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).
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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)...
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??
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...
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...
+(3/2) standard deviations from the mean = \frac {a+b}{12} + \frac{\sqrt3}{4} (b-a)
-(3/2) standard deviations from the mean = \frac {a+b}{12} - \frac{\sqrt3}{4} (b-a)
\frac {1}{b-a} \int_a^{\frac {a+b}{12} - \frac{\sqrt3}{4} (b-a)} dx = m_1= \frac {(-11+3\sqrt3)a + (1-3\sqrt3)b}{12(b-a)}...
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...
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...
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...
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...
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...
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...
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...
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...
(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...
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...
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...
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...
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...
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...
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...
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?
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...
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...
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...
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...
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 ...
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...