46 Mean of a distribution
The normal distribution is great, but…
It’s important to remember it’s approximate and that not everything follows a normal distribution. Firstly, recall that we can have discrete and continuous data. For discrete data, where there are only a finite number of variables a value can take, we usually represent probability distributions using tables or histograms.
For example, imagine we figured out the probability of the number of coins that would land on heads if we tossed four coins. Our table should end up looking something like this:
| [latex]X[/latex] | 0 | 1 | 2 | 3 | 4 |
| [latex]Pr(X)[/latex] | 0.0625 | 0.25 | 0.375 | 0.25 | 0.0625 |
On the top line we have the possible results (0 heads through to 4 heads) and on the bottom line we have the probability of this result occuring.
Note that a probability distribution will always add to one, whereas a frequency distribution will usually be whole numbers. A probability distribution indicates the likelihood of that particular value occurring, whereas a frequency distribution indicates the number of times it actually did occur, and sometimes we will estimate probability distributions based on frequency distributions of collected data (by turning the frequencies into percentages).
We can determine the mean of a discrete distribution by multiplying each of the probabilities by the values they correspond with. So in the above example, we would have:
0(0.0625) + 1(0.25) + 2(0.375) + 3(0.25) + 4(0.0625)
= .25 + 0.75 + 0.75 + 0.25
= 2
This is the same idea as a frequency distribution, e.g. if we tossed 4 coins 16 times and obtained the exact number of expected results, we would have
| [latex]X[/latex] | 0 | 1 | 2 | 3 | 4 |
| [latex]Freq[/latex] | 1 | 4 | 6 | 4 | 1 |
In other words, we got 0 once, we got 1 head 4 times, we got 2 heads 6 times, 3 heads 4 times and 4 heads once, then writing these all out we have the sixteen trials: 0, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 4. We should find that the average is 2 here as well.
Animal shelter
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An animal shelter charity considers paying a marketing company for an online advertisement.
The charity knows that from past experience, 70% of those who visit their website (by following an advertising link) won’t donate anything, 20% will make the minimum donation of $10 and 10% will donate $50. What is the average donation expected to be made by people clicking on the advertisement? How many clicks would be required for it to be worth paying $5000 for the ad? |
