Understanding Standard Errors: Mean, Estimate, and Measure | Study notes Psychology

PSY 3430

Standard Error

Standard errors are like standard deviations for samples, estimates, and measures.

Standard Error of the Mean- (SE Mean) If you take a sample from a population,

measure any trait (ht, wt, siblings, intelligence, etc.) and compute the mean, you have an

“unbiased estimate of the population” or a single value that best represents the overall

mean of the population. But how representative is it really???? That depends on a lot of

factors. If you randomly take the sample and obtain a mean you get a number. Then you

randomly take another sample and measure the same variable and compute the mean, the

value may be the same or it may be different (higher or lower). Then randomly take

another sample and take the mean, the value may be the same or it may be different from

the last two (higher or lower), etc. If this process is repeated many times (usually

thousands) you will get many different numbers representing the mean of the population.

In fact if you plot these “sampling means” you will see a normal distribution with the

majority of scores clustered around the center and more dispersed toward the tails, and

symmetrical in shape. The value at the center of the distribution will likely be the

population mean.

Usually, we don’t have the time or resources to take thousands of random samples in

order to compile a distribution of sampling means. The best we can do is estimate how

good the mean of our sample is at reflecting the population by making some adjustments.

Two pieces of information are necessary, the sample size, and the standard deviation of

the sample. The standard deviation is divided by the square root of the sample size. The

quotient is the “Standard Error of the Mean”. It is the theoretical standard deviation of

the sampling distribution (the thousand random samples of the population). As with any

standard deviation, it can be used to create a range in which the likelihood of a true

population mean falls can be calculated using what we know about the normal

distribution.

I can be 68% confident that the true population mean ranges between the 1 standard error

below the sample mean and 1 standard error above the sample mean. Z= +1.0 (SEMean)

I can be 95% confident that the true population mean ranges between 2 times the standard

error below the sample mean and 2 times the standard error above the sample mean. Z=

+2.0 (SEMean)

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PSY 3430

Standard Error Standard errors are like standard deviations for samples, estimates, and measures. Standard Error of the Mean- (SE Mean) If you take a sample from a population, measure any trait (ht, wt, siblings, intelligence, etc.) and compute the mean, you have an “unbiased estimate of the population” or a single value that best represents the overall mean of the population. But how representative is it really???? That depends on a lot of factors. If you randomly take the sample and obtain a mean you get a number. Then you randomly take another sample and measure the same variable and compute the mean, the value may be the same or it may be different (higher or lower). Then randomly take another sample and take the mean, the value may be the same or it may be different from the last two (higher or lower), etc. If this process is repeated many times (usually thousands) you will get many different numbers representing the mean of the population. In fact if you plot these “sampling means” you will see a normal distribution with the majority of scores clustered around the center and more dispersed toward the tails, and symmetrical in shape. The value at the center of the distribution will likely be the population mean. Usually, we don’t have the time or resources to take thousands of random samples in order to compile a distribution of sampling means. The best we can do is estimate how good the mean of our sample is at reflecting the population by making some adjustments. Two pieces of information are necessary, the sample size, and the standard deviation of the sample. The standard deviation is divided by the square root of the sample size. The quotient is the “Standard Error of the Mean”. It is the theoretical standard deviation of the sampling distribution (the thousand random samples of the population). As with any standard deviation, it can be used to create a range in which the likelihood of a true population mean falls can be calculated using what we know about the normal distribution. I can be 68% confident that the true population mean ranges between the 1 standard error below the sample mean and 1 standard error above the sample mean. Z= +1.0 (SEMean) I can be 95% confident that the true population mean ranges between 2 times the standard error below the sample mean and 2 times the standard error above the sample mean. Z= +2.0 (SEMean)

I can be 99% confident that the true population mean ranges from 3 times the standard error below the sample mean and 3 times the standard error above the sample mean. Z= +3.0 (SEMean) The value of the standard error decreases as the sample size increases (better representes the population) and the standard deviation decreases (more consistent scores are easier to estimate) Standard Error of the Estimate- SE Estimate. In any given regression equation, an estimation of the DV by the known values of the IV is of interest (what is your likely weight based on your height). Similarly, you may wish to predict something using scores on a test (success in school based on ACT scores). These types of variables are called predictors (ones on which the prediction is based [IV’s]) and criterion (the thing being predicted [DV’s]). The accuracy of these predictions is determined in part by how good the test is at measuring what it is supposed to be measuring (validity). You could give the test thousands of times and measure how accurately it predicted the criterion each time. You will likely encounter different predicted values of the criterion for each time you make a prediction. These different values will vary in regard to how accurate they are. Some will be really close to the actual value of the criterion; some will miss it by a long way. If you plot how much each prediction missed the true value (the residual value), the results will be normally distributed with most misses being relatively close and fewer toward the extreme values. The standard deviation of this distribution is called the Standard Error of the Estimate and can be used to calculate confidence intervals for the predictions made. I can be 68% confidence that the actual value of the criterion will fall between 1 standard error of the estimate below and one standard error of the estimate above the predicted value. + SEEstimate If my regression equation predicts that someone 6’tall would weigh 208 and the standard error of the estimate is 10, then I am 68% confident that someone who is indeed 6’ tall would weigh between 198 and 218 based on my sample. I can be 95% confident that his/her weight would range between 2 times the Standard Error of the Estimate below to 2 times the Standard Error of the Estimate above the estimated value (someone 6’ tall would weigh between 188 and 228 based on my sample). + 2(SEEstimate) I can be 99% confident that his/her weight would range between 3 times the Standard Error of the Estimate below and 3 times the Standard Error of the Estimate above the predicted value (someone 6’ tall would weigh between 178 and 238 based on my sample). + 3(SEEstimate) The higher the correlation between my predictor and criterion, the smaller the Standard Error of the Estimate, and the more confident I am in my predictions.

Understanding Standard Errors: Mean, Estimate, and Measure, Study notes of Psychology

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PSY 3430