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positive & negative skewness - please help

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#1 lboogy



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Posted 15 April 2007 - 06:31 PM

Hi everyone,

I'm a little confused about positive & negative skewness. From what I've read, this is the picture I have, please tell me if it is correct:

Positive Skewness
This is when more data on a graph is towards the left of the graph (i.e. the lower values on horizontal axis). Even though the mass of the values are towards the left of the graph - the 'tail' is said to be greater on the right and the distribution is said to be right-skewed (if the majority of the data is on the left then wouldn't it make sense to think of the tail as being greater on the left and skewness to be left skewed?).

Negative Skewness
More data on a graph towards the right (i.e. the higher values on horizontal axis). More values to the right - but the 'tail' is said to be greater on the left and distribution is said to be left-skewed.

thanks in advance!
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#2 lboogy



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Posted 15 April 2007 - 06:44 PM

furthermore, if you look at this link http://en.wikipedia.org/wiki/Skewness and look at the introduction.

they give a definition of positive skewness and a definition of negative skewness. but then they have 2 graphs which explain positive & negative skewness in the opposite way round. i.e. in the definition they say positive skewness is when the mass distribution is concentrated on the left. fair enough. but then under the graph that is positively skewed they say the most data is in the right.

Am I really confused or has wikipedia got it wrong?
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#3 D H


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Posted 16 April 2007 - 02:01 AM

You are confused, but then again, that left and right stuff is a bit confusing. The skewness itself is just a number, so the terms positive and negative skewness are straightforward. One way to remember the left/right stuff is that it corresponds with the orientation of the numberline. Since negative numbers are to the left of zero, negative skewness is the same as left-skewed. The same goes for positive skewness and right-skewed.

The first thing to remember is that skewness is measured with respect to the mean. The mean divides the PDF into two equal partitions:

\int_{-\infty}^{\infty}(x-\mu) f(x)\,dx=0

This is the first of a family of moments u_i about the mean. The skewness is just the ratio u_3/u_2^{3/2}. The third moment and the skewness share the same sign since the second moment is always positive.

Think of what that means when one tail is "longer" than the other. The weight (x-\mu)^3 will have more impact on the long tail. The skewness will be positive if the long tail is on the right (positive) side of the mean, and negative if the long tail is on the left side of the mean. The left/right nomenclature thus refers to which side has the longer tail.

Some distributions with long tails include salaries (people don't get negative salaries, but the sky is the limit for CEOs) and distances (non-zero by definition, but points can be very far removed from the origin). Distributions of salaries and distance are skewed to the right.

Part of the reason you are confused is that the median of a distribution usually lies on the opposite side of the long tail. For example, consider househole income, which is skewed to the right. Over half the households made less than $44,000/year in 2004. Because the sky is the limit with respect to salaries, the mean household income was considerably higher: over $60,000/year.
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