What is Standard deviation: Definition and 276 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 sigma σ, 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 are 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. I Calculating Statistical Likelihood Knowing Standard Deviations

Let’s say that we know, with 95% confidence, that something is likely to occur when the Universe is between 1058 and 10549 years old. What is the statistical likelihood that it has already occurred in the first 13.8 billion years of the Universe’s existence? (1.38 X 1010 years) I know the...
2. B Is there a relationship between the harmonic mean and the standard deviation?

I was doing a thought experiment on the harmonic mean. Let's say we have a sequence of 4 numbers: 100, 110, 90, 100 The arithmetic mean (AM) of this sequence is 100 but the harmonic mean (HM) is 99.4975. Let's imagine that the initial and final values remain constant but the middle ones get...
3. B How to statistically calculate the final value?

My steps would be that I would first calculated ##a_1## and ##a_2##, determined by using that formula with partial derivatives its errors, and then I would made an arithmetic mean of ##a_1## and ##a_2##. I am not sure how to determine the final error, but I think I can use this formula ##\Delta...
4. I Error bars in Excel (standard deviation, standard error, percentage)

Hello, I was working in excel on data reported in two columns, X data and Y data. I measured both X and Y, i.e. Y was not calculated using a function and the X data. The Y an X data was plotted as Y vs X using a scatterplot. The Excel graph offers the option to add error bars over each point...
5. 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...
6. 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!
7. Understanding the meaning of "expected fraction" (Statistics)

The first part of the question asked me to calculate the mean and standard deviation for the number of remain votes in the simple binomial model consisting of total sample size of 2091 people. I believe this is fairly straightforward, it was simply ##E(X) = \mu = 2091(0.5) = 1045.5## votes and...
8. 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...

10. I Alternate expressions for the uncertainty propagation

I have seen that there are two different formulas that we can use when calculating the propagation of uncertainty in a measurement. If ##X=f(A, B, C, \ldots)## is the quantity whose uncertainty we want to estimate, which depends on the quantities ##A,B,C,...##, then we could calculate the...
11. 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...
12. 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...
13. 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...
14. 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...
15. A Introduction of a factor Δℓ when summing equal distants 𝐶ℓ

Hello, In the context of Legendre expansion with ##C_\ell## quantities, below the following formula which is the error on a ##C_\ell## : ##\sigma_(C_{\ell})=\sqrt{\frac{2}{(2 \ell+1)\Delta\ell}}\,C_{\ell}\quad(1)## where ##\Delta\ell## is the width of the multipoles bins used when computing...
16. 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. 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...
18. 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...
19. 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...
20. I Need clarification on the margin of error in two different cases

I need this in simple terms. Here's what I think I know so far (assuming 95% confidence level): MOE from a sample proportion is 2√(p(1-p)/n) and I think this is assuming the sample proportion is close to the population proportion. But then there is a sampling distribution of sample proportions...
21. 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...
22. 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...
23. 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...
24. 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)...
25. 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??
26. 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...
27. 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...
28. Uniform distribution and standard deviation

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29. How to take into consideration errors in the Mann-Whitney test?

I have 2 independent samples (groups), each group contains 10 experimental values of variable T and their standard deviations (errors). I need to do intergroup statistical analysis, i.e. to elucidate whether there is significant difference between 2 groups in terms of variable T. I chose...
30. I Error propagation and standard deviation

Hey there, First time on this forums, looking forward to some interesting discussions :) I am currently trying grasp the concepts of error propagation and standard deviation in relation to experimental physics. I have some data set and i want to determine the difference between the measured and...
31. B Chebyshev inequality, confidence intervals, etc

Hello. I am bewildered by so many different notions of probability distribution percentages, i.e. the proportion of values that lie within certain standard deviations from the mean. (1) There is a Chebyshev's inequality: - for any distribution with finite variance, the proportion of the...
32. 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...
33. 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...
34. 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...
35. 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...
36. Intensity characterization of laser pulses

754/5000 I am using the fluroescence spectroscopy technique to obtain the fluorescence spectrum of exposed neuroblasts to uv radiation (355 nm) from a Nd:YAG source and I need to characterize the radiation pulses of the laser. Specifically, I use a spectrometer (Avantes brand) with integration...
37. 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...
38. 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...
39. A Sample Test | Component Lifetime

Hi, I'm currently working on a project for which I have to determine the Life-Time of a certain mechanical component within a certain confidence interval. By sampling a small number (let's say n = 10) of these components and measuring the number of hours until failure, I want to determine this...
40. I Understanding the transformation of skewness formula

Hello. Here is the formula that computes the sample skewness: Sk = [ (n / [(n-1)(n-2)] ] x [ ∑ (Xi - X)3 / s3 ] , where n is the number of elements in the sample, Xi is the specific i-th value of the sample, where i starts from 0 and ends at i=n, X is the arithmetic mean, s - standard...
41. 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...
42. 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...
43. Standard deviation of V_x of He gas

Homework Statement Homework EquationsThe Attempt at a Solution ## \frac 1 2 m<v_x^2> =\frac 1 2 k_BT , \\ \sqrt{ <v_x^2>} = 556~ m/s ## So, I guess that the standard deviation should be less than rms speed. So, the option is (a) ## \left< ax + b \right> = a\left<x\right> + b...
44. I Standard Deviation Versus Sample Size & T-Distribution

I don't understand why does the standard deviation of a t-Distribution decreases as the degree of freedom (and, thus, also the sample size) increases when the sample standard deviation underestimates the population standard deviation?
45. T

I The Standard Deviation of Sample vs. Population in Probability Calculations

I know that the standard deviation of sample = standrad deviation of population divided by sqrt(n) ... However , in the following question , i don't know how to identify whether the standard deviation given is standard deviation of sample or standard deviation of population ... Can anyone help...
46. What's the standard deviation of values in the histogram bar

Homework Statement If one bar of a histogram has been generated with ##n## entries from a total of ##x## measurements, i.e. the event occurs randomly ##n## times in the ##x## event interval, then what is the standard deviation of values in this bar? Let ##k## be the range of values that could...
47. MHB How Do I Calculate Standard Deviation with Changed Values in a Set?

Can someone guide me through the process of solving the following? Many thanks. "Given the set {11, 12, 13, 14, 15}, by approximately how much does the standard deviation change if the least value is decreased by 2 and the greatest value is increased by 2?" These questions are all over the...
48. MHB Standard Deviation: 10 & 11 Consecutive Positive Multiples of n

Hi, All. I'm trying to re-familiarize myself with standard deviations. Any resources? I'm reading through the threads here and trying to figure out the following: "n is a positive integer. What is the standard deviation of 10 consecutive positive multiples of n. And what is the standard...
49. T

Standard deviation change problem

Homework Statement A supplier converts the weights on the cement packages she sends out from Ounces to kilograms (1kg = 35.27 oz). How does this affect the standard deviation of the weights? Homework EquationsThe Attempt at a Solution [/B] My ans is the standard deviation remain constant ...
50. T

Calculating Changes in Standard Deviation without Given Summation Values

Homework Statement There are 15 numbers on a list, and the mean is 25. The smallest number on the list is changed from 12.9 to 1.29. a. Is it possible to determine by how much the mean changes? If so, by how much does it change? b. Is it possible to determine the value of the mean...