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# Standard Error Of The Mean Non Normal Distribution

## Right Skewed Distribution

Join them; it only takes a minute: Sign up Here's how it works: Anybody can ask a question Anybody can answer The best answers are voted up and rise to the top What does standard deviation tell us in non-normal

## Confidence Interval Non Normal Distribution

distribution up vote 16 down vote favorite 6 In a normal distribution, the 68-95-99.7 rule imparts standard deviation a lot of meaning. But what would standard deviation mean in a non-normal distribution (multimodal or skewed)? Would all data values still fall within 3 standard deviations? Do we have rules like the 68-95-99.7 one for non-normal distributions? normal-distribution standard-deviation share|improve this question asked Jul 20 '14 at 7:54 Zuhaib Ali 181115 8 Have a look at Chebyshev's inequality. –COOLSerdash central limit theorem Jul 20 '14 at 8:08 @COOLSerdash great. This perfectly answers my question. –Zuhaib Ali Jul 20 '14 at 8:38 1 @COOLSerdash's point is on-target here, but be aware that the standard statement of Chebyshev's inequality pertains to the true SD known a-priori, not an SD estimated from your sample. It may help to read this excellent CV thread: Does a sample version of the one-sided Chebeshev inequality exist? –gung Jul 20 '14 at 16:15 Also, you should probably not settle for Chebyshev right away--you can probably do a lot better, skewed or not. –Steve S Jul 20 '14 at 16:34 I'm not interested in specifics... just need to understand what purpose does the SD concept universally serve regardless of type of distribution... and with Chebyshev's inequality I can grasp much of the concept, I think. –Zuhaib Ali Jul 20 '14 at 16:38 | show 1 more comment 3 Answers 3 active oldest votes up vote 4 down vote The standard deviation is one particular measure of the variation. There are several others, Mean Absolute Deviation is fairly popular. The standard deviation is by no means special. What makes it appear special is that the Gaussian distribution is special. As Pointed out in comments Chebyshev's inequality is useful for getting a feeling. However there are a more. share|improve this answer answered Jul 20 '14 at 17:25 Keith 16112

the mean of a variable, we are presumably trying to focus on what happens "on average," or perhaps "typically". The mean is very appropriate for this purpose when the distribution is symmetrical, and especially when it is "mound-shaped," such as

## Median Absolute Deviation

a normal distribution. For a symmetrical distribution, the mean is in the middle; if bimodal distribution the distribution is also mound-shaped, then values near the mean aretypical. But if a distribution is skewed, then the mean is usually standard error formula not in the middle. Example: The mean of the ten numbers 1, 1, 1, 2, 2, 3, 5, 8, 12, 17 is 52/10 = 5.2. Seven of the ten numbers are less than the mean, with http://stats.stackexchange.com/questions/108578/what-does-standard-deviation-tell-us-in-non-normal-distribution only three of the ten numbers greater than the mean. A better measure of the center for this distribution would be the median, which in this case is (2+3)/2 = 2.5. Five of the numbers are less than 2.5, and five are greater. Notice that in this example, the mean is greater than the median. This is common for a distribution that is skewed to the right (that is, bunched up toward the https://www.ma.utexas.edu/users/mks/statmistakes/skeweddistributions.html left and with a "tail"stretching toward the right). Similarly, a distribution that is skewed to the left (bunched up toward the right with a "tail"stretching toward the left) typically has a mean smaller than its median. (See http://www.amstat.org/publications/jse/v13n2/vonhippel.html for discussion of exceptions.) (Note that for a symmetrical distribution, such as a normal distribution, the mean and median are the same.) For a practical example (one I have often given my students): Suppose a friend is considering moving to Austin and asks you what houses here typically cost. Would you tell her the mean or the median house price? Housing prices (in Austin, at least -- think of all those Dellionaires) are skewed to the right. Unless your friend is rich, the median housing price would be more useful than the mean housing price (which would be larger than the median, thanks to the Dellioniares' expensive houses). In fact, many distributions that occur in practical situations are skewed, not symmetric. (For some examples, see the Life is Lognormal! website.) Implications for Applying Statistical Techniques How do we work with skewed distributions when so many statistical techniques give information about the mean? First, note that most of these techniques assume that the random variable in question has a distribution that is normal. Many of these techniques are somew