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. Lotto

    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. chwala

    Find the standard deviation of the values of ##y##

    This is the question; This is the solution as received; I am not familiar with the approach used in the solution...my thinking was as follows The frequencies are the same...the only thing changing are the discrete variables thus; Let ##[x= 2,4,6]## and ##[y=7,13,19]## form a...
  3. Graham87

    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. A

    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. mopit_011

    Deviation of Plumb Bob In Uniform Circular Motion

    I started by making my coordinate system so that the x-axis aligned with the radius of the circle at a certain latitude L and the positive direction was facing away from the center of the circle, and the y-axis was parallel to the vertical axis of the Earth. Then, I wrote the equations for the...
  6. Zuzana

    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...
  7. J

    Very High Standard Deviation in Excitation Emission Matrix Measurement

    Hi, I obtain really high standard deviations in Excitation-Emission Spectra mainly for the phenolic compounds in olive oil (Em: 290-350nm). Method: I weigh 0.05g of olive oil and dilute it up to 25ml with cyclohexane to remain in the range of linearity for absorbance measurements to correct...
  8. chwala

    Find the mean and standard deviation of the heights of 13 boys

    Find the textbook problem here; Find the textbook solution here: Now, to my question, did the textbook guys make an error on the value of ##σ?##, see my working; Mean (##13## boys)=##\dfrac{153.4+(148.8×12)}{13}=149.15## We know that, ##29.16##=##\dfrac{\sum x^2}{12}##-##(148.8)^2## ##\sum...
  9. Twigg

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    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...
  10. L

    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...
  11. P

    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...
  12. M

    MHB Minimal mean square deviation

    Hey! :giggle: We consider a double roll of the dice. The random variable X describes the number of pips in the first roll of the dice and Y the maximum of the two numbers. The joint distribution and the marginal distributions are given by the following table Using : For all $a,b\in...
  13. JD_PM

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    Let's focus on obtaining how much will the ball deviate from a straight path and assume the spin of the ball to be ##11600## rpm and ##C_L = 0.4## to be the lift coefficient A pitcher is able to make a baseball follow a curved path by impinging spin on the ball (which triggers nonsymmetric...
  14. P

    Mathematical equation for the deviation from ideality of real gases

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  15. karush

    MHB -14.1 ....standard deviation of 15......

    Given IQ scores are approximately normally distributed with a mean of 100 and standard deviation of 15, the proportion of people with IQs above 130 is: $ a.\ 95\% \quad b.\ 68\% \quad c.\ 5\% \quad d.\ 2.5\% \quad e.\ 12\%$ ok its been some moons since I looked...
  16. I

    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...
  17. N

    Measurement of frequency deviation in FM signals

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  18. mfb

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  19. tanaygupta2000

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    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 =...
  20. iVenky

    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...
  21. C

    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...
  22. A

    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...
  23. bluemystic

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    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...
  24. G

    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...
  25. JorgeM

    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...
  26. J

    A Low value standard deviation bias

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  27. K

    I Standard deviation of the Hamiltonian?

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  28. A

    MATLAB Plotting standard deviation in Matlab

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  29. arcTomato

    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...
  30. Buzz Bloom

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    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...
  31. olgerm

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    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...
  32. R

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  33. Jim Hasty

    I Calculating Acceleration of Gravity w/ Geodesic Deviation: Troubleshooting

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  34. J

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  35. S

    I Error propagation and standard deviation

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  36. Flabbergast

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  37. M

    A Using standard deviation values as independent variables

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  38. E

    Calculations using Standard Deviation and Mean

    Homework Statement Homework Equations Chebyshev's Theorem: The percentage of observations that are within k standard deviations of the mean is at least 100(1 - (1/k2))% Chebyshev's Theorem is applicable to ANY data set, whether skewed or symmetrical. Empirical Rule: For a symmetrical...
  39. Roger Dodger

    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...
  40. S

    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...
  41. T

    Calculate the bulk modulus and Derive the mean standard deviation

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 Relevant equation, K=E/3(1-2v) Inputting our known values...
  42. T

    Determine the standard deviation of these results

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  43. C

    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...
  44. nomadreid

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  45. Sheldon11

    B Why do we have minimum deviation of a prism?

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  46. S

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  47. B

    I First order deviation from circularity

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  48. Pushoam

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  49. J

    Deviation in refraction and TIR

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