Adding averages

Question

I am working with data on the feeding behavior of beef cattle. The data provides from the electronic trough system are the total daily time spent in the trough (in seconds), time spent in the daily trough with food consumption (in seconds), time spent in the daily trough without food consumption (in seconds) and the total number of visits to the trough in the day, number of visits to the trough with feed consumed in the day, number visits to the trough without feed consumption in the day. My objective is to obtain the duration of trough visits per visit for each of the three variables (total, with feed consumption, and without feed consumption). To do so, I made a simple average between the time spent in the trough and the number of visits for each variable and obtained the following results:

average of total time spent in the trough per visit:$139.466666666667$ seconds $(= 4184/30)$

average of time spent in the trough with consumption per visit: $143.793103448276$ seconds $(= 4170/29)$

average of time spent in the trough without consumption per visit: $14$ seconds $(= 14/1)$

My question is: Why is the sum of the averages with and without consumption $(143.79 + 14)$ greater than the average of total time spent in the trough per visit $(139.47)$?

Answer 1

As a weighted average of the average times, you will see $\dfrac{T_{\text{total}}}{n_{\text{total}}} $ between $\dfrac{T_{\text{no cons}}}{n_{\text{no cons}}} $ and $\dfrac{T_{\text{cons}}}{n_{\text{cons}}}$, so less than or equal to the larger of them and thus less than their sum.

More generally, for positive $a,b,c,d$ you have $\frac{a+b}{c+d} \le \max\left(\frac ac, \frac bd\right) < \frac ac+ \frac bd$ for the same reason.