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proportion of samples that would fall between 0, 1, 2, and 3 standard deviations above and below the actual value. The standard error normal distribution formula (SE) is the standard deviation of the sampling distribution of a

What Is Normal Distribution

statistic,[1] most commonly of the mean. The term may also be used to refer to an estimate normal distribution examples of that standard deviation, derived from a particular sample used to compute the estimate. For example, the sample mean is the usual estimator of a population mean. However, different normal distribution standard deviation samples drawn from that same population would in general have different values of the sample mean, so there is a distribution of sampled means (with its own mean and variance). The standard error of the mean (SEM) (i.e., of using the sample mean as a method of estimating the population mean) is the standard deviation of those sample

Standard Error Of The Mean

means over all possible samples (of a given size) drawn from the population. Secondly, the standard error of the mean can refer to an estimate of that standard deviation, computed from the sample of data being analyzed at the time. In regression analysis, the term "standard error" is also used in the phrase standard error of the regression to mean the ordinary least squares estimate of the standard deviation of the underlying errors.[2][3] Contents 1 Introduction to the standard error 1.1 Standard error of the mean (SEM) 1.1.1 Sampling from a distribution with a large standard deviation 1.1.2 Sampling from a distribution with a small standard deviation 1.1.3 Larger sample sizes give smaller standard errors 1.1.4 Using a sample to estimate the standard error 2 Standard error of the mean 3 Student approximation when σ value is unknown 4 Assumptions and usage 4.1 Standard error of mean versus standard deviation 5 Correction for finite population 6 Correction for correlation in the sample 7 Relative standard error 8 See also 9 Refer

for Normality Checks The histogram and the normal probability plot are used to check whether or not it is reasonable to assume that the random errors inherent in the process have been http://www.itl.nist.gov/div898/handbook/pmd/section4/pmd445.htm drawn from a normal distribution. The normality assumption is needed for the error rates we are willing to accept when making decisions about the process. If the random errors are not from a normal distribution, incorrect decisions will be made more or less frequently than the stated confidence levels for our inferences indicate. Normal Probability Plot The normal probability plot is constructed by plotting the sorted values of the residuals versus the normal distribution associated theoretical values from the standard normal distribution. Unlike most residual scatter plots, however, a random scatter of points does not indicate that the assumption being checked is met in this case. Instead, if the random errors are normally distributed, the plotted points will lie close to straight line. Distinct curvature or other signficant deviations from a straight line indicate that the random errors are probably not normally distributed. A few points that normal distribution formula are far off the line suggest that the data has some outliers in it. Examples Normal probability plots for the Pressure/Temperature example, the Thermocouple Calibration example, and the Polymer Relaxation example are shown below. The normal probability plots for these three examples indicate that that it is reasonable to assume that the random errors for these processes are drawn from approximately normal distributions. In each case there is a strong linear relationship between the residuals and the theoretical values from the standard normal distribution. Of course the plots do show that the relationship is not perfectly deterministic (and it never will be), but the linear relationship is still clear. Since none of the points in these plots deviate much from the linear relationship defined by the residuals, it is also reasonable to conclude that there are no outliers in any of these data sets. Normal Probability Plot: Temperature / Pressure Example Normal Probability Plot: Thermocouple Calibration Example Normal Probability Plot: Polymer Relaxation Example Further Discussion and Examples If the random errors from one of these processes were not normally distributed, then significant curvature may have been visible in the relationship between the residuals and the quantiles from the standard normal distribution, or there would be residuals at the upper and/or lower ends of the l

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