Standard Deviation Calculator

Your Data
Select Population if the data contains all measurable values or all values you are interested in.
Select Sample if the data is a sample of a large or unlimited population and you wish to make a statement about the entire population.
Input the data separated by comma, space or enter. To calculate standard deviation, you need at least two values in your dataset.
Choose the number of decimal places.

How to Calculate Sample Standard Deviation: A Step-by-Step Guide

Statistical analysis software typically calculates standard deviation automatically. You can also quickly obtain it if you select Sample in our Standard Deviation Calculator. But computing it manually can aid in understanding how the formula works.

To calculate the sample standard deviation of a dataset, you can follow these steps:

  1. Calculate the sample’s mean value:

    X¯=i=1nXin

  2. Calculate the deviations from the sample mean:

    (XiX¯)

  3. Square each deviation from the sample mean:

    (XiX¯)2

  4. Find the sum of the squared deviations:

    i=1n(XiX¯)2

  5. Divide the sum of squared deviations by n – 1 to obtain the variance (

    s2
    ):

    s2=i=1n(XiX¯)2n1

  6. Take the square root of the sum of the squared deviations to calculate the sample standard deviation:

    s=i=1n(XiX¯)2n1

How to Calculate Population Standard Deviation: A Step-by-Step Guide

If, you’re looking for a quick calculation, select Population in our Standard Deviation Calculator to obtain the value. Yet, it’s useful to know how to do it by hand as well.

To find the population standard deviation manually, follow these steps:

  1. Calculate the population’s mean value:

    μ=i=1nXin

  2. Calculate the deviations from the population mean:

    (Xiμ)

  3. Square each deviation from the population mean:

    (Xiμ)2

  4. Find the sum of the squared deviations:

    i=1n(Xiμ)2

  5. Divide the sum of squared deviations by n to obtain the population variance (

    σ2
    ):

    σ2=i=1n(Xiμ)2n

  6. Take the square root of the sum of the squared deviations to calculate the population standard deviation:

    σ=i=1n(Xiμ)2n

Standard Deviation Calculator

What Is Standard Deviation?

Standard deviation measures the amount of variation or dispersion in a dataset. Specifically, it shows us what is the average distance of each observation from the mean. Standard deviation is often abbreviated as SD. In mathematical terms, we denote the population standard deviation with the lowercase Greek letter sigma σ and the lowercase s for the sample standard deviation.

You can find both with our Standard Deviation Calculator. But to interpret the results, you must fully grasp the definition.

What Does Standard Deviation Mean?

To reiterate, standard deviation measures how dispersed a set of values is in regard to the mean. These values can either be below or above the mean, depending on the standard deviation curve. In that sense, a dataset with a low standard deviation indicates that data is closer to the mean, whereas a high standard deviation indicates that observations are more spread out.

But what if there is no dispersion? In some cases, the standard deviation in a dataset is 0, meaning all of the values are identical. Think of a dataset of aggregated information on people’s height. If every subject has exactly the same height as all the others, then there’s no variability in the dataset.

The following image depicts two graphs with a standard deviation different than 0:

Sample Standard Deviation graph. Population Standard Deviation graph.

In the upper graph, the data points are more dispersed, therefore the standard deviation is higher. Meanwhile, the graph below shows a distribution with observations that are more clustered around the mean, resulting in a lower standard deviation.

The 68–95–99.7 Rule

Standard deviation is a particularly useful measure for the normal distribution (also known as Gaussian distribution).

Because the shape of the normal distribution is symmetrical and bell-like, the probability of observing a value to the left or right of the mean is equal. This means that the distribution is balanced and most of its values are in the center, with no long tail on either side of the mean. Put simply, both sides are mirror images.

Normal distribution graph.

Many variables in life follow the normal distribution, including people’s weights and heights, standardized test scores, errors in measurement, or manufacturing defects.

You can use the Standard Deviation Calculator and Mean Calculator to quickly find out how clustered the data values are around the mean. According to the 68-95-99.7 rule, also known as the empirical rule, the percentage of values will fall within three standard deviations, more specifically:

  • Approximately 68% of all data points fall in the interval μ ± σ (within 1 SD of the mean).
  • Approximately 95% of all data points fall in the interval μ ± 2σ (within 2 SD of the mean).
  • Approximately 99.7% of all data points fall in the interval μ ± 3σ (within 3 SD of the mean).
Standard Deviation of the mean graph. The three Standard Deviation of the mean graph.

Suppose that you have a dataset with student exam scores; the values follow a normal distribution with a mean of 80 and a standard deviation of 10. According to the 68-95-99.7 rule, the scores that fall within different intervals of the distribution are as follows:

  • Approximately 68% of the scores will fall between 70 and 90 SD.
  • Approximately 95% of the scores will fall between 60 and 100 SD.
  • Approximately 99.7% of the scores will fall between 50 and 110 SD.

Standard Deviations in a Normal Distribution.

Why Is Standard Deviation Important?

Standard deviation is an essential statistic for many types of analysis, used by statisticians, data analysts, and data scientists across the board. In this article, we’ll look at just some of the widespread applications of this measure.

Measuring Dispersion

Standard deviation tells you how widely values are spread out from the mean. For example, consider the following two populations:

  • {2,2,16,16}
  • {8,8,10,10}

Both of them have a mean of 9 and their standard deviations are 7 and 1, respectively. From that, we can tell that the second population has a much smaller standard deviation because its values are closer to the mean.

Evaluating Data Quality

Standard deviation is useful when identifying data points at an abnormal distance away from the mean; otherwise known as outliers whose existence can detriment any statistical findings.

Some practitioners argue that observations that lie three standard deviations away from the mean are considered outliers.

Making Statistical Inferences

We use standard deviation in calculate confidence intervals and hypothesis testing . It provides information on the data’s variability and helps to make statistical inferences about the population.

Modeling

Standard deviation plays a crucial role in many statistical models, such as linear regression and normal distributions. It is often used to quantify the variation of errors or residuals in a model and when evaluating model performance.

How to Calculate the Standard Deviation

Our Standard Deviation Calculator is a useful resource when you want to streamline your work. But knowing how to calculate it yourself will help you better interpret the results.

The standard deviation calculation varies depending on whether your data is from a population or a sample.

Population standard deviation, denoted by σ, measures the variability of the entire population. To obtain it we sum, the squared differences of each data point from the population mean and divide by the total number of data points. The population standard deviation is the square root of the result.

Use the following formula to calculate the population standard deviation:

σ=i=1n(Xiμ)2n

Here μ, represents the population mean and n is the number of observations in the dataset.

Meanwhile, we calculate the sample standard deviation, denoted by , as follows:

s=i=1n(XiX¯)2n1

Here

X¯
is the sample mean and n is the number of observations in the sample.

Please note that when calculating sample variance, we divide the sum of the squared deviations of each observation from the population mean by n-1, instead of n. Known as Bessel's correction, this measure allows us to correct the bias in the population standard deviation estimatе due to the finite sample size.

Put simply, by using n-1 instead of n in the denominator, we can produce unbiased estimates of our population. In practice, it’s more common to calculate the sample standard deviation than the population standard deviation since sample data is easier to collect. (Still, our Standard Deviation Calculator allows you to find both.)

You can also note that in both equations we square the differences between each data point and the mean. Squaring the differences has two main purposes:

  • Non-negative dispersion
  • Larger observation sets

First, by squaring the numbers, we always get non-negative computations. In essence, dispersion is about distance, which cannot be negative. If we don’t elevate to the second degree, we obtain both positive and negative values. In turn, both types of values cancel out each other when summed, leaving us with no information about the dispersion.

Second, squaring amplifies the effect of large differences. This means that outliers (observations that are far from the mean) have a greater impact on the variance than those closer to the mean. For example, if the mean is 0 and you have an observation of 100, then the squared spread is 10,000. But without squaring, this observation would contribute a spread of only 100, which might not fully capture the magnitude of its deviation from the mean.

Standard Deviation vs Variance

Standard deviation and variance are two of the most used statistical measures of variability. Variance is equal to the average of the squared deviations around the mean, whereas the standard deviation is equal to the square root of variance.

When it comes to units of measurement, the two are stated as follow:

  • Variance is expressed in squared units (e.g. ounces squared).
  • Standard deviation is expressed in the same units of measurement as the original units (e.g. ounces).

Unlike variance, we measure standard deviation in the same units as the observations. This makes it the preferred method to assess variability, especially when it comes to practical applications such as data analysis, decision-making, and reporting.

Still, you can find both using our Variance Calculator and Standard Deviation Calculator.

What is the standard deviation of 5 5 9 9 9 10 5 10 10?

You can find the standard deviation of 5 5 9 9 9 10 5 10 10 using our Standard Deviation Calculator. But knowing how to manually calculate gives you a definite advantage when running statistical analysis. Learn how to do it by following these steps:

We begin by calculating the sample’s mean value:

X¯=i=1nXin=5+5+9+9+9+10+5+10+1010=8

Then, we calculate the deviations from the sample mean. Start with the first variable, subtracting it from the mean value:

(XiX¯)=58=3

Now, square each deviation from the sample mean. Our first variable becomes:

(XiX¯)2=32=9

Find the sum of the squared deviations:

i=1n(XiX¯)2=i=1n(58)2+(58)2+...+(108)2=42

Divide the sum of squared deviations by n – 1 to obtain the variance (s2)

s2=i=1n(XiX¯)2n1=4291=5.25

Take the square root of the sum of the squared deviations to calculate the sample standard deviation:

s=i=1n(XiX¯)2n1=5.25=2.29

So, the standard deviation of 5 5 9 9 9 10 5 10 10 is 2.29.


How do we calculate standard deviation in Excel?

Excel has six built-in functions that allow us to calculate the standard deviation of a dataset¾ three for sample data and three for population data.

We find the population standard deviation in Excel with the formula STDEV.P(number1,[number2],…). This is the newer version, available from Excel 2010 onward, that has improved accuracy for estimation. You can still use the old formula, STDEVP( number1,[number2],…), as it’s still available for backward compatibility.

Meanwhile, STDEVPA(value1, [value2], …) calculates the standard deviation based on a population. Its biggest advantage over STDEVP and STDEV.P is that it includes text and logical values in the calculation; it reads all FALSE values as 0, whereas text and TRUE values are represented by 1. When dealing with numerical data, STDEVPA will return the same result as the STDEV.P function.

As for the sample standard deviation, we calculate it using the Excel 2010 formula STDEV.S(number1,[number2],…). It offers better accuracy and calculates the sample standard deviation on numeric values. Similarly to the population SD, this new formula replaces the old STDEV(number1,[number2],…), which you can still find available in older Excel versions for the sake of backward compatibility.

Meanwhile, STDEVA(value1, [value2], …) calculates the sample standard deviation and considers text and logical values. The function assigns 1 to TRUE arguments and 0 to FALSE ones. In this case, text values are treated as 0.

Which formulas you should use depends on the type of data (sample or population; numerical or logical and text) and the version of Excel you have (pre-2010 or newer).

For example, if you’re evaluating numerical sample data on Excel 2010 or a later version, then STDEV.S is the right choice. However, if your data contains text or logical values, then you should go for STDEVA. Here is a summary of all six functions:

STDEV functions table.

To illustrate how to calculate standard deviation in Excel, suppose you have the stock returns of Company AXY over the past 10 months. Now, you’d like to estimate their standard deviation:

Stock returns.

Since this is just a sample of returns and we only have numerical data, we can go for the STDEV.S formula. Type the function, select the data, and close the parentheses to obtain 11.58%:

STDEV.S formula in Excel.

If you use Excel 2007 or prior, you’ll have to use STDEV instead:

STDEV formula in Excel. STDEV.S and STDEV formula in Excel.

The result is the same: 11.58%.