As Sample Size Increases The Standard Error
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As Sample Size Increases The Standard Deviation
Standard Error Statistics Essentials For Dummies Statistics For Dummies, 2nd Edition SPSS Statistics for Dummies, 3rd Edition Statistics II as sample size increases the standard error decreases for Dummies Load more EducationMathStatisticsHow Sample Size Affects Standard Error How Sample Size Affects Standard Error Related Book Statistics For Dummies, 2nd Edition By Deborah J. Rumsey The size (n)
As Sample Size Increases The Standard Error Of M
of a statistical sample affects the standard error for that sample. Because n is in the denominator of the standard error formula, the standard error decreases as n increases. It makes sense that having more data gives less variation (and more precision) in your results.
Distributions of times for 1 worker, 10 workers, and 50 workers. Suppose X is the as sample size increases the standard error of the mean time it takes for a clerical worker to type and send one letter of recommendation, and say X has a normal distribution with mean 10.5 minutes and standard deviation 3 minutes. The bottom curve in the preceding figure shows the distribution of X, the individual times for all clerical workers in the population. According to the Empirical Rule, almost all of the values are within 3 standard deviations of the mean (10.5) -- between 1.5 and 19.5. Now take a random sample of 10 clerical workers, measure their times, and find the average, each time. Repeat this process over and over, and graph all the possible results for all possible samples. The middle curve in the figure shows the picture of the sampling distribution of Notice that it's still centered at 10.5 (which you expected) but its variability is smaller; the standard error in this case is (quite a bit less than 3 minutes, the standard deviation of the individual times). Looking at the figure, the average times for samples of 10 clerical workers are closer to the mean (10.5) than thusing Windows 95, 98 or NT. When asked if you want to install the sampling control, click on Yes. When we draw a sample from a population, and calculate a sample statistic such as the mean,
What Happens To Standard Error When Sample Size Increases
we could ask how well does the sample statistic (called a point estimate) represent
As N Increases The Standard Error Decreases
the same value for the population? That is, if we calculate the mean of a sample, how close will it be what happens to the distribution of the sample means if the sample size is increased? to the mean of the population? Of course, the answer will change depending on the particular sample that we draw. But could we develop a measure that would at least give us an indication of http://www.dummies.com/education/math/statistics/how-sample-size-affects-standard-error/ how well we expect the sample mean to represent the population mean? We could subtract the sample mean from the population mean to get an idea of how close the sample mean is to the population mean. (Technically, we don't know the value of the population mean -- if we knew the population mean, then there would be no sense in calculating the sample mean. But in theory, it http://academic.udayton.edu/gregelvers/psy216/activex/sampling.htm is possible to get an arbitrarily good estimate of the population mean and we can use that estimate as the population mean.) That is, we can calculate how much the sample mean deviates from the population mean. But is this particular sample representative of all of the samples that we could select? It may or may not be. So, we should draw another sample and determine how much it deviates from the population mean. In fact, we might want to do this many, many times. We could then calculate the mean of the deviates, to get an average measure of how much the sample means differ from the population mean. The standard error of the mean does basically that. To determine the standard error of the mean, many samples are selected from the population. For each sample, the mean of that sample is calculated. The standard deviation of those means is then calculated. (Remember that the standard deviation is a measure of how much the data deviate from the mean on average.) The standard deviation of the sample means is defined as the standard error of the mean. It is a measure of how well the point estimate (e.g. the sample mean) represents the population parameter (e
Tour Start here for a quick overview of the site Help Center Detailed answers to any questions you might have Meta Discuss the workings and policies of this site About Us Learn more about Stack Overflow the company http://stats.stackexchange.com/questions/129885/why-does-increasing-the-sample-size-lower-the-variance Business Learn more about hiring developers or posting ads with us Cross Validated Questions Tags http://demonstrations.wolfram.com/DistributionOfNormalMeansWithDifferentSampleSizes/ Users Badges Unanswered Ask Question _ 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 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 Why sample size does increasing the sample size lower the variance? up vote 14 down vote favorite 4 Big picture: I'm trying to understand how increasing the sample size increases the power of an experiment. My lecturer's slides explain this with a picture of 2 normal distributions, one for the null-hypothesis and one for the alternative-hypothesis and a decision threshold c between them. They argue that increasing sample size will lower variance and thereby cause a higher kurtosis, sample size increases reducing the shared area under the curves and so the probability of a type II error. Small picture: I don't understand how a bigger sample size will lower the variance. I assume you just calculate the sample variance and use it as a parameter in a normal distribution. I tried: googling, but most accepted answers have 0 upvotes or are merely examples thinking: By the law of big numbers every value should eventually stabilize around its probable value according to the normal distribution we assume. And the variance should therefore converge to the variance of our assumed normal distribution. But what is the variance of that normal distribution and is it a minimum value i.e. can we be sure our sample variance decreases to that value? variance sampling power share|improve this question asked Dec 21 '14 at 0:01 user2740 3391213 Your thought experiment concerned normally distributed data but it also applies to data drawn from many other distributions (as noted by @Aksakal, not all! The Cauchy is a commonly cited example of such bad behaviour). For binomial data there is good discussion of how power and standard error vary with sample size at stats.stackexchange.com/q/87730/22228 –Silverfish Dec 21 '14 at 2:26 1 As you are new to CrossValidated, allow me to point out that if you received what you consider a satisfactory answer,
mean 52 and standard deviation 14. The distribution of sample means for samples of size 16 (in blue) does not change but acts as a reference to show how the other curve (in red) changes as you move the slider to change the sample size. Distributions of sample means from a normal distribution change with the sample size. This Demonstration lets you see how the distribution of the means changes as the sample size increases or decreases. Contributed by: David Gurney THINGS TO TRY Automatic Animation SNAPSHOTS DETAILS The population mean of the distribution of sample means is the same as the population mean of the distribution being sampled from. Thus the mean of the distribution of the means never changes. The standard deviation of the sample means, however, is the population standard deviation from the original distribution divided by the square root of the sample size. Thus as the sample size increases, the standard deviation of the means decreases; and as the sample size decreases, the standard deviation of the sample means increases.Reference:Michael Sullivan, Fundamentals of Statistics, Upper Saddle River, NJ: Pearson Education, Inc., 2008 pp. 382-383. RELATED LINKS Central Limit Theorem (Wolfram MathWorld)Normal Distribution (Wolfram MathWorld)Sample Size (Wolfram MathWorld) PERMANENT CITATION "Distribution of Normal Means with Different Sample Sizes" from the Wolfram Demonstrations Projecthttp://demonstrations.wolfram.com/DistributionOfNormalMeansWithDifferentSampleSizes/Contributed by: David Gurney Share:Embed Interactive Demonstration New! Just copy and paste this snippet of JavaScript code into your website or blog to put the live Demonstration on your site. More details »Download Demonstration as CDF »Download Author Code »(preview »)Files require Wolfram CDF Player or Mathematica.Related DemonstrationsMore by AuthorImpact of Sample Size on Approximating the Normal DistributionPaul Savory (University of Nebraska-Lincoln)Sampling Distribution of the Sample MeanJim R LarkinSampling Distribution of the Mean and Standard Deviation in Various PopulationsIan McLeodNormal Approximation to a Poisson Random VariableChris BoucherStatistics Associated with Normal SamplesPierre-Emmanuel RobertExploring the Tails of the Normal DistributionJoe O'HaraThe p-Value in One-Sample Tests for the MeanIan McLeodSimulating a Normal Process from Sums of Uniform DistributionsJim GerdyRobustness of Student t in the One-Sample