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Hello and welcome to the third module.

Today we are going to travel together to the Swiss village called Sunnydorf.

Sunnydorf is a small village by the Swiss lake, inhabited by 16 men and women.

Well obviously these 16 men and women are different in their height.

Some are bigger, some are smaller.

Another difference of the inhabitants of this village is the amount of money that

they have in their bank account, some of them are very rich,

some of them have just very little money.

So today we're going to take data about the height and wealth of the inhabitants

of Sunnydorf and learn a little bit more about random variables.

If we go to our table here, on the right of our document we have the information

about the height and the wealth of the inhabitants in Sunnydorf.

We have an inhabitant ID for each of those, and

additionally we have some data here about the wealth in US dollars of these

inhabitants after winning the lottery.

This is something that we will use for a later question.

So, as we see we have different heights for

different inhabitants, and different amounts of the money in the bank account.

Some have very few, some have very much.

Now let's use the information to answer some questions.

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The first question asks us to compute the arithmetic mean of the height and

the the wealth of their population.

Well in Excel,

we can calculate the arithmetic mean by using the formula AVERAGE as we see here.

If we want to calculate the average in height we would just take the command

AVERAGE here, and select the corresponding column in the data that I just showed.

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If we would like to do this with the wealth, we would do exactly the same

command, just AVERAGE, but instead of using column K4 to K23 in Excel,

we would move to the column L4 to L23, where we have the data about the wealth.

These would be the reasons.

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Now we have an extra person.

Compute the Standard Deviation and Variance in height and wealth for

the population in Sunnydorf.

Well to calculate the Standard Deviation and the Variance, we can use these

formulas, STDEV.P or .S for the Standard Deviation.

The only difference in these two formulas is that we would use the one with .P if

in our data we would have the whole population, which is the case now,

we have the whole population of the village.

If we would only have one sample meaning some inhabitant of the village,

we would use the same formula but with .S for sample.

Similarly we would calculate the variance for

the population with the .P and the variance for the sample with .S.

So, if we apply the formula here, STDEV.P for the same data.

Meaning for the height in the inhabitant,

we would have a standard deviation of 13.003 for

the height and a standard deviation

of 12078299 for the wealth.

So as we see the standard deviation in the height is much,

much smaller than the standard deviation in the wealth.

Similarly, we would get the results for

the variance, which as well, differ very much in the absolute value.

The new question is an interpretation question into which we will use

the standard deviation, the variance that we just calculated.

And the question is if a new inhabitant was about to move to Sunnydorf, and

you would have to make a guess about her height and

wealth, for which of both variables would you

think that your error (measured in the units of each variable) would be smaller?

Well here we know that the intuition about the standard deviation is that,

if the standard deviation is small, then the values are much,

much more concentrated around the mean.

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This is what we see in the case of the height.

And this is also a logical thing.

The height of a man or a woman is constrained between much,

much smaller values than the wealth of a man, of a woman.

So, if we would do a prediction about the height and the wealth of a new inhabitant,

we would probably do a much, much smaller error in predicting the height.

Although we would have no idea about the height of this person, than

in predicting the wealth, due to these differences in the standard deviation.

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Now, we can play a little bit with our data to answer another question.

Assume that all the inhabitants of Sunnydorf would buy a lottery ticket

together, and decide to split the prize in equal shares if they win.

Well, congratulations to them because they won the prize.

Now, each inhabitant receives a share of $10,000

additionally to the wealth that they had at the beginning of this exercise.

The question is, how would this affect the Average, Variance and

Standard Deviation previously calculated?

Well, you learned some rules about transformations with

Carl in the theory slots.

We now just need to apply these rules, so we know that applying these calculation

rules, adding a constant to an expected value which is what we are doing here,

we're just adding 10,000 to the existing wealth, in this case,

to the mean, increases the value by this constant.

So if we would calculate the new average,

as we do here, we could compare this value with the previous value that we

calculated, average value and see that this value is 10,000 units,

in this case US dollars, higher than the one before.

Similarly, if we want to apply the calculation rules,

we know that the variance and the standard deviation

would not be affected by just adding a constant to the values.

Such to the standard deviation, and

obviously also the variance would remain constant.

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Thank you very much for having attended the third exercise.

I hope that now you know a little bit more about random variables and

how to cope with the different rules,

that you enjoyed, and hopefully see you in the fourth module.

Have fun.