Two sets can share an average and be nothing alike.
Compare the two
Set A is 48, 49, 50, 51, 52. Set B is 10, 30, 50, 70, 90. What is true?
Both sets total 250 over five values, so both have a mean of 50. But A's range is 52 − 48 = 4 while B's is 90 − 10 = 80. An average alone would report these as identical, which is exactly why a measure of spread is needed alongside it.
range = highest − lowest
The range uses only the two extremes and ignores everything between them. Add one unusual value and the range changes completely, while the data as a whole has barely moved.
4, 6, 8, 10, 12 — mean 8
find the MAD
distances: 4, 2, 0, 2, 4
always positive
MAD = 12 ÷ 5 = 2.4
the average distance
Without the absolute values the distances would be −4, −2, 0, 2, 4 and sum to zero — every time, for every data set. That is why the signs are dropped.
variance = the average of the squared distances
| Value | Distance from 8 | Squared |
|---|---|---|
| 4 | −4 | 16 |
| 6 | −2 | 4 |
| 8 | 0 | 0 |
| 10 | 2 | 4 |
| 12 | 4 | 16 |
| total | 40 |
variance = 40 ÷ 5 = 8
population variance
Squaring also removes the signs, and it punishes far-away values much more heavily than near ones.
standard deviation = √variance
variance = 8
from step 3
SD = √8 ≈ 2.83
back in the original units
Squaring the distances also squared the units — a variance of heights is in square centimetres, which means nothing physically. Taking the root undoes that, which is why the standard deviation is the figure usually reported.
Two bus routes can average 20 minutes. One is 20 minutes every day; the other is 5 minutes sometimes and 40 others. If you need to arrive on time, the spread matters more than the average.
Adding the same number to every value moves the data but not its spread. Multiplying every value multiplies the spread.
Start with 3, 5, 10 (mean 6). Compare it with every value plus 10, and every value doubled.
| Data | Mean | Range | MAD |
|---|---|---|---|
| 3, 5, 10 | 6 | 7 | (3 + 1 + 4) ÷ 3 = 83 |
| 13, 15, 20 (each + 10) | 16 | 7 | 83 |
| 6, 10, 20 (each × 2) | 12 | 14 | 163 |
Adding 10 slides every value along together, so every distance from the mean stays the same — and so do the range, MAD and standard deviation. Doubling stretches every distance to twice its size.
Adding 10 to each value raises the mean by 10, but the range is still 7. Answering "the range goes up by 10" mixes up where the data sits with how spread out it is.
The same recipe — mean, distances, average — at its edges.
7, 7, 7, 7
mean = 7, every distance = 0
range = MAD = variance = SD = 0
Zero is the smallest a spread can ever be, and it happens only when every value is the same.
Find the standard deviation of 1, 2, 3, 6.
mean = 12 ÷ 4 = 3
squared distances: 4, 1, 0, 9
from −2, −1, 0, 3
variance = 14 ÷ 4 = 3.5
SD = √3.5 ≈ 1.87
not a perfect square
Most standard deviations are not whole numbers. Round only at the last step.
Find the range and MAD of −4, −1, 2, 3.
range = 3 − (−4) = 7
highest minus lowest
mean = 0 ÷ 4 = 0
distances: 4, 1, 2, 3 → MAD = 10 ÷ 4 = 2.5
distances are positive
Subtracting a negative lowest value adds: the range runs all the way from −4 up to 3.
Items ask for the range or MAD of a short list, the standard deviation from a given variance, or which set, machine or player is more consistent. The fastest route is mean first, then the distances from it — absolute for the MAD, squared for the variance, and a square root for the SD. For a 10-second check, a spread is never negative, adding a constant to every value never changes it, and identical values give 0.
Find the mean absolute deviation
Find the MAD of 2, 5, 6, 7, 10.
The mean is 30 ÷ 5 = 6. The distances are 4, 1, 0, 1, 4, which total 10, so the MAD is 10 ÷ 5 = 2. C is the mean itself, not the deviation, and D is the range (10 − 2). Always take the distances as positive — with signs they would cancel to zero.
Hint: find the mean first, then how far each value sits from it, ignoring direction. Average those distances.
