8. Statistics
Mean/median/SD intuition. Evenly spaced sets: mean = median = average of endpoints.
Core ideas
- Mean = sum / count; you can always recover the sum as mean * count, which is the workhorse move.
- In any evenly spaced set, mean = median = average of the endpoints. No adding required.
- Standard deviation measures spread from the mean; adding a value equal to the mean shrinks SD, adding extreme values grows it, and adding a constant to every term leaves SD unchanged.
- Median only cares about the middle position, so extreme values can move the mean without touching the median.
Worked example 1
What is the sum of all multiples of 4 from 12 to 40, inclusive?
Show solution
The set is evenly spaced, so the mean is the average of the endpoints: (12 + 40)/2 = 26. The count is (40 - 12)/4 + 1 = 7 + 1 = 8 terms. Sum = mean * count = 26 * 8 = 208. No term-by-term adding needed.
Worked example 2
The average of five test scores is 80. If a sixth score of 80 is added, what happens to the mean and to the standard deviation?
Show solution
The original sum is 5 * 80 = 400; adding 80 gives 480 over 6 scores, so the mean stays 80. The new value sits exactly at the mean, contributing zero spread while increasing the count. So the standard deviation decreases (unless all five scores were already 80, in which case SD stays 0). Mean unchanged, SD down: that is the signature of adding a value at the mean.
Practice set
Question 1
A set of 5 positive integers has an average (arithmetic mean) of 10 and a median of 12. What is the greatest possible value of the largest integer in the set?
To maximize one value, push every other value as low as its constraints allow.
The five integers sum to 50 and, in order, the third is 12. To maximize the fifth value, minimize the others: the first two can be 1 and 1, and the fourth must be at least the median, so 12. That leaves 50 - 1 - 1 - 12 - 12 = 24 for the largest integer. Choosing anything smaller elsewhere is impossible, so 24 is the maximum.
Question 2
What is the median of the list 3, 7, 7, 12, 15, 20?
With an even number of terms, the median averages the two middle values.
The list is already in order and has six terms, so the median is the average of the third and fourth terms: (7 + 12)/2 = 9.5.
Question 3
What is the average (arithmetic mean) of all the integers from 17 to 41 inclusive?
For an evenly spaced set, the mean is just the average of the endpoints.
Consecutive integers form an evenly spaced set, so the mean equals the average of the first and last terms: (17 + 41)/2 = 58/2 = 29. No summing needed.
Question 4
What is the sum of all even integers from 20 to 60 inclusive?
Sum of an evenly spaced set = (average of endpoints) x (number of terms).
The even integers from 20 to 60 are evenly spaced with average (20 + 60)/2 = 40. The count is (60 - 20)/2 + 1 = 21 terms. The sum is 40 x 21 = 840.