How a 5% Quantile Relates to the Mean and Skewness

Summary

The document explains how to interpret the gap between a distribution’s mean and its 5% quantile. A quantile marks a value below which a specified fraction of observations is expected to fall; the example given is that the 5% quantile of a standard normal distribution is approximately -1.65. The mean describes the distribution’s average, so their distance depends on the distribution’s shape and spread.

For unimodal distributions, the response suggests that a relatively close mean and lower-tail quantile may indicate right skew, and recommends plotting the distributions under study to build intuition. This is a qualitative observation rather than a general formula: the gap alone does not determine skewness across all possible distributions. The discussion also distinguishes value at risk (VaR), a tail quantile used in risk measurement, from variance, a measure of dispersion, since their abbreviations are easily confused.

Key ideas

• A 5% quantile is a threshold with 5% of the distribution below it.
• The mean and a lower-tail quantile describe different features of a distribution.
• Their distance depends on distribution shape and spread.
• For unimodal distributions, the response associates a relatively close lower quantile and mean with right skew.
• Plotting the distribution can help clarify the relationship, but the gap is not a universal measure of skewness.
• VaR means value at risk, while variance is a separate statistical measure.

Tags

# the relationship between VaR(0.05) and mean?


# the relationship between VaR(0.05) and mean?












What is the meaning of the difference between the quantile of prob=0.05 and mean for a sample form a specific distribution?

In other words, I would like to understand the relationship between quantile and mean? Thanks a lot.

Question:

## Answer by Stelios Kounis (score 1)

https://quant.stackexchange.com/a/54760

I think first understanding what the mean is and what a quantile is would be helpful. The 0.05 quantile is the value for which for a given distribution only 5% of the values are expected to be lower, for example in the standard normal distribution this quantile is roughly -1.65. Now to understand a relationship between those two you have to look at the graphs of distributions and think if those two values are close this means that a lot of mass is "near" 5% quantile roughly speaking and for unimodal distributions, this means that our distribution is right-skewed. Find a way to plot all those distributions that you have and this will help you get a better understanding of the topic.

P.S VaR i.e Value-at-Risk and Var i.e variance can be easily confused so try to distinguish them more clearly.

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