly, to find the standard deviation, take the square root of the answer for the variance from step 7.  Here, I’ll round the answer to 4 decimal places.

$$ s = \sqrt{30.7} = 5.5408 $$

Since our data is initially in units of inches, the standard deviation is 5.5408 inches.

That’s it!  Not so bad, huh?  It’s a great idea to use the standard deviation calculator above to guide you in solving more problems.  Try manually working out the solutions on your own and check your work against the worked out solution from the calculator. You GOT this!

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