Lesson · Grade 11 · Foundations of Mathematics
Standard deviation: how spread out the data are
- Grade 11
- Foundations of Mathematics
- 2 min 20 s
- British Columbia
- BC11-MA-32
Eight quiz scores (2, 4, 4, 4, 5, 5, 7, 9, mean 5) are shown as dots above a mean line, with each deviation from the mean drawn as a bar; the deviations are squared (total 32), averaged into the variance (32 ÷ 8 = 4) and square rooted to give σ = 2. Then a wider group with the same mean gives σ = 3, and adding 1 to every score raises the mean to 6 but leaves σ at 2; the video ends by showing that a calculator does it in one step, that a sample divides by n − 1 (about 2.14), and with a one-line recap.
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In one sentence
The question this lesson answers
Group A scored 2, 4, 4, 4, 5, 5, 7 and 9 on a quiz out of 10. How spread out are they?
They add up to 40, so the mean is 40 divided by 8, which is 5.
Transcript
What the lesson says
Group A scored 2, 4, 4, 4, 5, 5, 7 and 9 on a quiz out of 10. How spread out are they?
They add up to 40, so the mean is 40 divided by 8, which is 5.
Standard deviation measures spread: roughly, how far a typical score is from the mean.
Step 1: a deviation is how far a score is above or below the mean. They are minus 3, minus 1, minus 1, minus 1, 0, 0, 2 and 4.
Step 2: the deviations add up to 0, so square each one: 9, 1, 1, 1, 0, 0, 4 and 16.
Step 3: the squares add up to 32, and 32 divided by 8 is 4. This average is called the variance.
Step 4: take the square root to get back to marks. The square root of 4 is 2.
So the standard deviation, sigma, is 2: a typical score is about 2 marks from the mean.
Group B scored 0, 2, 4, 4, 6, 6, 8 and 10. Its mean is also 5.
Its deviations are bigger: the squares add up to 72, 72 divided by 8 is 9, and the square root of 9 is 3.
A common mistake is thinking a bigger mean means more spread. Add 1 to every score in group A: the mean rises to 6, but the deviations stay the same, so sigma is still 2.
A calculator or spreadsheet does all this in one step. For a whole group, it divides by n, the number of values, here 8.
For a sample of a bigger group, it divides by n minus 1, here 7, giving about 2.14.
Recap: the standard deviation is the square root of the average of the squared deviations, and a bigger one means more spread.
Curriculum
Where this fits
- British Columbia, Grade 11
Foundations of Mathematics 11, Statistics: applications of statisticsBC11-MA-32
- British Columbia, Grade 12
Statistics 12, Describing data: use of summary statistics to describe variationBC12-MA-199
This lesson is also listed in Grade 12.
The outcomes and strands follow the British Columbia curriculum.
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