As A Sample Size Increases Standard Error
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As Sample Size Increases The Standard Error Decreases
Essentials For Dummies Statistics For Dummies, 2nd Edition SPSS Statistics for Dummies, 3rd Edition Statistics as sample size increases the standard deviation II for Dummies Load more EducationMathStatisticsHow Sample Size Affects Standard Error How Sample Size Affects Standard Error Related Book Statistics For Dummies, 2nd as sample size increases the expected value of m Edition By Deborah J. Rumsey The size (n) 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
As Sample Size Increases The Standard Error Of The Mean
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 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 timesAs Sample Size Increases The Standard Error Of M ____
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 the individual times are. That's because average times don't vary as much from sample to sample as individual times vary from person to person. Now take all possible random samples of 50 clerical workers and find their means; the sampling distribution is shown in the tallest curve in the figure. The standard error of You can see the average times for 50 clerical workers are even closer to 10.5 than the ones for 10 clerical workers. By the Empirical Rule, almost all of the values fall between 10.5 - 3(.42) = 9.24
using 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, we could ask how as the sample size increases the standard error of the mean increases. true false well does the sample statistic (called a point estimate) represent the same value for
As The Sample Size Increases The Standard Error Of The Mean Gets
the population? That is, if we calculate the mean of a sample, how close will it be to the mean of the as the sample size increases the standard error also increases 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 how well we expect the sample http://www.dummies.com/education/math/statistics/how-sample-size-affects-standard-error/ 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 is possible to get an arbitrarily good estimate of the http://academic.udayton.edu/gregelvers/psy216/activex/sampling.htm 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.g. the population mean.) If the standard error of the mean is close to zero, then the sample mean is
test of goodness-of-fit Power analysis Chi-square test of goodness-of-fit G–test of goodness-of-fit Chi-square test of independence G–test of independence Fisher's exact test Small numbers in chi-square and G–tests Repeated G–tests of goodness-of-fit Cochran–Mantel– Haenszel test Descriptive statistics Central http://www.biostathandbook.com/standarderror.html tendency Dispersion Standard error Confidence limits Tests for one measurement variable One-sample http://demonstrations.wolfram.com/DistributionOfNormalMeansWithDifferentSampleSizes/ t–test Two-sample t–test Independence Normality Homoscedasticity Data transformations One-way anova Kruskal–Wallis test Nested anova Two-way anova Paired t–test Wilcoxon signed-rank test Tests for multiple measurement variables Linear regression and correlation Spearman rank correlation Polynomial regression Analysis of covariance Multiple regression Simple logistic regression Multiple logistic regression Multiple tests Multiple comparisons Meta-analysis Miscellany sample size Using spreadsheets for statistics Displaying results in graphs Displaying results in tables Introduction to SAS Choosing the right test ⇐ Previous topic|Next topic ⇒ Table of Contents Standard error of the mean Summary Standard error of the mean tells you how accurate your estimate of the mean is likely to be. Introduction When you take a sample of observations from a population and calculate the sample mean, you are sample size increases estimating of the parametric mean, or mean of all of the individuals in the population. Your sample mean won't be exactly equal to the parametric mean that you're trying to estimate, and you'd like to have an idea of how close your sample mean is likely to be. If your sample size is small, your estimate of the mean won't be as good as an estimate based on a larger sample size. Here are 10 random samples from a simulated data set with a true (parametric) mean of 5. The X's represent the individual observations, the red circles are the sample means, and the blue line is the parametric mean. Individual observations (X's) and means (red dots) for random samples from a population with a parametric mean of 5 (horizontal line). Individual observations (X's) and means (circles) for random samples from a population with a parametric mean of 5 (horizontal line). As you can see, with a sample size of only 3, some of the sample means aren't very close to the parametric mean. The first sample happened to be three observations that were all greater than 5, so the sample mean is too high. The second sample has three observations that were less
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 Di