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A short class with Sarah and her students. The other ways are under More.
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Our marks are 3, 5, 7, 5 and 10. What is the mean?
They add up to 30, and there are 5 marks, so the mean is 6.
Now subtract 6 from each mark.
−3, −1, 1, −1 and 4.
Add those deviations up. What do you notice?
They add up to zero, so their average would be zero. That can't measure spread.
That's exactly why the next step is to square them.
Square the deviations and add them.
9 + 1 + 1 + 1 + 16 = 28.
This is a sample, so divide by n − 1.
28 divided by 4 is 7. That's the variance.
And the standard deviation?
The square root of 7, about 2.65 marks.
So a typical mark is roughly 2.65 marks from the mean of 6.
Another book divides 28 by 5 and gets a standard deviation of 2.37. Which is right?
Both, for different purposes. If these five marks are the whole group, divide by n to get σ ≈ 2.37.
And if they're a sample from a bigger group, we divide by n − 1 and get 2.65?
Yes. Dividing by n − 1 corrects for the fact that a sample tends to underestimate the spread.
My calculator shows both, σx and sx.
Good. σx divides by n and sx divides by n − 1, so read the question to choose.
mean, deviations, squares, divide, square root.
that is why they are squared before being averaged.
divide by n − 1 for a sample (s ≈ 2.65 here) or by n for a whole population (σ ≈ 2.37 here).
the standard deviation is measured in marks, while the variance is in marks squared.
adding a constant leaves the standard deviation unchanged, multiplying by k multiplies it by k, and outliers increase it.
Marks 3, 5, 7, 5, 10: mean 6, squared deviations 9, 1, 1, 1, 16 with sum 28, so s² = 28 ÷ 4 = 7 and s = √7 ≈ 2.65.
The same marks treated as a whole population: σ² = 28 ÷ 5 = 5.6 and σ ≈ 2.37.
Marks 5, 6, 6, 6, 7: mean 6, squared deviations sum to 2, so s² = 0.5 and s ≈ 0.71, much more consistent than the first group.
Replacing the 10 with 20: mean 8, squared deviations 25, 9, 1, 9, 144 with sum 188, so s² = 47 and s ≈ 6.86.
Find the mean, then take away; square each gap so signs can't play; add them up, divide with care, square root shows the spread that's there.
Five friends stand along a line at 3, 5, 7, 5 and 10 metres. Their meeting point is 6 metres. The standard deviation, about 2.65 metres, is roughly how far a typical friend must walk to reach it.
"My Dog Sits Down Sweetly" - Mean, Deviations, Square, Divide, Square root.
M.D.S.D.R. - Mean, Deviate, Square, Divide, Root.
Term
What is the mean of 3, 5, 7, 5, 10?
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What do the deviations from the mean always add up to?
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What is the sample variance of 3, 5, 7, 5, 10?
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What is the sample standard deviation of 3, 5, 7, 5, 10?
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When do you divide by n instead of n − 1?
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The sum of the values divided by the number of values, x̄ = Σx ÷ n.
The difference between a value and the mean, x − x̄.
s² = Σ(x − x̄)² ÷ (n − 1), used when the data are a sample from a larger population.
s, the square root of the sample variance.
σ = √(Σ(x − x̄)² ÷ n), used when the data are the whole population.
A value far from the rest of the data, which can greatly increase the standard deviation.
A subset of a population used to estimate its features.
Example 1 finds the variance and standard deviation of the sample of marks 3, 5, 7, 5 and 10. The mean is 6, the squared deviations add up to 28, the sample variance is 28 ÷ 4 = 7 and the sample standard deviation is √7 ≈ 2.65 marks.
This worked example takes five test marks out of 10 (3, 5, 7, 5 and 10) through every step of a standard deviation calculation: find the mean (6), find each deviation from the mean, square the deviations and add them (28), divide by n − 1 to get the sample variance (7), and take the square root to get the sample standard deviation (about 2.65). The lesson explains why the deviations are squared, why a sample uses n − 1, and how the answer changes to σ ≈ 2.37 when the five marks are treated as a whole population instead. It also checks the result with a shortcut formula, interprets it, and shows how shifting, scaling and outliers affect the standard deviation.
The data. Five students' marks out of 10: 3, 5, 7, 5, 10. Here n = 5.
Step 1: the mean. x̄ = (3 + 5 + 7 + 5 + 10) ÷ 5 = 30 ÷ 5 = 6.
Step 2: deviations from the mean. Subtract the mean from each value:
The deviations always add up to 0 (here −3 − 1 + 1 − 1 + 4 = 0), which is why we cannot simply average them.
Step 3: square the deviations and add. 9 + 1 + 1 + 1 + 16 = 28. Squaring makes every term positive and gives extra weight to values far from the mean.
Step 4: the sample variance. For a sample, divide by n − 1:
s² = 28 ÷ (5 − 1) = 28 ÷ 4 = 7
Step 5: the sample standard deviation. s = √7 ≈ 2.65 marks.
Interpreting the answer. The marks typically lie about 2.65 marks away from the mean of 6. The standard deviation is in the same units as the data (marks), which makes it easier to interpret than the variance (marks squared).
Sample or population? The divisor depends on what the data represent.
A check with the shortcut formula. The sum of squared deviations also equals Σx² − n x̄². Here Σx² = 9 + 25 + 49 + 25 + 100 = 208 and n x̄² = 5 × 36 = 180, so the sum is 208 − 180 = 28, as before.
How changes to the data affect the standard deviation.