 # Would you expect distributions of these variables to be uniform unimodal or bimodal? Symmetric or skewed? Explain why.

• Ages of all the people at some league game.
• No. of siblings of all of your classmates
• Heart rates of college-age male students
• No. of times each face shows up in a hundred tosses of a die

The aim of this question is to understand the different statistical properties of data. For example, whether the data is uniform, uni-modal or bi-modal, whether symmetric or skewed, etc.

When a distribution of data is plotted, its peak represents the average value of that sample. If there is only one peak (average value), then the distribution is called unimodalIf there are two distinct peaks, then the distribution is called bimodal. If there is no distinct peak and all data values are equally likely, then the distribution is called uniform.

If the negative and positive tails of the distribution are of equal length, then the data is said to be symmetric. If they are not equal, it’s called skewed.

Part (a): Ages of all the people at some league game.

Since a league game may be attended by people from all age groups with equal likelihood, we can conclude that their ages will form a uniform distribution. By definition, all uniform distributions are symmetric, so their ages will also be symmetric.

Part (b): No. of siblings of all of your classmates

Most people have zero, one, or two siblings. Therefore we could expect one clear peak for the distribution of no. of siblings in any population group. Therefore it’s uni-modal. Also, we can note that the tail of this distribution is more extended towards the higher no. of siblings compared to the lower ones, so this distribution is also skewed.

Part (c): Heart rates of college-age male students

All heart rate values will vary around some average value, so we can expect a single clear peak. Therefore, the distribution is uni-modal. Since there is an equal likelihood of heart rate falling slightly below or above this average value, the distribution is also symmetric.

Part (d): No. of times each face shows up in a hundred tosses of a die

If the die is fair, every face has an equal likelihood of showing up, so the distribution will be uniform and symmetric.

## Numerical Result

– The distribution of ages of all the people at some league game would be uniform and symmetric.

– The distribution of no. of siblings of all of your classmates would be uni-modal and skewed.

– The distribution of heart rates of college-age male students would be uni-modal and symmetric.

– The distribution of no. of times each face shows up in a hundred tosses of a die would be uniform and symmetric.

## Example

Would you expect the distribution of heights of adult humans to be uniform, unimodal, bimodal, symmetric, or skewed?

We know that there are two distinct types of adult humans with different average heights i.e. men and women. Therefore the distribution would have two distinct peaks and the data would be bimodal. There is an equal likelihood that the height of a man or a woman may fall below or above their respective average heights, so the data distribution would also be symmetric.

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