# Calculating an average

Hi I’ve got a quick question. On statistics. I ran a survey in which I asked respondents how many hours of quality time they spend with their family per week . I also asked them what percentage of that quality time is genuine in another question. When I calculated the average for the number of Hours of quality time, I got 20.8. I then also calculated the average for the second question I.e percentage of that time that is genuine and got 52.5%. If I wanted to then get an overall average of how many hours respondents think is genuine of the quality time they spend, can I simply multiple 0.52 by 20.8? I've tried calculating it using 2 methods which I will describe below.

Q1 - asks how many hours of quality time respondents spend Q2 - asks what percentage of that quality time stated in Q1 is genuine

Method A

Respondent A stated 20 hours in Q1 and 50% in Q2 Respondent B stated 30 hours in Q1 and 25% in Q2

Average of Q1 = (20+30)/2 = 25 hours Average of Q2 = 37.5% Therefore average number of genuine hours is 37.5% of 25 hours = 9.375 hours

Method 2

Respondent A stated 20 hours in Q1 and 50% in Q2 Respondent B stated 30 hours in Q1 and 25% in Q2

Respondent A genuine hours = 50% of 20 hours = 10 hours Respondent B genuine hours = 25% of 30 hours = 7.5 hours Average of respondent A and B genuine hours = (10 + 7.5)/2 = 8.75 hours

How come each method gives me a different answer? And which method is the correct one? I'd greatly appreciate if someone could let me know which is the correct method and why. Thanks in advance.

## 1 Answer

The second method is the correct one to compute the arithmetic mean number of hours of quality time.

If you want the average number of hours of quality time you need to compute the mean of those values.

For instance, in a different example, if you had a variable $$X$$ with the measurements $$x_1 = -1$$ and $$x_2 = 1$$ and a variable $$Y$$ with the measurements $$y_1 = -3$$ and $$y_2 = 3$$, the mean of $$X$$ would be $$\overline{X} = 0$$ and the mean of $$Y$$ would also be $$\overline{Y} = 0$$.
If you consider a new variable $$Z$$ equal to the product, $$Z = XY$$, then $$z_1 = (-1)\times(-3) = 3$$ and $$z_2 = (1)\times(3) = 3$$.

Hence, the mean of the product would be different from the product of the means (similarly to what happens in your example): $$\overline{Z} = \frac{3 + 3}{2} = 3 \quad \neq \quad 0 = \overline{X} \ \overline{Y}$$

This happens because, in general, the expected value (the arithmetic mean) of the product is different from the product of the expected values, i.e.: $$\mathbb{E}(X Y) \neq \mathbb{E}(X) \mathbb{E}(Y) \quad \textrm{in general}$$

Moreover, the difference between these quantities is called covariance, $$\operatorname{Cov}(X,Y)$$.

Note that if $$X$$ and $$Y$$ are independent, then $$\mathbb{E}(X Y) = \mathbb{E}(X) \mathbb{E}(Y)$$ and the covariance is zero, $$\operatorname{Cov}(X,Y) = 0$$.