Foundation lesson
Adding and averaging
Add the values, then divide by how many there are. The average keeps the total and the count, and loses the individual values.
Written for this edition, not translated from Einstein.
What does an average keep, and what does it lose?
Four jars hold 3, 1, 1 and 3 marbles. Pour them together and there are 8. Share the 8 equally between the four jars and each holds 2. Two is the average, although no jar held 2 to begin with.
The average keeps the total and the count and nothing else, so 3, 1, 1, 3 and 2, 2, 2, 2 have the same average. Listing every value twice doubles both the total and the count, and the average stays at 2.
Worked example: Three mornings with the same average
- One morning the four jars hold 3, 1, 1 and 3 marbles. Add them: 8. Divide by the four jars: an average of 2.
- Another morning they hold 8, 0, 0 and 0. Add them: 8 again. Divide by four: 2 again, and now three jars are empty.
- A third morning every jar holds 2, which is the average itself. Add them: 8. Divide by four: 2.
- Write any of those three lists out twice over. The total doubles to 16 and the count doubles to 8 jars, and 16 divided by 8 is still 2.
- Four descriptions and one average. It keeps the total and the count, and the differences between the mornings are what it leaves behind.
The four jars, before and after sharing
The same 8 marbles in the same 4 jars, twice. On the left they are as they were found; on the right they have been poured together and shared out equally.
Count them: 8 marbles on the left and 8 on the right, in 4 jars both times. The average is the right-hand picture, and it is 2 even though no jar on the left held 2.
What it shows, in words
On the left, four jars holding 3, 1, 1, 3 marbles: two are filled to three and two hold a single marble. On the right, the same 8 marbles in the same 4 jars, every one holding 2. The total is 8 in both pictures and the count of jars is 4 in both, which is everything the average keeps. What the right-hand picture has lost is which jar was full and which was nearly empty, and no amount of looking at it will bring that back.
Where this lesson stops
An average is the total shared equally among the observations, and two very different lists can have the same one.
If you came here from a passage, Back returns you to the exact place you left.