# What is deviation: Definition and 400 Discussions

In statistics, the standard deviation is a measure of the amount of variation or dispersion of a set of values. A low standard deviation indicates that the values tend to be close to the mean (also called the expected value) of the set, while a high standard deviation indicates that the values are spread out over a wider range.
Standard deviation may be abbreviated SD, and is most commonly represented in mathematical texts and equations by the lower case Greek letter σ (sigma), for the population standard deviation, or the Latin letter s, for the sample standard deviation.
The standard deviation of a random variable, sample, statistical population, data set, or probability distribution is the square root of its variance. It is algebraically simpler, though in practice less robust, than the average absolute deviation. A useful property of the standard deviation is that, unlike the variance, it is expressed in the same unit as the data.
The standard deviation of a population or sample and the standard error of a statistic (e.g., of the sample mean) are quite different, but related. The sample mean's standard error is the standard deviation of the set of means that would be found by drawing an infinite number of repeated samples from the population and computing a mean for each sample. The mean's standard error turns out to equal the population standard deviation divided by the square root of the sample size, and is estimated by using the sample standard deviation divided by the square root of the sample size. For example, a poll's standard error (what is reported as the margin of error of the poll), is the expected standard deviation of the estimated mean if the same poll were to be conducted multiple times. Thus, the standard error estimates the standard deviation of an estimate, which itself measures how much the estimate depends on the particular sample that was taken from the population.
In science, it is common to report both the standard deviation of the data (as a summary statistic) and the standard error of the estimate (as a measure of potential error in the findings). By convention, only effects more than two standard errors away from a null expectation are considered "statistically significant", a safeguard against spurious conclusion that is really due to random sampling error.
When only a sample of data from a population is available, the term standard deviation of the sample or sample standard deviation can refer to either the above-mentioned quantity as applied to those data, or to a modified quantity that is an unbiased estimate of the population standard deviation (the standard deviation of the entire population).

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1. ### B What does this distance of an atom mean?

I can look at it as if a vibrational motion of the atoms was a simle harmonic motion. So I can consider one of the two atoms to be at rest and the second one to vibrate. Its deviation can be written as ##x(t)=r(t)-r_0##. When I know that the hydrogen molecule stops exiting when the range of...
2. ### Find the standard deviation of the values of ##y##

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3. ### I Basic standard deviation calculation

I don’t get how they got the equation for the standard deviation. Why do they only square with the time in the denominator? Thanks!
4. ### Uncertainty of the Standard Deviation

Using this error propagation formula: I expressed the standard deviation (s) and the partial derivatives of s w.r.t. each data point as: This gives me an uncertainty of: , where m is the mean. Does this seem reasonable for the uncertainty of the standard deviation? I also found the thread...
5. ### Deviation of Plumb Bob In Uniform Circular Motion

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7. ### I Standard deviation and count rate

Hello, I watched MIT course on Nuclear physics (13. Practical Radiation Counting Experiments on ytb) and I do not understand why 2*sigma (standard deviation) = 0.05* countRate. As far as I know, integral of normal distribution from -2sigma to 2 sigma gives 95 % probability, but how can 2*sigma...
8. ### Very High Standard Deviation in Excitation Emission Matrix Measurement

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9. ### Find the mean and standard deviation of the heights of 13 boys

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10. ### A Software for Converting Spectral Density to Modified Allan Deviation

Mathematically, you can convert between a power spectral density (PSD) and the modified allan variance as follows: $$\sigma_y^2 (\tau) = \int_0^{\infty} \frac{G_\nu(f)}{\nu^2} \times 32 \frac{(\sin(\pi f \tau/2))^4 \times |\sin(\pi f \tau)|^2}{(\pi \tau f)^4} df$$ I was wondering if anyone knew...
11. ### MHB Variance / standard deviation

Hi all - I wonder if you can help please. Watching a video on youtube to help me understand about the mean, variance and standard deviation but last part of video left me confused. The speaker said the following for the formula for standard deviation: Consider if the variance is 200 for the...
12. ### A Re-writing the geodesic deviation eqn in matrix notation (3d only)

This is my attempt to re-write the geodesic deviation equation in the special case of 3 dimensions and +++ signature in matrix notation. We start with assuming an orthonormal basis. Matrix notation allows one to express vectors as column vectors, and dual vectors as row vectors, but by...

17. ### Standard deviation and standard error

The mean of some data was 21.2°C, the standard deviation was 2, and the standard error was 0.8. My textbook says that using one standard deviation, we would report the temperature of the substance as 21.2 ± 2°C, while using the standard error, the temperature would be reported as 21.2 ± 0.8°C...
18. ### Measurement of frequency deviation in FM signals

Hi all Is there any way to measure frequency deviation in a recorded frequency modulated time domain data? The little research i have done on net mostly concerns with frequency deviation measurement using spectrum analyzer in real time i.e. data continuously received on spectrum analyzer while...
19. ### A Spectroscopy of positronium fine structure: 4.5 sigma deviation

Precision Microwave Spectroscopy of the Positronium n=2 Fine Structure A nice compact abstract, so I'll just quote it here: Positronium with its two light leptons is the dream of every theorist, that keeps the uncertainties small. The 0.61 MHz experimental uncertainty are the sum of 0.57 MHz...
20. ### Angle of deviation from a magnetic field

The beam of protons are directed towards the axis of the cylinder, perpendicular to the direction of the field. While traveling through the cross-section of the cylinder, the proton beam experiences a magnetic force, which tends to move the beam in a circular orbit of the radius given by: r =...
21. ### I Confidence interval on Standard deviation

I read that confidence interval on standard deviation can be found using chi-square distribution. If I have a sample size N=500, and sample standard deviation= 3 with mean=0, and I need a 95% confidence, I wasn't sure what to set for degrees of freedom in chi-square formula. Is the degrees of...
22. ### I Correctly Scaling the Standard Deviation for Scaled Measurements

We're working on a project that plots flux density of a light curve with respect to time. To do this, we had to scale data from different wavelengths so we had just the one variable for the flux. Essentially we took each value for flux density and multiplied it by three over the frequency raised...
23. ### Recursively calculate the standard deviation

#include<stdio.h> #include<math.h> double foo(int n, int mean){ double square; if(n==1){ return(1); } if(n!=0){ square=pow(n-mean,2); return( (sqrt(square)+foo(n-1,mean))/(sqrt(n-1)) ); }} int main(){ int num; double mean; int i; int...
24. ### Standard deviation vs measurement uncertainty

Using the above formulas, we can arrive at an unbiased estimate of the standard deviation of the sample, then divide by sqrt(N) to arrive at the standard deviation of the average. What I'm confused about it where the measurement uncertainty comes into the equation. Is it being ignored? Say I...
25. ### B Eddington's 1919 Eclipse: Photon Deviation & Redshift

Hi all, I've been wondering: Thinking of Arthur Eddington's relativistic oriented 1919 eclipse observation, would the photon deviation due to the Sun's gravitational imposition have caused the photons to exhibit a qualitative redshift due to the time photons had spent within the Sun's...
26. ### I Is this weighted mean and standard deviation correct?

The expression I have found is this one. https://ibb.co/kqG24L3 I have been looking for information because I could not to realize what is the value that "alpha" has to have. If any of you do know what this alpha value is supposed to represent or if you have seen it before I would be really...
27. ### A Low value standard deviation bias

Excuse me, I am not too well versed in statistics. I am in engineering. Let's say I have an expected measurements(grand mean of values) and then I take another measurement of two different samples. Each sample is measured a few times to get it's own standard deviation and expected value. I...
28. ### I Standard deviation of the Hamiltonian?

I am currently reading Griffiths Introduction to Quantum Mechanics, 2nd Edition. I am aware that, in light of considering potential functions independent of time, the Schrödinger equation has separable solutions and that these solutions are stationary states. I am also aware (If I stand correct)...
29. ### MATLAB Plotting standard deviation in Matlab

I have a data set 1st coloumn containing ##x## values 2nd coloumn containing ##y## values and 3rd coloumn containing standard deviation of ##y## associated with each ##x## value...I want to plot these 3 values together in a MATLAB plot...can anyone please help me in this regard??
30. ### How to show the root mean square deviation

Hi I tried like this. ##σ^2=<(λ_1+λ_2+,,,+λ_i)^2>=<λ_1^2>+<λ_2^2>+,,,<λ_i^2>## And I know ##σ^2=Σ_in_iλ_i^2##from equation (4-12) (so this is cheat 😅). So I know also ##<λ_i^2>=n_iλ_i^2##, But why?? I know if I take ##λ=1 ,σ^2=n##,But I don't understand ##λ≠1## version. Sorry my bad...
31. ### I Calculating a standard deviation involving weighted samples

Below is the spreadsheet image. The source of the data in the two columns, Ho and +/-, is the following article http://planck.caltech.edu/pub/2015results/Planck_2015_Results_XIII_Cosmological_Parameters.pdf . Let AHo be the conjoined weighted average values for Ho and the corresponding error...
32. ### I Standard deviation question -- population std vs sample std

I know that standard deviation of whatever data is defined as sqaure root of square difference from mean value: ##\sigma(data)=\frac{\sum_{x \in data}((x-x_{mean\ of\ data})^2)}{|data|}=\frac{\sum_{x \in data}((x-\sum_{y \in data}(y)/|data|)^2)}{|data|}## but sometimes formula...
33. ### Uniform distribution and standard deviation

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34. ### I Calculating Acceleration of Gravity w/ Geodesic Deviation: Troubleshooting

I have tried twice now to calculate acceleration of gravity using the general relativistic equation of geodesic deviation and both times my solution is twice the correct answer. What am I doing wrong? As an example here is one problem: Calculate the acceleration of gravity g at the earth’s...
35. ### Uncertainties and the angle of minimum deviation

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36. ### I Error propagation and standard deviation

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37. ### Standard deviation and probability for decay

Homework Statement (b) A nuclear research reactor produces radiation for neutron scattering measurements. A safety procedure shuts the reactor down if a radiation level monitoring detector measures more than 3 counts per minute. In a test, 156 counts are recorded during a random 24 hour...
38. ### A Using standard deviation values as independent variables

Hey. I am planning on doing some research, where I predict a change based on different types of risk. The question is simple. Can I use values of standard deviation as independent variables in a linear regression analysis (OLS)? The standard deviation values over time will be calculated in...
39. ### Calculations using Standard Deviation and Mean

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40. ### B Standard Deviation as Function of Sample Size

In high school, I was taught that the standard deviation drops as you increase the sample size. For this reason, larger sample sizes produce less fluctuation. At the time, I didn't question this because it made sense. Then, I was taught that the standard deviation does not drop as you increase...
41. ### I Geodesic Deviation: Definition of Connecting Vector

Hello! I read a derivation for the geodesic deviation: you have 2 nearby geodesics and define a vector connecting points of equal proper time and calculate the second covariant derivative of this vector. I understand the derivation but I am a bit confused about the actual definition of this...
42. ### Calculate the bulk modulus and Derive the mean standard deviation

(apologies for re-posting, I am unable to find my original thread.) A material which has a Young's modulus of elasticity of 250 GN m-2 and a poisons ratio of 0.32, calculate: (a) the bulk modulus of the material   Relevant equation, K=E/3(1-2v) Inputting our known values...
43. ### Determine the standard deviation of these results

I've managed to work out the below for the sample standard deviation. Any error's that I should be aware of? Homework Statement The ultimate tensile strength of a material was tested using 10 samples. The results of the tests were as follows. 711 N 〖mm〗^(-2),732 N 〖mm〗^(-2),759 N...
44. ### I The deviation of Universe expansion from general relativity

What is the deviation in the expansion of the universe exactly quantified, when I would assume general relativity and project it backwards? As a statistician I am asking for data, for either the backwards projected general relativity case and either the real expansion case, as it is...
45. ### I Infinite or undefined standard deviation in HUP

If we measure one conjugate variable in an uncertainty relation precisely , i.e., so its standard deviation is zero, then by the HUP the sd of the other one is either infinite or undefined. But what about the cases when the other conjugate variable has limits: e.g., there cannot be an infinite...
46. ### B Why do we have minimum deviation of a prism?

Why do we have minimum deviation of prism? physical explanation is needed!
47. ### Random Variables: Mean and Standard Deviation

Homework Statement The same potato chip company reports that their bags of family sized chips each follows an approx. Normal distribution with a mean of 10.72 ounces and a standard deviation of 0.2 ounces. If the company wants to ship these chips into boxes that contain 6 bags, what would be...
48. B

### I First order deviation from circularity

This is regarding the Bertrand theorem in the book Classical Mechanics by Goldstein. It is said that for more than first order deviations from circularity the orbits are closed only for inverse square law and hooke's law. What does first order deviation mean ?
49. ### Standard deviation of V_x of He gas

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50. ### Deviation in refraction and TIR

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