dispersion of the values ββof a random variable around its expected value
Wikipedia
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Standard deviation - Wikipedia
20 hours ago - Standard deviation may be abbreviated SD or std dev, and is most commonly represented in mathematical texts and equations by the lowercase Greek letter Ο (sigma), for the population standard deviation, or the Latin letter s, for the sample standard deviation.
Statistics LibreTexts
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Symbols - Statistics LibreTexts
March 12, 2023 - https://stats.libretexts.org/@app/auth/3/login?returnto=https://stats.libretexts.org/Bookshelves/Introductory_Statistics/Mostly_Harmless_Statistics_(Webb)/zz%3A_Back_Matter/24%3A_Symbols
Confused when to use Population vs Sample standard deviation in engineering testing - Cross Validated
When I run an test for something (say 10 trials) and want to find the standard deviation of all 10 trials, I am getting confused if I should use the sample or population standard deviation. My init... More on stats.stackexchange.com
Population standard deviation vs. Sample standard deviation?
I think part of the problem is that the terms "population standard deviation" and "sample standard deviation" are confusing, because almost everyone seems to think that the "sample standard deviation" is the standard deviation of the sample. It's actually a different concept entirely; it's an "estimator" for the population standard deviation. Imagine that we're making precisely calibrated rulers, and we want to make sure that the lengths of all 1 million of the rulers we made today have a very small standard deviation. That is, we want to know the population standard deviation of the lengths of the rulers. However, no one has the time to literally measure 1 million rulers, so our only choice is to draw a small random sample and figure out how to guess the population standard deviation from the limited data we have. That's what we use the sample standard deviation for. It's a value that we calculate from the sample in order to estimate the true value of the population standard deviation. So, why don't we divide by N in the sample standard deviation? Why isn't the standard deviation of the sample a good estimate of the standard deviation of the population? It turns out that it's biased to give values that are too small. The problem is that we want to find the population standard deviation, which measures variation around the true mean, but the standard deviation of the sample only gives us variation around the sample mean. Naturally, the members of a sample are biased to be closer to their sample mean, so they tend to have a smaller standard deviation than the whole population. That's why we need a different formula for the sample standard deviation. We use N-1 because that's the value that turns the sample standard deviation into an "unbiased estimator", whose average value approaches the true population standard deviation. More on reddit.com
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May 21, 2019Sample vs. population standard deviation
"Know" is not the correct word, "predict" is a better word. Remember, when you are talking about samples, all of the parameters you come across are just estimates. The thing is that in that case, the mean itself (which is used in the standard deviation formula) is estimated, which makes the standard deviation look less that it probably is. Therefore, a correction is made to increase the std estimation (the smaller the sample size, the stronger the correction is). This should be obvious for small samples. For large enough samples, the effect is tiny. Most samples stay somewhere in the middle, and some really smart guys proved that the formula you were given is the best one to use. (tried to do a simple explanation, someone with more formal knowledge than me can throw math at it to complement my answer) More on reddit.com
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June 14, 2013Sample vs Population Standard Deviation in Propagation of Uncertainty
You're making a pretty big error in your first paragraph. You don't "convert" between the population variance and the sample variance like that -- that's not a thing. The formula you're referencing is the unbiased estimator of the population variance. There are a bunch of proofs online that if you take sum(x_i - x-bar)^2 / n to estimate the population variance, you get in expectation (n-1)/n * sigma_P^2 using your notation. Multiplying by n / (n-1) makes it unbiased. I'm really unsure of what the second paragraph is about, nor do I understand what the point is about margin of error vs standard deviation. Nevertheless, hopefully the above clarification sets you on a more correct path. More on reddit.com
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December 13, 2020Videos
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ThoughtCo
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Population vs. Sample Standard Deviations
May 11, 2025 - The population standard deviation is a parameter, which is a fixed value calculated from every individual in the population. A sample standard deviation is a statistic. This means that it is calculated from only some of the individuals in a population. Since the sample standard deviation depends upon the sample, it has greater variability.
BrownMath
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Symbol Sheet / SWT
Here are symbols for various sample statistics and the corresponding population parameters. They are not repeated in the list below. ΞΌ and Ο can take subscripts to show what you are taking the mean or standard deviation of.
Uedufy
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Population vs Sample Standard Deviation Formula: Complete Guide
March 22, 2022 - And this is how we read the above equation: sample standard deviation (s) is equal to the square root of the sum of (Ξ£) the squared differences between every data point (xi) in the sample and the sample mean (xΜ), divided by population N β 1.
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The two forms of standard deviation are relevant to two different types of variability. One is the variability of values within a set of numbers and one is an estimate of the variability of a population from which a sample of numbers has been drawn.
The population standard deviation is relevant where the numbers that you have in hand are the entire population, and the sample standard deviation is relevant where the numbers are a sample of a much larger population.
For any given set of numbers the sample standard deviation is larger than the population standard deviation because there is extra uncertainty involved: the uncertainty that results from sampling. See this for a bit more information: Intuitive explanation for dividing by 
when calculating standard deviation?
For an example, the population standard deviation of 1,2,3,4,5 is about 1.41 and the sample standard deviation is about 1.58.
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My question is similar pnd1987's question. I wish to use a standard deviation in order to appraise the repeatability of a measurement. Suppose I'm measuring one stable thing over and over. A perfect measuring instrument (with a perfect operator) would give the same number over and over. Instead there is variation, and let's assume there's a normal distribution about the mean.
We'd like to appraise the measurement repeatability by the SD of that normal distribution. But we take just N measurements at a time, and hope the SD of those N can estimate the SD of the normal distribution. As N increases, sampleSD and populationSD both converge to the distribution's SD, but for small N, like 5, we get only weak estimates of the distribution's SD. PopulationSD gives an obviously worse estimate than sampleSD, because when N=1 populationSD gives the ridiculous value 0, while sampleSD is correctly indeterminate. However, sampleSD does not correctly estimate the disribution's SD. That is, if we measure N times and take the sampleSD, then measure another N times and take the sampleSD, over and over, and average all the sampleSDs, that average does not converge to the distribution's SD. For N=5, it converges to around 0.94Γ the distribution SD. (There must be a little theorem here.) SampleSD doesn't quite do what it is said to do.
If the measurement variation is normally distributed, then it would be very nice to know the distribution's SD. For example, we can then determine how many measurements to take in order tolerate the variation. Averages of N measurements are also normally distributed, but with a standard deviation 1/sqrt(N) times the original distribution's.
Note added: the theorem is not so little -- Cochran's Theorem
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Population and sample standard deviation review (article)
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Statology
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Population vs. Sample Standard Deviation: When to Use Each
August 23, 2021 - The formula to calculate a sample standard deviation, denoted as s, is: ... 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 ...
Math Vault
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List of Probability and Statistics Symbols | Math Vault
April 11, 2025 - A comprehensive collection of the most common symbols in probability and statistics, categorized by function into charts and tables along with each symbol's term, meaning and example.
Math is Fun
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Standard Deviation and Variance
Looks complicated, but the important change is to divide by N-1 (instead of N) when calculating a Sample Standard Deviation. If we just add up the differences from the mean ... the negatives cancel the positives: So that won't work. How about we use absolute values? That looks good (and is the Mean Deviation), but what about this case:
Quizlet
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Standard Deviation Symbols and Units Study Guide
May 7, 2024 - Quizlet makes learning fun and easy with free flashcards and premium study tools. Join millions of students and teachers who use Quizlet to create, share, and learn any subject.
The Math Doctors
themathdoctors.org βΊ formulas-for-standard-deviation-more-than-just-one
Formulas for Standard Deviation: More Than Just One! β The Math Doctors
July 25, 2025 - The traditional symbol for the sample standard deviation is S (lowercase or uppercase; there is a slight difference between the two) and the equivalent Greek letter sigma (which looks like an o with a little tail sticking out from the top) is commonly used to denote the population standard ...
ScienceDirect
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Population Standard Deviation - an overview | ScienceDirect Topics
There are actually two different (but related) kinds of standard deviation: the sample standard deviation (for a sample from a larger population, denoted S) and the population standard deviation (for an entire population, denoted Ο, the lowercase Greek sigma).
Socratic
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What is the difference between the population standard ...
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Stat Trek
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Statistics Notation
Ο refers to the standard deviation of a population; and s, to the standard deviation of a sample. By convention, specific symbols represent certain population parameters.
Reddit
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r/learnmath on Reddit: Population standard deviation vs. Sample standard deviation?
May 21, 2019 -
why is the population standard deviation the square root of the sum of the (values - means)^2 Γ· n , while the sample standard deviation is all that over n - 1? I don't understand why you have to subtract 1 from the number of things.
Top answer 1 of 4
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I think part of the problem is that the terms "population standard deviation" and "sample standard deviation" are confusing, because almost everyone seems to think that the "sample standard deviation" is the standard deviation of the sample. It's actually a different concept entirely; it's an "estimator" for the population standard deviation. Imagine that we're making precisely calibrated rulers, and we want to make sure that the lengths of all 1 million of the rulers we made today have a very small standard deviation. That is, we want to know the population standard deviation of the lengths of the rulers. However, no one has the time to literally measure 1 million rulers, so our only choice is to draw a small random sample and figure out how to guess the population standard deviation from the limited data we have. That's what we use the sample standard deviation for. It's a value that we calculate from the sample in order to estimate the true value of the population standard deviation. So, why don't we divide by N in the sample standard deviation? Why isn't the standard deviation of the sample a good estimate of the standard deviation of the population? It turns out that it's biased to give values that are too small. The problem is that we want to find the population standard deviation, which measures variation around the true mean, but the standard deviation of the sample only gives us variation around the sample mean. Naturally, the members of a sample are biased to be closer to their sample mean, so they tend to have a smaller standard deviation than the whole population. That's why we need a different formula for the sample standard deviation. We use N-1 because that's the value that turns the sample standard deviation into an "unbiased estimator", whose average value approaches the true population standard deviation.
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Here is an article explaining it. https://www.statisticshowto.datasciencecentral.com/bessels-correction/
Statistics LibreTexts
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6.1: The Mean and Standard Deviation of the Sample Mean - Statistics LibreTexts
March 27, 2023 - Suppose random samples of size \(n\) are drawn from a population with mean \(ΞΌ\) and standard deviation \(Ο\).
YouTube
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STATS 1024 (Western University) - Population vs Sample Standard Deviation Example - YouTube
Give me a shout if you have any questions at [email protected] :)Course Website - Introduction to Statisticswww.STATS1024.comOther Western Uni...
Published Β September 9, 2019
Macroption
macroption.com βΊ population-sample-variance-standard-deviation
Population vs. Sample Variance and Standard Deviation - Macroption
When I calculate population variance, I then divide the sum of squared deviations from the mean by the number of items in the population (in example 1 I was dividing by 12). When I calculate sample variance, I divide it by the number of items in the sample less one. In our example 2, I divide by 99 (100 less 1). As a result, the calculated sample variance (and therefore also the standard deviation) will be slightly higher than if we would have used the population variance formula.