If a distribution is symmetrical, each side of the distribution is a mirror image of the other. For a symmetrical, bell-shaped distribution (known as the normal distribution), the mean, median, and mode of the distribution are equal. The normal distribution can be completely described by its mean and variance.

A distribution is skewed if one of its tails is longer than the other (that is, it is not symmetrical). A symmetrical distribution has no skewness, (the skewness is zero). Skewness refers to the degree of asymmetry of a distribution. It occurs due to the existence of extremely large or small values in the data set. It allows us to see if large positive or negative deviations dominate.

A positively skewed distribution means that it has a long tail in the positive direction (a long right tail). It is sometimes called "skewed to the right." This type of distribution is characterized by many small losses and a few extreme gains.

For a positively skewed distribution, the mode is less than the median, which is less than the mean.

Recall that the mean is affected by outliers. In a positively skewed distribution, there are large positive outliers which will tend to "pull" the mean upward. An example of a positively skewed distribution is that of housing prices. Suppose that you live in a neighborhood with 100 homes. Ninety-nine of those homes sell for $600,000 and there is one house that sells for $3,000,000. The median and the mode will be $600,000, but the mean will be $1,300,000. The mean has been "pulled" upward by the existence of one distinctive home in the neighborhood.

A negatively skewed distribution has a long tail in the negative direction (a long left tail). It is sometimes called "skewed to the left." It is characterized by many small gains and a few extreme losses.

For the negatively skewed distribution, the mean is less than the median, which is less than the mode. In this case, there are large negative outliers which tend to "pull" the mean downward.

Distributions with positive skew are more common than distributions with negative skew. One example is the distribution of income. Most people make under $60,000 a year, but some make quite a bit more, with a small number making many millions of dollars per year. The positive tail therefore extends out quite a long way, whereas the negative tail stops at zero.

In a more psychological example, a distribution with a positive skew typically results if the time it takes to make a response is measured. The longest response times are usually much longer than typical response times, whereas the shortest response times are seldom much less than typical response times.

Negatively skewed distributions do occur, however.

Tips on how to remember these relative locations:

• The mean is always in the direction of the skew. For example, a positively (negatively) skewed distribution skews to the right (left), so its mean is on the right (left). This is because the mean is unduly influenced by extreme values.
• The median is always in the middle.

a typical exam question: There is a certain probability distribution with the characteristics described below:

• Mean = 100
• Highest possible value = 200
• Lowest possible value = 20

What type of distribution is this?

When a distribution is normal, the dispersion to the left of the mean is the same as the dispersion to the right of the mean. The highest number above (200) is 100 units larger than the mean, whereas the lowest number (20) is 80 units below the mean. Thus, the distribution is not symmetrical. It is skewed to the right.

We've just discussed skewness, which refers to the deviation from a normal distribution due to asymmetry. A distribution may deviate from the normal distribution by being more or less peaked than a normal distribution.

Kurtosis

Kurtosis is based on the size of a distribution's tails. It is the statistical measure that tells us when a distribution is more or less peaked than a normal distribution.

• Distributions with small tails (that is, less peaked than normal) are called "platykurtic." If a return distribution has more returns with large deviations from the mean, it is platykurtic.
• Distributions with relatively large tails (that is, more peaked than normal) are called "leptokurtic." If a return distribution has more returns clustered closely around the mean, it is leptokurtic.
• A distribution with the same kurtosis as the normal distribution is called "mesokurtic."

Kurtosis is critical in a risk management setting. Most research on the distribution of securities returns has shown that returns are not normal. Actual securities returns tend to exhibit both skewness and kurtosis (sounds like fungus!). Skewness and kurtosis are critical for risk management because if securities returns are modelled after a normal distribution, predictions from those models will take into consideration the potential for extremely large negative outcomes. In fact, most risk managers put very little emphasis on the mean and standard deviations of a distribution and focus more on the distribution of returns in the tails of the distribution, since this is where the risk is.

To calculate skewness (when N is large):

where μ is the mean and σ is the standard deviation.

The normal distribution has a skew of 0, since it is a symmetric distribution.

If skewness is positive, the average magnitude of positive deviations is larger than the average magnitude of negative deviations.

To calculate excess kurtosis (when N is large):

The kurtosis equals excess kurtosis + 3.

For all normal distributions, kurtosis is equal to 3; excess kurtosis is equal to 0.

A leptokurtic distribution has an excess kurtosis greater than 0, and a platykurtic distribution has an excess kurtosis less than 0.