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Standard Deviation Calculator
This free standard deviation calculator computes the standard deviation, variance, mean, sum, and error margin of a given data set.
Khan Academy
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Population and sample standard deviation review (article)
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sampling - What is the standard error of the sample standard deviation? - Cross Validated
There is no general exact formula for this standard error, as @Alecos Papadopoulos pointed out. However, one can drive an approximate (large sample) standard error by means of the delta method. More on stats.stackexchange.com
ELI5: Why is the formula for standard deviation different for a sample and a population?
When you compute the standard deviation from a sample, you almost always have to compute it "around" the observed mean of the sample (not the true mean of the population) because the true mean of the population is unknown. The difference between the observed mean and the true mean causes a bias in the standard deviation which can be corrected by using a different formula.
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May 25, 2017Why does the standard deviation formula have an (n-1) instead of n?
Essentially, you want to try and find the population standard deviation from a smaller sample. If the population is finite and you know all the relevant information (eg, you know everyone's age in the US), then you can compute the exact population standard deviation, and this has an 'N' in the bottom rather than 'N-1'. But this works because if we know absolutely everything about a population, then we can compute the actual mean of the population. If we only have a sample of the population, then the mean itself is an estimation. So every time you use the mean of the sample, you're introducing error. In fact, using the sample mean and dividing by 'N', you will always underestimate the actual standard deviation you would get if you knew and computed with the actual mean. Wikipedia has a pretty good explanation of why this happens. So computing the standard deviation using the sample mean (and dividing by N) will always give you a smaller standard deviation than you should get. Luckily, we can compare this computed standard deviation to the standard deviation you would get using the population mean instead, and the way to correct for the error that you get for using the sample mean is to multiply your result by N/(N-1), which is the same as just dividing by N-1 at the get-go. More on reddit.com
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August 30, 2017[Statistics] Standard deviation formulas
The difference is due to Bessel's correction. You can read more about it here.
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October 2, 2014Videos
How to Find Sample Standard Deviation in Minutes! - YouTube
02:06
How To Calculate Sample Standard Deviation (Step By Step) - YouTube
How to Find Sample Standard Deviation - YouTube
07:14
How To Calculate The Standard Deviation - YouTube
05:00
Statistics - How to calculate the standard deviation - YouTube
05:23
How To Calculate The Standard Deviation - Clearly Explained! - YouTube
dispersion of the values โโof a random variable around its expected value
Wikipedia
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Standard deviation - Wikipedia
1 day ago - Most often, the standard deviation is estimated using the corrected sample standard deviation (using N โ 1), defined below, and this is often referred to as the "sample standard deviation", without qualifiers. However, other estimators are better in other respects: the uncorrected estimator ...
JMP
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Standard Deviation
The sample variance is shown in formulas as s2. The standard deviation makes sense for continuous data. This data is measured on a scale with many possible values.
CalculatorSoup
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Standard Deviation Calculator
November 4, 2025 - When working with data from a complete population the sum of the squared differences between each data point and the mean is divided by the size of the data set, n. When working with a sample, divide by the size of the data set minus 1, n - 1.
DataCamp
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Sample Standard Deviation: The Key Ideas | DataCamp
September 26, 2024 - The sample standard deviation is calculated as the square root of the sample variance, where the sample variance is the sum of the squared differences from the mean, divided by the sample size minus one. We use the sample standard deviation when we want to know how much the data points in a ...
Statology
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Population vs. Sample Standard Deviation: When to Use Each
August 23, 2021 - From the formulas above, we can see that there is one tiny difference between the population and the sample standard deviation: When calculating the sample standard deviation, we divided by n-1 instead of N.
. Then, the formula for the SE of 
is:
This is an exact formula, valid for any sample size and distribution, and is proved on page 438, of Rao, 1973, assuming that the 
is finite. The formula you gave in your question applies only to Normally distributed data.
. You want to find the SE of 
, where 
.
where 
is the first derivative.
National Library of Medicine
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Standard Deviation - Finding and Using Health Statistics - NIH
The first step is to subtract the mean from each data point. Then square the value before adding them all together. Now divide by 9 (the total number of data points) and finally take the square root to reach the standard deviation of the data:
BYJUS
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Standard Deviation Formula
September 16, 2020 - This formula is given as: \(\begin{array}{l}\sigma=\frac{1}{N}\sqrt{\sum_{i=i}^{n}f_{i}x_{i}^{2}-(\sum_{i=1}^{n}f_{i}x_{i})^{2}}\end{array} \) Also Check: Difference Between Variance and Standard Deviation
Statistics LibreTexts
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4.3: Standard Deviation - Statistics LibreTexts
October 22, 2024 - One which is subsumed into the standard deviation formula is known as the sum of squares within (often referred to simply as the โsum of squaresโ or SS). The sum of squares within is the sum of squared deviations from the mean within a sample. Its name is just a simplified version of its definition and operations.
Laerd Statistics
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Standard Deviation | How and when to use the Sample and Population Standard Deviation - A measure of spread | Laerd Statistics
Therefore, you would normally calculate the population standard deviation if: (1) you have the entire population or (2) you have a sample of a larger population, but you are only interested in this sample and do not wish to generalize your findings to the population.
University of Southampton Library
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Maths and Stats - Variance, Standard Deviation and Standard Error - LibGuides@Southampton at University of Southampton Library
November 10, 2025 - Standard deviation is denoted by the Greek letter sigma and, being the square root of variance, is written as: ... Standard error is another measure of spread. The most common standard error is the standard error of the mean, and used to measure sampling error as it measures how accurately the mean of a sample distribution represents the mean of the population.
Investopedia
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Standard Deviation Formula and Uses, vs. Variance
June 5, 2025 - Standard deviation is calculated by taking the square root of a value derived from comparing data points to a collective mean of a population. The formula is:
wumbo.net
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Sample Standard Deviation Formula
The variable differentiates the sample standard deviation from the population standard deviation which is denoted using (sigma). Comparing the two formulas, the sample standard deviation subtracts one from the sample size which can be seen in the expression .
Cuemath
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Standard Deviation - Formula | How to Calculate Standard Deviation?
To adjust this, the denominator of the sample standard deviation is corrected to be n-1 instead of just n. This is known as Bessel's correction. There are two types of data sets: populations and samples. A population is an entire group that we are interested in studying, while a sample is a smaller group of individuals that is taken from the population. The formulas to calculate the standard deviations of population and sample differ a little.
Statistics LibreTexts
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6.1: The Mean and Standard Deviation of the Sample Mean - Statistics LibreTexts
March 27, 2023 - The standard deviation of the sample mean \(\bar{X}\) that we have just computed is the standard deviation of the population divided by the square root of the sample size: \(\sqrt{10} = \sqrt{20}/\sqrt{2}\). These relationships are not coincidences, ...
Khan Academy
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Standard deviation: calculating step by step (article)
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Scribbr
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How to Calculate Standard Deviation (Guide) | Calculator & Examples
March 28, 2024 - Since weโre working with a sample size of 6, we will use n โ 1, where n = 6. To find the standard deviation, we take the square root of the variance. From learning that SD = 13.31, we can say that each score deviates from the mean by 13.31 ...
Standard Deviation Calculator
standarddeviationcalculator.io
Standard Deviation Calculator - Sample/Population
Step 5: Take the square root of โ(xi - xฬ
)2/(N-1) to get the standard deviation. ... The above sample standard deviation calculator can be used to check the result of this example.