Skewness

A measure of a frequency distribution's tendency away from symmetry. The normal curve has 0 skewness, since it is symmetric about the mean.

Skewness is a measure of how asymmetrical the spread of a set of data is. It can be a positive, negative or zero value. A positive skewness tells us that most of the data are found before the median, as in the approximate diagram below:

               ....
             ..    ..
            .        .
           .          .
          .            .
         .              .
        .                ..
       .                   ..
      .                      ...
     .                          .......
   ..                                  ...........
...                                               ...................

In this example, mode < median < mean. Also, the semi-quartile range between the lower quartile and the median is less than between the median and the upper quartile:
(Q₂ - Q₁) < (Q₃ - Q₂).

When the skewness has a value of 0, the graph of the data should be symmetrical, as the following diagram almost is.

                                         .....
                                       ..     ..
                                     ..        ..
                                    ..          ..
                                   ..            ..
                                  ..              ..
                                ...                ...
                              ...                   ...
                          ....                       ....
                  .......                                .......
          ........                                              ........
.........                                                               ........

In this example, (in theory) mode = median = mean. Likewise, (Q₂ - Q₁) = (Q₃ - Q₂). The Normal Distribution exhibits these properties.

When the skewness is negative, exactly the opposite is true as for positive skewness, as in the diagram below.

                                                    .....
                                                  ..     .
                                                 .        .
                                                .          .
                                               .            .
                                             ..              .
                                          ...                 .
                                       ...                     .
                                   ....                         ..
                             ......                              ...
                    ........                                      ....
....................                                                ....

In this diagram (you guessed it) mean < median < mode, and (Q₂ - Q₁) > (Q₃ - Q₂).

But wait! I haven't told you how to find a value for the skewness yet! An approximate formula is:

3(mean - median
^{_________________}
standard deviation

The bigger the number, the more asymmetric the data. However, this is very approximate. A better formula, using population notation, is:

√n Σ(x - μ)^{3
______________}
(Σ (x - μ)²)^3/2

Skewness - a case study -or- "Why you should all upvote my writeups - XP whoring for a better class of person."

My node reputations are very positively skewed, and are a classic example of the first diagram when viewed in CowofDoom's E2 Node Tracker. Here are the basic statistics. Modal reputation: 1 Median reputation: 6. Mean reputation: 7.96. From this, we can see that the mode is indeed less than the median, which is less than the mean. Applying the first formula, we get a value of 1.03 for the skewness. This shows that my reputations are rather skewed and you should all go and upvote them to prevent this travesty from continuing any longer, and to give me a nice bell curve to look at.

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Sources:

• http://engineering.uow.edu.au/Courses/Stats/File15152.html
• https://www.whack.org/~conform/e2info/
• Pat Savill, the best darn stats teacher in Kent.