# Standard Deviation & Variance Calculator

## A Guide to Understanding the Concepts & Calculations

The calculator below provides an interactive example, and will guide you through the calculation.

## Calculate Standard Deviation & Variance:

Enter your data set below. Each number can be separated by a comma, space, or a new line break.
e.g. enter 10,000 as 10000

Paste in as many values as you want!

 Standard Deviation (Based on a population): - Standard Deviation (Based on a sample) - Variance (Population) - Variance (Sample) - Average - Total Numbers - Standard Error of the Mean -

Empirical rule distributions
(if sampling distribution of the mean follows normal distribution)

 68% of the population is between: - 95% of the population is between: - 99.7% of the population is between: -
 68% of the sample is between: - 95% of the sample is between: - 99.7% of the sample is between: -

#### How to Calculate Standard Deviation

Below is an example of 6 test scores from a class to walk through the calculation:

Test scores: 78, 88, 81, 92, 65, 58
(1)
Find the average:
\begin{align*} \frac{78+88+81+92+65+58}{6} = 77 \end{align*}
(2)
Calculate the deviations of each data point from the mean, and square the result of each
\begin{align*} (78-77)^2 = 1 \end{align*} \begin{align*} (88-77)^2 = 121\\ \end{align*} \begin{align*} (81-77)^2 = 16\\ \end{align*} \begin{align*} (92-77)^2 = 225\\ \end{align*} \begin{align*} (65-77)^2 = 144\\ \end{align*} \begin{align*} (58-77)^2 = 361\\ \end{align*}
(3)
We can compute the population variance by taking the average of these values
\begin{align*} \frac{1+121+16+225+144+361}{6} = 144.67 \end{align*}
(4)
The population standard deviation is equal to the square root of the variance
\begin{align*} \sqrt{{144.67^8}} = 12.028 \end{align*}

The above example can be condensed to the following formulas:

Population Standard Deviation
(All elements from a data set - e.g 20 out of 20 students in class)

The population standard deviation, is used when the entire population can be accounted for. It is calculated by taking the square root of the variance of the data set. The following equation can be used in this scenario:

\begin{align*} σ = \sqrt{\frac{ \sum_{}(x_i-μ)^2}{6} } \end{align*}

Where,

σ = Population standard deviation

= Sum of..

xi = An individual value..

μ = Population mean

n = Number of values in the population data set

Sample Standard Deviation
(One or more elements from a data set - but not 100% of elements - e.g 100 out of 300 students taking a computer class)

Sometimes, it is not possible to capture all data from a population. This requires the above equation be modified, and is used to calculate the sample standard deviation. This equation modifies the population standard deviation using the sample size as the size of the population. The “N-1” is referred to as degrees of freedom. The version below is used in most basic statistical courses. While it is a better estimate compared to just using the population started deviation, still has significant bias for small sample sizes (less than 10)

\begin{align*} s = \sqrt{\frac{\sum_{}(x_i-x̄)^2}{6} } \end{align*}

Where,

s = Sample standard deviation

= Sum of..

xi = An individual value..

x̄ = Population mean

n = Number of values in the sample data set

#### How to Interpret Standard Deviation

Standard deviation is a measure that is used to quantify the amount of variation or dispersion of a set of data values. The standard deviation is a description of the data's spread, how widely it is distributed about the mean. A smaller standard deviation indicates that more of the data is clustered about the mean. A larger one indicates the data are more spread out.

Generally speaking data is normally distributed. This is important as it can be inferred that normally distributed data follows a bell shaped curve. That bell shaped curve can give us further insights.

The above graph shows the rules for normally distributed data. 68.2% of responses are within 1 deviation of the mean, 95.4% of responses are within 2 deviations of the mean, while 99.6% of the data is within 3 deviations of the mean.

Example: If a question in your survey asks for annual income, the mean could be $35,000 with a standard deviation of$5,000. From the empirical rule, we could assume that 68% of total responses fall somewhere between $30,000 and$40,000. We could also assume 95% of the data falls between $25,000 and$45,000.

With this example, knowing your audience's income range can set you up for a successful marketing camping. You would now be able to create a campaign specific to your audience!

#### How to Interpret Variance

Variance also measures the amount of variation or dispersion of a set of data values from the mean.

As mentioned, variance takes the average of all the squared differences from the mean. Standard deviation takes the square root of that number. Thus, the only difference between variance and standard deviation, is the units. For example, if we took the times of 50 people running a 100-meter race, we would capture their time in seconds. When we compute the variance, we come up with units in seconds squared. Seconds squared aren’t extremely useful, so to get back to regular second units we take a square root of the variance.

The variance takes the squares of the difference compared to the mean (as opposed to the absolute value) for two important reasons: squaring always gives a positive value and squaring emphasizes larger differences.

#### Survey Questions that Use Standard Deviation

Single text boxes with number, dollar, or percent validation - Useful to gather income, age, or numbers which require analysis.

Continuous Sum gives deviation for each label - Useful to gather budget data, time allocated to projects, or other numerical allocation questions requiring analysis.

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