Mean, median, mode and range

The mean is a fair-sharing idea: pour all the values into one pile, then divide the pile into as many equal shares as there are values. The size of one share is the mean, the number that stands in for the whole data set. If every number in the list were replaced by the mean, the total would come out the same, which is what the mean really promises.

Mateo's mistake was dividing by the wrong count. He totaled 30 correctly, then divided by 4, landing on 7.5. The list 4, 8, 6, 5, 7 has five values, not four, so the 30 must be split five ways. Dividing by 4 means one value got a free ride, and the mean came out bigger than it should, well above the largest list value's neighborhood.

The fix is to count the data points as deliberately as you add them. Five values means five shares: 30 / 5 = 6. Halfway check: the mean must sit somewhere in the middle of the data range, and 6 sits between 4 and 8, so it is at least plausible. The total divided by the number of values is the whole formula, so the count matters as much as the sum.

Check by multiplying the mean by the count: 6 * 5 = 30, which is the original total. That reverse move always works for averages and catches count mistakes instantly, because the total is exact and unforgiving. If the multiplication does not return the original total, recount the values and redivide. The recount takes a second and saves a wrong answer.