V12 Sampling Distribution Theory

Welcome to part seven of our series in support of sampling distribution theory. I'm Renee Clark from the Swanson School of Engineering at Pitt. In this video, we are going to discuss the empirical, or the 68-95-99.7 rule, associated with the normal distribution.

Okay, so, this rule states the following. Let's say you have a normal probability curve as shown in this picture. Okay, it has a certain mean. And let's say you go to the right a distance of one Sigma, or one standard deviation. Sigma is the symbol for standard deviation. Let's say you also go one standard deviation to the left of the mean. Okay, so, over to the left one, Sigma to the left. Okay, the area under the normal curve that's bounded by those two values along the x-axis, or the area shown in green, represents approximately 68% of the area under the normal curve.

Okay, likewise, if we go out a distance of two standard deviations, or two Sigma, both to the right and to the left of the mean, okay, the area under the normal… normal curve that's bounded by those two values along the x-axis, or the area shown in pink, is approximately 95% of the area under the normal curve, or 0 95% of the probability under the normal curve.

And, finally, if we go out three standard deviations from the mean in either direction of the mean, three sigma or three standard deviations to the right, and we go out three standard deviations to the left, the area under the normal curve that's bounded by those two points along the x-axis, or the area in blue, represents approximately 99.7% of the area under the normal curve. 99%... or .9997 amount of probability under the normal curve. Okay, now, in the video on using the Z probability table in… in the back of the book, you'll recall that I had brought up the fact that, in that particular table, Z was as large as 3.49 and as small as negative 3.49. Okay, recall that Z is the number of standard deviations, or sigmas, from the mean, which for Z is zero. That's why most, or 99%, of the area under the normal curve is plus or minus 3 Sigma, or three standard deviations from the mean equal to mu.

Thank you to the National Science Foundation for supporting our work under Grant 233 5802. Thank you for watching.