# Standard error of the mean (Central limit theorem)

00:00 / 00:00

3 pages

of complete

of complete

### USMLE® Step 2 style questions USMLE

of complete

A group of students is trying to determine the average height of 9th graders at their high school. The students survey the entire 9th grade student population and are calculating a parameter that measures how spread out the data points are from the average. Which of the following best describes this parameter?

### Transcript

Watch video only

#### Contributors

Let’s say you ask 1000 men for their weights and then plot those weights on a histogram, which is a type of plot that shows the distribution of measurements or data.

So let’s say that the majority of men weighed the same as the average - which in this case might be 170 pounds, or around 77 kilograms - while fewer men weighed a little bit higher or a little bit lower than the average, and even fewer men weighed much higher or much lower than the average.

If we draw a curve over the top of our histogram, we get the normal distribution curve, which is also called the bell curve, because it’s shaped like a bell.

The bell curve is symmetrical, with half the data on the left of the average and half the data on the right side of the average.

The area under the bell curve is equal to 1, or 100%, with the highest percentage of data in the middle section and the lowest percentage of data in the outer tails of the curve.

Typically, for population data, the average point in a bell curve is labeled with the greek letter mu, and mu refers to the mean, median, and mode, because when data are normally distributed, the mean, median, and mode are all equal to each other.

The standard deviation is a measure of how spread out the data are from the average, and for population data it’s represented by the lowercase greek letter sigma.

For example, let’s say the standard deviation of weight for our sample of men is 29 pounds, or 13 kilograms.

In a normal distribution, 68 percent of the data are found within one standard deviation.

That means that 68 percent of men will weigh somewhere between 170 minus 29, or 141 pounds, and 170 plus 29, or 199 pounds.

Also, 95 percent of the data are found within two standard deviations - so, since 29 times 2 is 58, then 95 percent of men will weigh somewhere between 170 minus 58, or 112 pounds, and 170 plus 58, or 228 pounds.

Finally, 99.7 percent the data are found within three standard deviations, and since 29 times 3 is 87, 99.7% of men will weigh between 170 minus 87, or 83 pounds, and 170 plus 87, or 257 pounds.

This is called the empirical rule, or the 68-95-99.7 rule.

Now, the shape of the bell curve depends on the size of the standard deviation.

A small standard deviation, like if it was only 5 pounds, tells you that most of the data are clustered around the average - and this makes the bell curve very tall and skinny.

On the other hand, a large standard deviation, like if it was 50 pounds, tells you that most of the data are way above and way below the average - and this makes the bell curve look very wide and flat.

### Summary

Central limit theorem states that if the desired data is obtained repeatedly from random samples and the mean is calculated for each sample, these means will form a normal Gaussian curve.This curve will always be normal regardless to the shape of the original curve. The standard deviation of this curve is called the standard error of mean.The standard error of mean does not measure the dispersion of data but measures how much the sample represents the population. It is directly proportional to standard deviation and inversely proportional to sample size.

Copyright © 2024 Elsevier, its licensors, and contributors. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

Cookies are used by this site.

USMLE® is a joint program of the Federation of State Medical Boards (FSMB) and the National Board of Medical Examiners (NBME). COMLEX-USA® is a registered trademark of The National Board of Osteopathic Medical Examiners, Inc. NCLEX-RN® is a registered trademark of the National Council of State Boards of Nursing, Inc. Test names and other trademarks are the property of the respective trademark holders. None of the trademark holders are endorsed by nor affiliated with Osmosis or this website.