Comparing the shape of two probability distributions

Question

How do we compare the two probability distributions based on their means and standard deviations ?

suppose we have the following.

1st probability distribution

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mean = 0.6645 SD = 0.02345

2nd probability distribution

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mean = 0.832009 SD = 0.00341

Are the shapes of the above two distributions equal or not ? How do we compare it ?

Answer 1

For different distrbutions, mean and standard deviation control different features of the pdf. For the Normal distribution:

• the larger the mean, the more to the right the spike will be, and
• the larger the variance, the more steeply it will come down from the spike (i.e., the spike will be less spread out horizontally).

EDIT For the Beta distribution $B(\alpha, \beta)$, the impact graph can be seen here:

Answer 2

You can use the coefficient of variation (CV) to compare the homogeneity/ variability between the two distributions. The CV is given by $$CV=\frac{s}{\bar{x}}$$ where $s$ denotes the standard deviation of the sample and $\bar{x}$ the mean of the sample. The $CV$ is measured in percent units, which means that it is a number (free of specific measurement units) and can be used for comparison between different data sets. Sets with values of CV less than $10\%$ tend to be interpreted as homogenous sets.

Another way to compare two data sets based on the given values is to proceed in two steps

1. Compare the means, which give you an indication of the location or central tendency of the data. According to your values for example, the second sample has a higher mean.
2. Compare the standard deviations, which give you an indication of the variability of the distribution and the extend to which the data (or the distribution) is dispersed around the mean. The second distribution shows again less variability meaning that the observations are concentrated around the mean (0.832) in contrast to first disrtibution.
3. Better yet, is to calculate the CV's of both and compare them. The second distribution will be more homogenous than the first.