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 MATLAB: An Introduction with Applications Publisher:John Wiley & Sons Inc Probability and Statistics for Engineering and th... Publisher:Cengage Learning Statistics for The Behavioral Sciences (MindTap C... Author:Frederick J Gravetter, Larry B. Wallnau Publisher:Cengage Learning Elementary Statistics: Picturing the World (7th E... Author:Ron Larson, Betsy Farber The Basic Practice of Statistics Author:David S. Moore, William I. Notz, Michael A. Fligner Introduction to the Practice of Statistics Author:David S. Moore, George P. McCabe, Bruce A. Craig MATLAB: An Introduction with Applications ISBN:9781119256830 Author:Amos Gilat Publisher:John Wiley & Sons Inc Probability and Statistics for Engineering and th... ISBN:9781305251809 Author:Jay L. Devore Publisher:Cengage Learning Statistics for The Behavioral Sciences (MindTap C... ISBN:9781305504912 Author:Frederick J Gravetter, Larry B. Wallnau Publisher:Cengage Learning Elementary Statistics: Picturing the World (7th E... ISBN:9780134683416 Author:Ron Larson, Betsy Farber Publisher:PEARSON The Basic Practice of Statistics ISBN:9781319042578 Author:David S. Moore, William I. Notz, Michael A. Fligner Publisher:W. H. Freeman Introduction to the Practice of Statistics ISBN:9781319013387 Author:David S. Moore, George P. McCabe, Bruce A. Craig Publisher:W. H. Freeman What Is the Standard Error?The standard error (SE) of a statistic is the approximate standard deviation of a statistical sample population. The standard error is a statistical term that measures the accuracy with which a sample distribution represents a population by using standard deviation. In statistics, a sample mean deviates from the actual mean of a population; this deviation is the standard error of the mean. Key Takeaways
Standard ErrorUnderstanding Standard ErrorThe term "standard error" is used to refer to the standard deviation of various sample statistics, such as the mean or median. For example, the "standard error of the mean" refers to the standard deviation of the distribution of sample means taken from a population. The smaller the standard error, the more representative the sample will be of the overall population. The relationship between the standard error and the standard deviation is such that, for a given sample size, the standard error equals the standard deviation divided by the square root of the sample size. The standard error is also inversely proportional to the sample size; the larger the sample size, the smaller the standard error because the statistic will approach the actual value. The standard error is considered part of inferential statistics. It represents the standard deviation of the mean within a dataset. This serves as a measure of variation for random variables, providing a measurement for the spread. The smaller the spread, the more accurate the dataset. Standard error and standard deviation are measures of variability, while central tendency measures include mean, median, etc. Formula and Calculation of Standard ErrorThe standard error of an estimate can be calculated as the standard deviation divided by the square root of the sample size: SE = σ / √n where
If the population standard deviation is not known, you can substitute the sample standard deviation, s, in the numerator to approximate the standard error. Requirements for Standard ErrorWhen a population is sampled, the mean, or average, is generally calculated. The standard error can include the variation between the calculated mean of the population and one which is considered known, or accepted as accurate. This helps compensate for any incidental inaccuracies related to the gathering of the sample. In cases where multiple samples are collected, the mean of each sample may vary slightly from the others, creating a spread among the variables. This spread is most often measured as the standard error, accounting for the differences between the means across the datasets. The more data points involved in the calculations of the mean, the smaller the standard error tends to be. When the standard error is small, the data is said to be more representative of the true mean. In cases where the standard error is large, the data may have some notable irregularities. The standard deviation is a representation of the spread of each of the data points. The standard deviation is used to help determine the validity of the data based on the number of data points displayed at each level of standard deviation. Standard errors function more as a way to determine the accuracy of the sample or the accuracy of multiple samples by analyzing deviation within the means. Standard Error vs. Standard DeviationThe standard error normalizes the standard deviation relative to the sample size used in an analysis. Standard deviation measures the amount of variance or dispersion of the data spread around the mean. The standard error can be thought of as the dispersion of the sample mean estimations around the true population mean. As the sample size becomes larger, the standard error will become smaller, indicating that the estimated sample mean value better approximates the population mean. Example of Standard ErrorSay that an analyst has looked at a random sample of 50 companies in the S&P 500 to understand the association between a stock's P/E ratio and subsequent 12-month performance in the market. Assume that the resulting estimate is -0.20, indicating that for every 1.0 point in the P/E ratio, stocks return 0.2% poorer relative performance. In the sample of 50, the standard deviation was found to be 1.0. The standard error is thus: SE = 1.0/√50 = 1/7.07 = 0.141 Therefore, we would report the estimate as -0.20% ± 0.14, giving us a confidence interval of (-0.34 - -0.06). The true mean value of the association of the P/E on returns of the S&P 500 would therefore fall within that range with a high degree of probability. Say now that we increase the sample of stocks to 100 and find that the estimate changes slightly from -0.20 to -0.25, and the standard deviation falls to 0.90. The new standard error would thus be: SE = 0.90/√100 = 0.90/10 = 0.09. The resulting confidence interval becomes -0.25 ± 0.09 = (-0.34 - -0.16), which is a tighter range of values. What Is Meant by Standard Error?Standard error is intuitively the standard deviation of the sampling distribution. In other words, it depicts how much disparity there is likely to be in a point estimate obtained from a sample relative to the true population mean. What Is a Good Standard Error?Standard error measures the amount of discrepancy that can be expected in a sample estimate compared to the true value in the population. Therefore, the smaller the standard error the better. In fact, a standard error of zero (or close to it) would indicate that the estimated value is exactly the true value. How Do You Find the Standard Error?The standard error takes the standard deviation and divides it by the square root of the sample size. Many statistical software packages automatically compute standard errors. The Bottom LineThe standard error (SE) measures the dispersion of estimated values obtained from a sample around the true value to be found in the population. Statistical analysis and inference often involves drawing samples and running statistical tests to determine associations and correlations between variables. The standard error thus tells us with what degree of confidence we can expect the estimated value to approximate the population value. How do you find the smallest standard error?You can decrease standard error by increasing sample size. Using a large, random sample is the best way to minimize sampling bias.
What makes standard errors smaller?The standard error is also inversely proportional to the sample size; the larger the sample size, the smaller the standard error because the statistic will approach the actual value.
How do you calculate the estimated standard error?Steps to Calculate Standard Error
Standard error is calculated by dividing the standard deviation of the sample by the square root of the sample size.
What does the estimated standard error measure quizlet?The estimated standard error in the denominator measures how much difference is reasonable to expect between a sample mean and the population mean.
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Which set of characteristics will produce the smallest value for the estimated standard error?
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