Solved – Skewness – why is this distribution right skewed

distributionsnormal distributionqq-plotskew-normal-distributionskewness

If I interpret the accepted answer here correctly, I would say that the distribution below is right skewed (because of the points above the straight line on the far right).

I don't really understand this however.

My thinking is that:

The sample distribution (on the y-axis) 'wins quantiles faster' on the far right than the theoretical distribution (on the x-axis).
Hence there is higher density on the right for the sample distribution than for the theoretical distribution.
Hence the distribution is left skewed (like on the image below).

I have two questions:

What is wrong about my reasoning? Can anyone please give me a step by step argument?
Could anyone please give me an example of what would happen if the points on the far left would deviate from the straight line (i.e. when would the distribution then be left/right skewed)?

Thanks in advance.

Best Answer

Your notion that the distribution on the right "wins quantiles faster" doesn't seem to fit the plot. Steeper slopes correspond to more spread, and the slope is steeper for the right half.

If we look at the region from -2 to -1 on the theoretical (t) quantiles, and the corresponding interval above 0 (from 1 to 2), then the first corresponds to a much narrower interval than the second:

More spread out in the right tail is "right skew".

Related Solutions

Statistical Plots – How to Read the X-Axis of a Q-Q Plot?

This seems to be a qqplot of the data compared with a standard normal distribution, so I would have thought the $x$ values should the typical values of the population quantiles of a standard normal distribution

So with $105$ observations I would have thought the extreme left $x$ value should be not far away from $\Phi^{-1}\left(\dfrac{0.5}{105}\right) \approx -2.59$ and the one next to it near $\Phi^{-1}\left(\dfrac{1.5}{105}\right) \approx -2.19$, with the extreme right values being the corresponding $\Phi^{-1}\left(\dfrac{104.5}{105}\right) \approx +2.59$ and $\Phi^{-1}\left(\dfrac{103.5}{105}\right) \approx +2.19$. Visually, this seems to be close to what you have in the charts

Solved – fitting a distribution to skewed data with negative values

Not a complete answer, but people will see why this is not a comment. I get 473 values here and suggest that the ambition to fit a smooth distribution here needs to surmount marked granularity in the data. What protocol lies behind this? The small moral that binning can conceal fine structure as well as noise should need little emphasis. The square root scale (compare J.W. Tukey's "rootograms") is chosen as a matter of convenience.

      score |      freq.           %      cum. %
------------+-----------------------------------
       -.45 |         14        2.96        2.96
       -.26 |         16        3.38        6.34
          0 |        186       39.32       45.67
        .15 |        119       25.16       70.82
        .21 |          1        0.21       71.04
        .35 |         46        9.73       80.76
        .55 |         36        7.61       88.37
        .65 |          1        0.21       88.58
        .74 |          7        1.48       90.06
        .75 |          6        1.27       91.33
        .94 |         13        2.75       94.08
        .95 |          3        0.63       94.71
       1.15 |          5        1.06       95.77
       1.35 |          7        1.48       97.25
       1.55 |         13        2.75      100.00
------------+-----------------------------------
      total |        473      100.00