Binomial Standard Error Formula
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Binomial Distribution Standard Error
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Binomial Sampling Error
_ Cross Validated is a question and answer site for people interested in statistics, machine learning, data analysis, data mining, and data visualization. Join them; it only takes a minute: Sign binomial standard deviation up Here's how it works: Anybody can ask a question Anybody can answer The best answers are voted up and rise to the top Standard error for the mean of a sample of binomial random variables up vote 21 down vote favorite 8 Suppose I'm running an experiment that can have 2 outcomes, and I'm assuming that the underlying "true" distribution of the standard error formula excel 2 outcomes is a binomial distribution with parameters $n$ and $p$: ${\rm Binomial}(n, p)$. I can compute the standard error, $SE_X = \frac{\sigma_X}{\sqrt{n}}$, from the form of the variance of ${\rm Binomial}(n, p)$: $$ \sigma^{2}_{X} = npq$$ where $q = 1-p$. So, $\sigma_X=\sqrt{npq}$. For the standard error I get: $SE_X=\sqrt{pq}$, but I've seen somewhere that $SE_X = \sqrt{\frac{pq}{n}}$. What did I do wrong? binomial standard-error share|improve this question edited Jun 1 '12 at 17:56 Macro 24.1k496130 asked Jun 1 '12 at 16:18 Frank 3561210 add a comment| 4 Answers 4 active oldest votes up vote 25 down vote accepted It seems like you're using $n$ twice in two different ways - both as the sample size and as the number of bernoulli trials that comprise the Binomial random variable; to eliminate any ambiguity, I'm going to use $k$ to refer to the latter. If you have $n$ independent samples from a ${\rm Binomial}(k,p)$ distribution, the variance of their sample mean is $$ {\rm var} \left( \frac{1}{n} \sum_{i=1}^{n} X_{i} \right) = \frac{1}{n^2} \sum_{i=1}^{n} {\rm var}( X_{i} ) = \frac{ n {\rm var}(X_{i}) }{ n^2 } = \frac{ {\rm var}(X_{i})}{
-- it is just scaled by a factor 1/n. From the properties of the binomial distribution, its distribution has mean and standard
Standard Error Formula Statistics
deviation Bias and standard error When the proportion p is used standard error formula regression to estimate , the estimation error is p-. The error distribution therefore has the same shape as that
Standard Error Of Estimate Formula
of p, but is shifted to have mean zero. The bias and standard error of the sample proportion are therefore Standard error from data Unfortunately, the standard error http://stats.stackexchange.com/questions/29641/standard-error-for-the-mean-of-a-sample-of-binomial-random-variables of p involves , and this is unknown in practical problems. To get a numerical value for the standard error, we must therefore replace with our best estimate of its value, p. Rice survey In the rice survey, a proportion p =17/36=0.472 of the n=36 farmers used 'Old' varieties. The number using 'Old' varieties should have a http://www-ist.massey.ac.nz/dstirlin/CAST/CAST/HestPropn/estPropn3.html binomial distribution, The diagram below initially shows this distribution with replaced by our best estimate, p = 0.472. Use the pop-up menu to display the (approximate) distributions of the sample proportion, p, and the estimation error. Observe that all three distributions have the same basic shape -- only the scale on the axis changes. We estimated that a proportion 0.472 of farmers in the region use 'Old' varieties. From the error distribution, it is unlikely that this estimate will be in error by more than 0.2. Normal approximation to the error distribution If the sample size, n, is large enough, the binomial distribution is approximately normal, so we have the approximation You will see later that it is often easier to use this normal approximation than the binomial distribution. Closeness of the normal approximation The diagram below shows the binomial distribution for the errors in simulations with probability of success (red) and its normal approximation (grey). Use the sliders to verify that The normal approximation improves as
test AP formulas FAQ AP study guides AP calculators Binomial Chi-square f Dist Hypergeometric Multinomial Negative binomial Normal Poisson t Dist Random numbers Probability Bayes rule Combinations/permutations Factorial Event counter Wizard http://stattrek.com/estimation/standard-error.aspx?Tutorial=AP Graphing Scientific Financial Calculator books AP calculator review Statistics AP study guides Probability http://stattrek.com/statistics/formulas.aspx Survey sampling Excel Graphing calculators Book reviews Glossary AP practice exam Problems and solutions Formulas Notation Share with Friends What is the Standard Error? The standard error is an estimate of the standard deviation of a statistic. This lesson shows how to compute the standard error, based on sample data. standard error The standard error is important because it is used to compute other measures, like confidence intervals and margins of error. Notation The following notation is helpful, when we talk about the standard deviation and the standard error. Population parameter Sample statistic N: Number of observations in the population n: Number of observations in the sample Ni: Number of observations in population i ni: Number standard error formula of observations in sample i P: Proportion of successes in population p: Proportion of successes in sample Pi: Proportion of successes in population i pi: Proportion of successes in sample i μ: Population mean x: Sample estimate of population mean μi: Mean of population i xi: Sample estimate of μi σ: Population standard deviation s: Sample estimate of σ σp: Standard deviation of p SEp: Standard error of p σx: Standard deviation of x SEx: Standard error of x Standard Deviation of Sample Estimates Statisticians use sample statistics to estimate population parameters. Naturally, the value of a statistic may vary from one sample to the next. The variability of a statistic is measured by its standard deviation. The table below shows formulas for computing the standard deviation of statistics from simple random samples. These formulas are valid when the population size is much larger (at least 20 times larger) than the sample size. Statistic Standard Deviation Sample mean, x σx = σ / sqrt( n ) Sample proportion, p σp = sqrt [ P(1 - P) / n ] Difference between means, x1 - x2 σx1-x2 = sqrt [ &s
test AP formulas FAQ AP study guides AP calculators Binomial Chi-square f Dist Hypergeometric Multinomial Negative binomial Normal Poisson t Dist Random numbers Probability Bayes rule Combinations/permutations Factorial Event counter Wizard Graphing Scientific Financial Calculator books AP calculator review Statistics AP study guides Probability Survey sampling Excel Graphing calculators Book reviews Glossary AP practice exam Problems and solutions Formulas Notation Share with Friends Important Statistics Formulas This web page presents statistics formulas described in the Stat Trek tutorials. Each formula links to a web page that explains how to use the formula. Parameters Population mean = μ = ( Σ Xi ) / N Population standard deviation = σ = sqrt [ Σ ( Xi - μ )2 / N ] Population variance = σ2 = Σ ( Xi - μ )2 / N Variance of population proportion = σP2 = PQ / n Standardized score = Z = (X - μ) / σ Population correlation coefficient = ρ = [ 1 / N ] * Σ { [ (Xi - μX) / σx ] * [ (Yi - μY) / σy ] } Statistics Unless otherwise noted, these formulas assume simple random sampling. Sample mean = x = ( Σ xi ) / n Sample standard deviation = s = sqrt [ Σ ( xi - x )2 / ( n - 1 ) ] Sample variance = s2 = Σ ( xi - x )2 / ( n - 1 ) Variance of sample proportion = sp2 = pq / (n - 1) Pooled sample proportion = p = (p1 * n1 + p2 * n2) / (n1 + n2) Pooled sample standard deviation = sp = sqrt [ (n1 - 1) * s12 + (n2 - 1) * s22 ] / (n1 + n2 - 2) ] Sample correlation coefficient = r = [ 1 / (n - 1) ] * Σ { [ (xi - x) / sx ] * [ (yi - y) / sy ] } Correlation Pearson product-moment correlation = r = Σ (xy) / sqrt [ ( Σ x2 ) * ( Σ y2 ) ] Linear correlation (sample data) = r = [ 1 / (n - 1) ] * Σ { [ (xi - x) / sx ] * [ (yi - y) / sy ] } Linear correlation (population data) = ρ = [ 1 / N ] * Σ { [ (Xi - μX) / σx ] * [ (Yi - μY) / σy ] } Simple Linear Regression Simple linear regressio