Calculating Standard Error Of The Mean Difference
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randomly find the standard error of the difference of the two sample means drawn from the same normally distributed source population, belongs to calculating error between sample mean and population mean a normally distributed sampling distribution whose overall mean is equal to zero and whose standard deviation ("standard
Standard Error Formula For Two Samples
error") is equal to square.root[(sd2/na) + (sd2/nb)] where sd2 = the variance of the source population (i.e., the square of the standard deviation); na = the size of sample A; and nb = calculating standard error of the mean example the size of sample B. To calculate the standard error of any particular sampling distribution of sample-mean differences, enter the mean and standard deviation (sd) of the source population, along with the values of na andnb, and then click the "Calculate" button. -1sd mean +1sd <== sourcepopulation <== samplingdistribution standard error of sample-mean differences = ± sd of source population sd = ± size of sample A = size of sample B = Home Click this link only if you did not arrive here via the VassarStats main page. ©Richard Lowry 2001- All rights reserved.
say 10, years ago? Suppose a random sample of 100 student records from 10 years ago yields a sample average GPA of 2.90 with a standard deviation of .40. A random sample of 100 current students today yields a sample average of 2.98 http://www.stat.wmich.edu/s216/book/node81.html with a standard deviation of .45. The difference between the two sample means is 2.98-2.90 = .08. Is this proof that GPA's are higher today than 10 years ago? Well....first we need to account for the fact that 2.98 and 2.90 are not the true averages, but are computed from random samples. Therefore, .08 is not the true difference, but simply an estimate of the true difference. Can this estimate miss by much? Fortunately, statistics has a way of measuring the standard error expected size of the ``miss'' (or error of estimation) . For our example, it is .06 (we show how to calculate this later). Therefore, we can state the bottom line of the study as follows: "The average GPA of WMU students today is .08 higher than 10 years ago, give or take .06 or so." We now show how to calculate the .06, the standard error of the estimate. But first, a note on terminology. The estimate .08=2.98-2.90 is a difference between standard error of averages (or means) of two independent random samples. "Independent" refers to the sampling luck-of-the-draw: the luck of the second sample is unaffected by the first sample. In other words, there were two independent chances to have gotten lucky or unlucky with the sampling. The likely size of the error of estimation in the .08 is called the standard error of the difference between independent means. We calculate it using the following formula: (7.4) where and . Note that and are the SE's of and , respectively. The formula looks easier without the notation and the subscripts. 2.98 is a sample mean, and has standard error (since SE= ). Similarly, 2.90 is a sample mean and has standard error . Summarizing, we write the two mean estimates (and their SE's in parentheses) as 2.98 (SE=.045) 2.90 (SE=.040) If two independent estimates are subtracted, the formula (7.6) shows how to compute the SE of the difference : 2.98 - 2.90 (SE= ) or .08 .06. Remember the Pythagorean Theorem in geometry? Think of the two SE's as the length of the two sides of the triangle (call them a and b). The SE of the difference then equals the length of the hypotenuse (SE of difference = ). We are now ready to state a confidence interval for the difference between two independent means. The correct z critical value for a 95% confidence interval is z=1.96. Therefore a 95% z-confidence interval