In Probability and Statistics, the standard deviation of a statistical population or data set is a measure of how much variation or dispersion exists from its average value. It is defined as the square root of the variance.

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Can the mean of the means of multiple sets from a single population give insight into the individual sets or the overall population?

This relates to environmental sampling: I have three composite samples consisting of nine sub-samples each collected from a waste lagoon. Assuming I treat each composite sample result as a "mean" (...
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Can I use mean and standard deviation to spot outliers?

I have a list of measured numbers (e. g. lengths of products). Of these I can easily compute the mean and the standard deviation. Now, when a new measured number arrives, I'd like to tell the ...
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standard deviation above or below the mean [on hold]

The following is data of 10 patients in a hospital 165,135,151,155,158,146,149,124,162,173 If a patient is selected randomly, find the probability his/her data exceeds one standard deviation above ...
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How to maximize Std Dev given a range of possible values, a number of values, and a specific mean?

(I'm asking here and not stats.stackexchange because I'd like a mathematical proof of this) In this question: Prove how to maximize Standard Deviation given a certain mean $\bar{x}$ and set of values;...
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Prove how to maximize Standard Deviation given a certain mean $\bar{x}$ and set of values

I'm talking specifically of population SD, where $$s = \sqrt{ \frac{1}{n} \sum_{i=1}^{n} (x_i - \bar{x})^2}$$ I have a hunch that $s$ is maximized for a certain mean $\bar{x}$ when the values in ...
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Prove that Standard Deviation is always $\geq$ Mean Absolute Deviation

Where $$s = \sqrt{ \frac{1}{n} \sum_{i=1}^{n} (x_i - \bar{x})^2}$$ and $$ M = \frac{1}{n} \sum_{i=1}^{n} |x_i - \bar{x}|$$ I came up with a sketchy proof for the case of $2$ values, but I would like ...
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Caclulate Standard Error when subtracting two effects

I am working on this issue that compares two regions with the McCrary Test (Regression Discontinuity Design). It basically measures how the density of a variable of interest develops accross a pre set ...
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Derivation of standard error of regression estimate with degrees of freedom

I am taking a course of Econometrics: I need help to understand as to how do we arrive at the formula for standard error of regression $$\hat{\sigma}^2=\frac{\sum{e_i^2}}{n-k}.$$ I understand the ...
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1answer
50 views

What is the uncertainty of a discrete sum given the uncertainty of an individual element?

I have a measurement $$X=\sum_{i=1}^nX_i,$$ and I am interested to know standard deviation $\sigma_X^2$ of measurement $X$, assuming I know $\sigma_i^2$, the standard deviation of all measurements $...
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45 views

The standard deviation is more stable than the mean?

In an introduction to the subject of hypothesis testing, a book on probability and statistics for engineering students has a statement asserting that "the standard deviation is more stable than the ...
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1answer
17 views

Does standardizing a random variable that is not normally distributed change the underlying distribution?

For my analysis I am standardizing Response Times, which are usually known to be skewed and are in my data set, using the "classic" standardization method of substracting the grand mean and dividing ...
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How do you find 1 standard deviation from the center of an ellipse along the 45-degree angle?

I have generated a normally-distributed elliptical cluster of data in Matlab. The center of the ellipse falls at (0,0) and it has a standard deviation of 1 along one principle axis and a standard ...
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Combining standard deviations - given mean, sample size, and SD of two sets?

I have two groups A and B. I know A and B's sample sizes, mean performances on a test (not their individual scores however), and the standard deviations of A and B. I know how to combine them to find ...
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37 views

Relevancy of standard deviation

I'm a maths student but I'm a bit of a retard when it comes to probability. I have this very basic question in mind : Why is standard deviation defined as the square root of the variance, when surely ...
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29 views

Calculating a three sigma limit on data

So I have a list of data points each with their own error (eg. 3 +/- 0.2 units) and I want to determine which of the data points are within 3 sigma. To do this do I just calculate the standard ...
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24 views

Variance of exponential moving average

I'm not quite sure what the form for the rolling recursive variance should look like. I have that the recursive average is $m_t = \alpha x_t +(1-a)m_{t-1}$. Then would the rolling recursive variance ...
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1answer
34 views

Finding the confidence interval of a normally distributed sample

Traffic police monitor the speed of vehicles as they travel over a new bridge. The average speed for a sample of 27 vehicles was 91.29 km/h, with the sample standard deviation being 4.94 km/h. We will ...
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26 views

Confidence interval

I have a set of three data that represent the breakdown voltage of a sample. 1. 140 2. 190 3. 140 From this, the sample size is 3, And I calculated the mean which is 156.6. I want to find a 90% ...
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1answer
11 views

Gamma function's mean and standard deviation through shape and rate

I'm currently learning how to program neural networks. In the most basic ones, you initialize your matrix with random numbers that are distributed using Gamma function. I'm using a math library, that ...
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14 views

$X$ and $Y$ are standardized r.v.s. Find $a,b,c,d$ such that $Z=aX+bY$ and $W=cX+dY$ are uncorrelated but still standardized.

Let $X$ and $Y$ be standardized r.v.s (i.e., marginally they each have mean $0$ and variance $1$) with correlation $\rho \in (−1, 1)$. Find $a, b, c, d$ (in terms of $\rho$) such that $Z = aX + bY$ ...
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Find the Average Score Given the Standard Deviation

I think I'm supposed to use $z = (X - \bar X)/S$ but I'm not sure how to apply it properly to this problem. Kyle got an 8.25 on his previous test while the average score was a 7.2 and the ...
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Why %RSD of execution times, while sorting hundreds of arrays, is lower for larger arrays of random integers?

I am experimenting with the sorting of arrays and their execution times. While using bubblesort, insertsort and ...
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Binomial theorem estimate for very large samples

I have around $2^{105}$ balls, of which 1 in 20 is white. I expect that when I draw a random sampling of them, roughly 5% of all balls drawn would be white. What is the probability that, if I draw $...
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36 views

sample standard deviation given population standard deviation

How do you find the sample standard deviation when given the population standard deviation? What formula do you use? If you can make up an example that would be great.
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38 views

How to find the standard deviation from the given information and what is $B(0)$ equal to?

Assume that the risk free rate is $0$ and that the stock price is given by the equation $S(t)=6e^{2t+2B(t)}$ where $B(t)$ is the standard Brownian motion. Determine the price at time $0$ of the ...
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1answer
27 views

Proof that there will always be data within 1 SD

So we just started stats at school, and our teacher told us that no matter the data, no matter how distorted or weird it is, there will always be data within 1 standard deviation of the mean. Is this ...
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1answer
37 views

Understanding the value of standard deviation

I have two datasets $\{10,10,2,2\}$ and $\{13,7,0,4\}$. Now, when I compute standard deviation for both the sets, I get $4$ and $4.74$ respectively. My question: what is the significance of $4.74$ or $...
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19 views

Questions about Ridge Regression, how to explain the penalty term?

I have read the introduction of ridge regression in https://tamino.wordpress.com/2011/02/12/ridge-regression/. As we know, the equation of OLS is ∥ y − X β &#...
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How does using one distribution as another's sample size affect variance?

How does using one distribution as another's sample size affect variance? For example, let's say I roll a 6-sided dice and record the number shown. Then, I roll 'that many' 6 sided dice more and ...
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Cumulative frequency curve effects on change of standard deviation

Given a cumulative frequency curve . If the standard deviation is lower and median remains the same . Is it true to say that the curve will of the above statement will be steeper and pass through ...
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If this mean time is estimated to be in excess of 7 days, a new process will be implemented to reduce production costs.

a) You have just graduated with a post graduate degree in business and have obtained a position with a large manufacturing firm. The director of marketing has asked you to estimate the mean time ...
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The prime minister will accept an error of $5 million in the estimate of µ.

Government officials in Canberra have recently expressed concern regarding overruns on military contracts. These unplanned expenditures have been costing Australians millions of dollars every year. ...
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How is the t-statistic computed in this context?

According to pg. 55 of Statistics in Plain English, the t-statistic for the sample mean is $$ t = \frac{\bar{x} - \mu}{s} $$ where $\bar{X}$ is the sample mean, $\mu$ is the population mean, and $s$ ...
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26 views

Applications of mean and standard deviation

If I have this set of numbers: [9,7,5,5,4,4,4,2] We know that: mean = 5 variance = 4 standard deviation = 2 But is there an application/formula of standard ...
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Volatility of investment (/w currency hedging)

I´ve been trying to compute a volatility of invesment with currency hedging and I have a question. Let's take this example. We have our money in a fond copying the S&P500 index, which has 16% ...
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How is this not a counter-example to the fact that the standard deviation of a standardized random variable must be 1?

Let $X$ be a random variable that sends all members of the sample space to some constant real number $c$. Let $Z$ be its standardized counter-part. The standard deviation of $X$ is $0$, since $X$ is ...
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The aptitude test scores of applicants to a university graduate program are normally distributed with mean 500 and standard deviation 60.

The aptitude test scores of applicants to a university graduate program are normally distributed with mean 500 and standard deviation 60. a) Applicants need a test score higher than 530 to be ...
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Calculating mean and standard deviation of frequency given time displacement data

For example, using this data set (sample rate = 200 Hz): https://my.mixtape.moe/mtyocd.gz If I take the power DFT of it (either unwindowed or with the Blackman-Harris window) and zoom in, I can see ...
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25 views

How do I find the sample standard deviation given the population standard deviation?

A large population has a mean of 150 and a standard deviation of 40. A sample of 100 observations is to be selected at random from the population. What is the standard deviation of the sample mean? ...
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Create Matrix with Specific Standard Deviation and Mean

Assume I have a matrix $X_1$ which is being generated randomly. For this example we can assume it has 10 rows and 8 columns but the question is general. I want to create a matrix $X_2$ that will ...
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standard deviation and coefficient of varriation

A sample of 18 employees at a company is monitored to see how long they break for lunch on a particular day. The following data is the lunch break (in minutes) for each worker: ,25 ,25 ,32 ,45 ,22 ,34 ...
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Generalised Standard Deviation for less than 50%-50%

I know that for 50%-50% situations like coin toss the standard deviation for N number of toss is given by $$\sigma = \sqrt{.25N} = \sqrt{N}/2$$ What is the type though that I can calculate the ...
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1answer
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Normal Distribution Problem Normal Graph [closed]

Maple tree diameters in a forest area are normally distributed with a mean of 10 inches and a standard deviation of 2.5 inches. In a forest of 2000 maple trees, how many would have a diameter ...
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variance decreases when x gets farther from the average x?

I just read the description of Studentized residual on wikipeida https://en.wikipedia.org/wiki/Studentized_residual I'm confused about what it says about variace, itt says that "the residuals, unlike ...
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How to calculate mean for standard deviation

I've been doing some standard deviation revision but I'm getting confused as different questions have the mean calculated differently. In some questions, the mean squared is calculated as the actual $...
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Practical intuition for standard deviation

Simply: What is the intuition you get about the data if you are given standard deviation? More detailed: It is easy to imagine some information about the data if you are being told for example mean ...
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deriving mean-variance form

I have a function $$W(x_1,x_2,...,x_n)=A\sum_{i=1}^{n}x_i^2+\sum_{i\ne j}^{n}x_ix_j,$$ where $A \in[0,1)$. And in a paper I am reading, $W$ can be written in the "mean-variance" form:$$n(n+A-1)[\mu^2(...
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1answer
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calculate the internally studentized residual

This is from my textbook: it says that ...an ordinary residual divided by an estimate of its standard deviation $s(e_{i})$ As we can see from the example that mean for four residuals is 0, so $s(e_{...
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Z-normalization equation - finding the vector before being applied by Z-normalization

I was trying to find a value $X$ that were used to evaluate $Z$ in this equation $$ Z = \frac{X-\mu_x}{\sigma_x} $$ where $X = \{x_1, x_2,..., x_N\}$ and $Z = \{ y_1, y_2,..., y_N\}$ and $$\sigma_x =...
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How to find n=# of samples

My question is what is the equation to find n if you know standard deviation, and the CI of 95%. Like what is the equation for n?