Concept 1 · free preview
Mean of raw data
Understand
The arithmetic mean is the single value that could replace every observation without changing the total. Add all the values and divide by how many there are.
Mean x̄ = Σx ÷ n
Σx = x̄ × n
Example
4, 8, 9, 11, 13: Σx = 45, n = 5 → mean = 45 ÷ 5 = 9.
The mean of 5 numbers is 12; four of them are 10, 14, 9, 15 (sum 48) → fifth = 5 × 12 − 48 = 60 − 48 = 12.
20 observations had a mean of 45; 64 was read as 46 → correct mean = 45 + (64 − 46) ÷ 20 = 45 + 0.9 = 45.9.
Rule
Mean of the first n natural numbers = (n + 1) ÷ 2
Mean of the first n even numbers = n + 1; of the first n odd numbers = n
Mean of an evenly spaced list (AP) = (first + last) ÷ 2
Mean of x, x + 2, x + 4 … = the middle term (for an even number of terms, the average of the two middle terms)
Remember
The mean always lies between the smallest and the largest value — drop any option outside that range.
Common mistake
Leaving an observation equal to 0 out of the count n. A 0 is still an observation.
Check yourself
The mean of the first n natural numbers is 10. n = ?