The arithmetic mean of the observations 12, 46, 83, 43, 97, 34, 31, 33 and 26…

Concept: The arithmetic mean (average) of a set of n observations is the sum of all the observations divided by the number of observations: Mean = (Sum of all values) ÷ n.

Application:

1. List the nine given observations: 12, 46, 83, 43, 97, 34, 31, 33, 26.

2. Add the first five values: 12 + 46 + 83 + 43 + 97 = 281.

3. Add the remaining four values: 34 + 31 + 33 + 26 = 124.

4. Add the two partial sums to get the grand total: 281 + 124 = 405.

5. Divide the grand total by the number of observations, 9: 405 ÷ 9 = 45.

Cross-check:

Subtracting 45 from each of the nine observations gives the deviations -33, 1, 38, -2, 52, -11, -14, -12 and -19. Adding these deviations gives exactly 0, confirming 45 is the mean, since the deviations of all observations from their true mean always sum to zero.