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The deviation table row by row - the part you are actually marked on.
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The Coddy math course teaches the method itself - you work each step on an interactive board and get told exactly where a move went wrong.
Take a list of numbers and its mean. Every value sits some distance from that mean, and the standard deviation is roughly the typical size of those distances. It cannot be their plain average, because the distances above and below the mean always cancel exactly to zero - so each one is squared first, the squares are averaged, and the square root at the end undoes the squaring so the answer is back in the original units.
That is the whole formula, and the part you are actually marked on is the middle: the table of every value minus the mean, and every one of those differences squared. This page prints that table in full, with the total that both variances divide. Competing calculators print sigma and stop, which is the one part a student could have worked out with a phone.
There are two divisors and the difference matters. Divide the total by n when your numbers ARE the whole group - the population. Divide by n − 1 when they are a sample standing in for something bigger, because a sample's own mean sits slightly too close to its own values and dividing by one less compensates. Both are shown here, always, so the divisor is a choice you make rather than one a page makes for you.
Add every value and divide by how many there are. Keep it exact - if the mean is 7/3, do not write 2.33, because that error is about to be squared.
These are the deviations. Some are negative, and that is fine; as a check, they should add up to exactly zero.
Squaring removes the signs and weights the big distances more heavily. Add the squares up: that total is the sum of squares.
By n if these values are the entire group you care about; by n − 1 if they are a sample from something larger. The result is the variance.
That is the standard deviation. It is in the same units as your original data, which the variance was not.
The same numbers under both divisors. Notice how the gap between the two answers narrows as the list gets longer.
| Values | Mean | Sum of squares | σ (population) | s (sample) |
|---|---|---|---|---|
| 2, 4, 4, 4, 5, 5, 7, 9 | 5 | 32 | 2 | 2.138 |
| 1, 2, 3, 4 | 2.5 | 5 | 1.118 | 1.291 |
| 10, 12, 23, 23, 16, 23, 21, 16 | 18 | 210 | 5.123 | 5.477 |
| 3, 3, 3 | 3 | 0 | 0 | 0 |
| 1, 2, 4 | 7/3 | 14/3 | 1.106 | 1.528 |
| 5, 15 | 10 | 50 | 5 | 7.071 |
| 2, 4, 6, 8, 10 | 6 | 40 | 2.828 | 3.162 |
| 100, 100, 100, 200 | 125 | 7500 | 43.301 | 50 |
2, 4, 4, 4, 5, 5, 7, 9The eight values total 40, so the mean is 5. The deviations are −3, −1, −1, −1, 0, 0, 2, 4 (summing to zero, as they must), and their squares total 32. Divided by n = 8 that is a variance of 4, and the square root is exactly 2. As a sample it would be 32/7, giving about 2.138.
1, 2, 3, 4The mean is 2.5 and the sum of squares is 5. Divided by 4 the variance is 5/4 and the standard deviation is √5/2 ≈ 1.118; divided by 3 it is 5/3 and about 1.291. The sample answer is always larger, because a sample's own mean hugs its own values a little too closely.
1, 2, 4The mean is 7/3, not 2.33. The deviations are −4/3, −1/3 and 5/3, and their squares are 16/9, 1/9 and 25/9, totalling 14/3. Rounding the mean first would have thrown away a digit before the squaring even began - which is why every row of the table here is a fraction.
3, 3, 3Every deviation is zero, so the sum of squares is zero and so is the standard deviation. That is the only way a standard deviation can be zero: not "the numbers are small" but "every value is the same".
100, 100, 100, 200Three values agree exactly and one does not, and the standard deviation is about 43 - larger than a quarter of the whole range. Squaring the distances is what gives that single far value so much weight, which is exactly why the measure is good at flagging outliers.
