B. Probability Distributions · Continuous Distributions and the Normal Distribution

Standard Deviation: Measuring Spread

B2.2About 70 minutes

Learning goals

I can explain why standard deviation is a useful measure of spread.
I can calculate the variance and standard deviation of a population and of a sample by hand.
I can use the shortcut form Σx² − n·x̄² and check my work with a calculator's statistics mode.
I can predict how adding a constant or multiplying by a constant changes the standard deviation.

Words to know

deviation
How far a value is from the mean: x − x̄ (or x − μ). It is positive above the mean and negative below it.

variance
The mean of the squared deviations. Population: σ² = Σ(x − μ)²/N. Sample: s² = Σ(x − x̄)²/(n − 1).

standard deviation
The square root of the variance. It measures the typical distance of the data from the mean, in the same units as the data. Symbols: σ (population), s (sample).

population
The whole group you want to know about, of size N, with mean μ (mu) and standard deviation σ (sigma).

sample
A part of the population that you actually measure, of size n, with mean x̄ (x-bar) and standard deviation s.

Σ (sigma notation)
Capital sigma means "add up". Σx is the sum of all the data values.

Same average, different story

Two towns both had 6 snowstorms last winter, and in both towns the mean snowfall per storm was 13 cm. Are the towns' winters the same? Not at all. In Town A the storms gave 9, 11, 12, 13, 15 and 18 cm. In Town B every storm gave 12, 13 or 14 cm. Town A's storms are much more spread out.

Remember from Grade 9: in the Grade 9 data unit you measured spread with the range (maximum minus minimum) and the interquartile range (Q3 − Q1). The range uses only two values, and the IQR ignores the top and bottom quarters of the data. The standard deviation uses every value, and it is the measure of spread you need for the normal distribution.

Both towns have a mean of 13 cm per storm (dashed line). The Town A storms sit far from the mean; the Town B storms cluster close to it. The standard deviation puts a number on that difference.

Building the standard deviation step by step

Find the mean.
Find each deviation, x − x̄: how far each value is from the mean.
You cannot just average the deviations: they always add to 0, because the values above the mean balance the values below it. So square each deviation. Squares are never negative, and big deviations count extra.
Add the squares and divide. This average of squared deviations is the variance.
Take the square root to get back to the original units. This is the standard deviation.

The six Town A storms. Each arrow is a deviation from the mean of 13 cm. The deviations −1, +2, −4, +5, −2 and 0 add to 0, so we square them before averaging.

	Population (all N values)	Sample (n values from a bigger population)
Mean	μ = Σx / N	x̄ = Σx / n
Variance	σ² = Σ(x − μ)² / N	s² = Σ(x − x̄)² / (n − 1)
Standard deviation	σ = √(σ²)	s = √(s²)
Calculator key (usual name)	σx	sx or Sx

Population and sample formulas. The only difference is dividing by N or by n − 1.

Why divide by n − 1 for a sample?

A sample's values tend to sit a little closer to their own mean x̄ than to the true population mean μ, so Σ(x − x̄)² is a bit too small. Dividing by n − 1 instead of n makes s² a better (unbiased) estimate of the population variance σ². You will use samples often in the units Organization of Data for Analysis and One-Variable Data Analysis. Rule of thumb: if your data are all of the group you care about, use σ; if they are a sample that stands for a larger group, use s.

Worked example

Example 1: Sample standard deviation by hand

The snowfall (cm) from 6 storms in Town A is a sample of all storms there: 12, 15, 9, 18, 11, 13. Find the sample variance and standard deviation.

Mean: x̄ = (12 + 15 + 9 + 18 + 11 + 13)/6 = 78/6 = 13.
Deviations x − 13: −1, 2, −4, 5, −2, 0. (Check: they add to 0.)
Squared deviations: 1, 4, 16, 25, 4, 0. Sum: Σ(x − x̄)² = 50.
Sample variance: s² = 50/(6 − 1) = 50/5 = 10.
Standard deviation: s = √10 ≈ 3.16.

s² = 10 cm², s ≈ 3.16 cm. (If these 6 storms were the whole population, σ = √(50/6) ≈ 2.89 cm.)

x	x − x̄	(x − x̄)²
12	−1	1
15	2	4
9	−4	16
18	5	25
11	−2	4
13	0	0
Σ = 78	0	50

A column layout keeps the work organized and shows the deviations adding to 0.

Worked example

Example 2: Population standard deviation

A litter has exactly 5 puppies. Their masses at birth are 410, 450, 390, 430 and 420 g. Treat the litter as the population and find σ.

μ = (410 + 450 + 390 + 430 + 420)/5 = 2100/5 = 420 g.
Deviations: −10, 30, −30, 10, 0. Squares: 100, 900, 900, 100, 0. Sum = 2000.
σ² = 2000/5 = 400.
σ = √400 = 20.

σ = 20 g. A typical puppy is about 20 g from the mean of 420 g.

Worked example

Example 3: The shortcut form (good for continuous data)

Five times (s) for saying the alphabet are a sample: 3.4, 3.1, 3.6, 3.9, 3.0. Use Σ(x − x̄)² = Σx² − n·x̄² to find s.

Σx = 17.0, so x̄ = 17.0/5 = 3.4.
Σx² = 3.4² + 3.1² + 3.6² + 3.9² + 3.0² = 11.56 + 9.61 + 12.96 + 15.21 + 9.00 = 58.34.
n·x̄² = 5 × 3.4² = 5 × 11.56 = 57.80.
Σ(x − x̄)² = 58.34 − 57.80 = 0.54.
s² = 0.54/4 = 0.135, so s = √0.135 ≈ 0.367.

s ≈ 0.37 s. (Check: the deviations 0, −0.3, 0.2, 0.5, −0.4 have squares adding to 0.54. ✓)

Using a calculator or spreadsheet

Most scientific calculators have a statistics (STAT) mode. Enter the data, then read x̄, σx (population SD) and sx (sample SD). In a spreadsheet, =AVERAGE(A1:A6), =STDEV.P(A1:A6) for σ and =STDEV.S(A1:A6) for s. Always know which one the question needs, and still be able to do it by hand for a small data set.

How changes to the data affect the standard deviation

Add the same number to every value (for example, every storm gives 2 cm more): the mean goes up by 2, but the values are just as spread out. The SD does not change.
Multiply every value by the same positive number k (for example, change cm to mm, k = 10): every deviation is multiplied by k, so the SD is multiplied by k. Town A's s ≈ 3.16 cm becomes about 31.6 mm.
The SD is 0 only when every value is the same.
The SD is never negative.

Just for fun

Why did the standard deviation refuse to join the party? It heard everyone would be standing close to the mean, and it likes a little space.

Hands-on activity

Standard deviation of your own times

You need: the 20 alphabet times you recorded in the last lesson (or time 10 new ones), a calculator, paper and a pencil.

Choose any 6 of your times. Make a table with columns x, x − x̄ and (x − x̄)².
Find x̄, then the sample standard deviation s by hand.
Enter the same 6 values in your calculator's STAT mode (or a free spreadsheet) and check s.
Now find s for all 20 times with the calculator.
Ask a family member to do 6 times as well. Whose times are more consistent?

What to look for: your hand and calculator values of s should match to 2 or 3 decimal places. The person with the smaller standard deviation is more consistent, even if their mean is not the fastest.

Your turn

Find the mean and the population standard deviation of 2, 4, 4, 4, 5, 5, 7, 9.

Find the sample standard deviation of 10, 12, 14.

Without calculating, which data set has the larger standard deviation: {5, 5, 5, 5} or {1, 5, 9}? Explain.

A sample of four bottles of maple syrup contain 498, 502, 501 and 499 mL. Find s.

Every value in a data set is increased by 3. What happens to the mean and to the standard deviation?

Every value in a data set is doubled. The old SD was 4.5. What is the new SD?

Explain why we square the deviations instead of just adding them.

More practice: warm-up

What does the standard deviation measure?

What is Σ(x − x̄) for any data set?

Can a standard deviation be negative?

What is the standard deviation of 7, 7, 7, 7?

A variance is 25 cm². What is the standard deviation?

More practice: on your own

A sample of 5 reaction times (s): 3.2, 3.8, 4.1, 3.5, 3.9. Find x̄ and s.

The daily low temperatures (°C) for one week in a northern town are −12, −15, −9, −14, −10, −13, −11. Treat the week as the population. Find μ and σ.

A sample has n = 4, Σx = 40 and Σx² = 420. Find s.

Two cross-country skiers have the same mean time on a 5 km loop. Alexis has s = 0.8 min and Jordan has s = 2.1 min. Who is more consistent? Explain.

For the same data, which is larger: σ (dividing by n) or s (dividing by n − 1)? Explain.

Challenge

The data 4, 8 and x have a mean of 6. Find x and the sample standard deviation.

Show that Σ(x − x̄)² = Σx² − n·x̄².

A sample of 10 values has x̄ = 50 and s = 4. An 11th value equal to 50 is added. Find the new mean and new s.

Check your understanding

Quiz

Which measure of spread uses every value in the data?

What is the population standard deviation of 3, 5, 7?

A sample's squared deviations add to 72 and n = 9. What is s?

Each value in a data set is increased by 10. The standard deviation was 6. What is it now?

A set of lengths in centimetres has s = 2.4 cm. The lengths are converted to millimetres. What is the new standard deviation?

When should you divide by n − 1 instead of N?

For parents and teachers

Big idea (B2.2): the standard deviation is the typical distance of data values from their mean. It uses every value, is in the same units as the data, and it is the key to the normal distribution later in this unit.

How to help: Ask your teen to talk you through one small example in a table (x, x − x̄, (x − x̄)²). Then have them check the answer with a calculator's STAT mode or a spreadsheet. Ask: "Is this a sample or the whole population? Which button do you need?"

Common difficulties:

Forgetting to square the deviations, or forgetting the final square root.
Mixing up σ (divide by N, population) and s (divide by n − 1, sample).
Calculator confusion: σx and sx are different keys.
Rounding the mean before squaring deviations. Keep full accuracy until the end.
Believing that adding a constant changes the spread (it does not), or that multiplying does not (it does).