The normal distribution plays a significant role in “mean-variance” models, and it has the following characteristics:
- Mean is equal to the median
- It is completely described by two parameters (mean and variance)
- 68% of observations lie between (+/-) 1 standard deviation, 95% between (+/-) 2 standard deviations, and 99% between (+/-) 3 standard deviations
Skew or skewness refers to the distribution being not symmetrical. The skews could either be: - Right or positive, where (a) the tail is depicted on the right, (b) mode < median < mean, and (c) it has frequent small losses and infrequent high gains
- Left or negative, where (a) the tail is depicted on the left, (b) the mean<median<mode, and (c) it has frequent large gains and infrequent small losses. Please note investors prefer positive skews because although frequent downside, the mean return falls above the median
- Sample Skewness Formula: {[n/(n-1)*(n-2)] * ∑(Xi – Xbar)3 /s3}; (Cubing preserves the sign of deviation from the mean, while making it a scale free measure). As a friendly reminder, long tail to the right means positive skewness where long tail to the left means negative skewness
Kurtosis measures the sharpness of the peaks (of the distribution). The following are quick charateristics of the degrees of Kurtosis. Higher the kurtosis, higher the deviation. - Mesokurtic - Defined by a standard normal peak, resembling the bell curve (typically kurtosis = 3) the most from the other two forms
- Leptokurtic - Defined by a more skinny and higher peak, with fatter tails. This is best used for T-Distributions and has a greater kurtosis than "Mesokurtic"
- Platykurtic - Defined by a more wider and smaller peak, with more skinny tails. It has a smaller kurtosis than "Mesokurtic"
2 Comments
Peter Westfall
10/18/2018 02:49:48 pm
Actually, kurtosis has virtually nothing to do with the peak of a distribution. That is outdated and incorrect information. You can have negative kurtosis (which supposedly has a "wider and smaller peak") when the peak is infinitely pointy. And you can have near infinite kurtosis (which is supposed to mean a "skinny and higher peak") when the peak is perfectly flat and covers 99.99999% of the data range. Kurtosis only measures heaviness of tails; or equivalently, the potential for observing rare, extreme observations.
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Thank you for your comment and for sharing your insights on the topic of kurtosis. It's always great to learn new information and gain a deeper understanding of statistical concepts. I actually had read your paper "Kurtosis as Peakedness, 1905-2014" and found it to be a fascinating read. I appreciate you taking the time to share your expertise on this topic.
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