D. Statistical Analysis · One-Variable Data AnalysisMeasures of Spread: Range, IQR, Variance and Standard Deviation
D1.1About 75 minutes
Learning goals- I can explain what a measure of spread tells me and why the mean alone is not enough.
- I can calculate the population variance σ² and standard deviation σ by hand using a table of deviations.
- I can calculate the sample variance s² and standard deviation s, and explain why the sample formulas divide by n − 1.
- I can interpret the standard deviation in context and compare the consistency of two data sets.
Words to know
deviation
The difference between a value and the mean: x − μ (or x − x̄). Deviations always add to 0.
population variance σ²
The mean of the squared deviations for a whole population: σ² = Σ(x − μ)² ÷ N.
population standard deviation σ
The square root of the population variance. It is in the same units as the data and measures a typical distance from the mean. σ is the lower-case Greek letter sigma.
sample variance s²
s² = Σ(x − x̄)² ÷ (n − 1), used when the data is a sample and you want to estimate the population's variance.
sample standard deviation s
The square root of the sample variance.
range
Maximum − minimum.
interquartile range (IQR)
Q3 − Q1, the spread of the middle 50% of the data.
Same mean, very different stories
Two brands of AA batteries are tested in the same flashlight. Both last an average of 11.0 hours. Brand A's batteries all last between 10.2 and 11.6 hours. Brand B's last anywhere from 8.0 to 14.0 hours. If you are packing for a winter camping trip, you want to know about that difference. The mean cannot show it; a measure of spread can.
You already know two measures of spread from Grade 9:
- Range = max − min. Easy, but it depends on only the two most extreme values.
- Interquartile range (IQR) = Q3 − Q1. Resistant to outliers, but it ignores half the data.
If you have already studied the unit Continuous Distributions and the Normal Distribution, you have met the standard deviation there; this lesson reviews it with real data sets and focuses on choosing σ or s and interpreting spread. The standard deviation uses every value. It is the most important measure of spread in statistics and the foundation of z-scores, the normal distribution and margins of error later in this unit.
Building the standard deviation step by step
The idea: measure how far each value is from the mean, then find a typical distance.
- Deviations: find x − μ for each value. Some are negative and some positive, and they always add to 0 (lesson 5), so their plain average is useless.
- Square each deviation. Now all are positive, and big deviations count extra.
- Average the squared deviations. This is the variance: σ² = Σ(x − μ)² ÷ N.
- Square root to return to the original units. This is the standard deviation: σ = √(Σ(x − μ)² ÷ N).
The variance is in squared units (km², marks², hours²), which are hard to interpret. The standard deviation is in the same units as the data.
Worked exampleExample 1: Population standard deviation by hand
Mira's family tracked the distance (km) Mira ran on each of the 5 days she ran last week: 3, 5, 6, 8, 13. Treat this week as the whole population of interest. Find σ² and σ.
- Mean: μ = (3 + 5 + 6 + 8 + 13) ÷ 5 = 35 ÷ 5 = 7.
- Deviations x − μ: −4, −2, −1, 1, 6. Check: they add to 0 ✓.
- Squared deviations: 16, 4, 1, 1, 36. Sum: Σ(x − μ)² = 58.
- Variance: σ² = 58 ÷ 5 = 11.6 km².
- Standard deviation: σ = √11.6 ≈ 3.41 km.
σ² = 11.6 km² and σ ≈ 3.41 km. A typical day's distance is about 3.4 km away from the mean of 7 km.
Each arrow is a deviation from the mean. The single long run (13 km) has a squared deviation of 36, more than all the others combined, so it has a big effect on the standard deviation.| x | x − μ | (x − μ)² |
|---|
| 3 | −4 | 16 |
| 5 | −2 | 4 |
| 6 | −1 | 1 |
| 8 | 1 | 1 |
| 13 | 6 | 36 |
| Σx = 35 | Σ = 0 | Σ = 58 |
Population or sample?
If your data is the entire population you care about (every student in your class, Mira's week), use the population formula with N.
If your data is a sample used to estimate something about a bigger population (40 batteries out of a factory's production), use the sample formulas:
s² = Σ(x − x̄)² ÷ (n − 1) and s = √s²
Why n − 1? The sample mean x̄ is calculated from the sample itself, so it sits right in the middle of the sample. The data is therefore a little closer to x̄ than to the true population mean μ, and the squared deviations come out slightly too small on average. Dividing by n − 1 instead of n makes the result a little larger and corrects this. Another way to see it: the deviations must add to 0, so once you know n − 1 of them, the last one is fixed. Only n − 1 deviations are free to vary; statisticians call n − 1 the degrees of freedom. You will meet this idea again in university statistics courses.
For large n the difference between dividing by n and by n − 1 is tiny.
Worked exampleExample 2: Comparing consistency with sample standard deviations
Samples of 5 batteries from each brand lasted (hours): Brand A: 10.2, 10.8, 11.0, 11.4, 11.6. Brand B: 8.0, 9.5, 11.0, 12.5, 14.0. Find x̄ and s for each brand and decide which is more consistent.
- Means: A: 55.0 ÷ 5 = 11.0 h. B: 55.0 ÷ 5 = 11.0 h. The same.
- Brand A deviations: −0.8, −0.2, 0, 0.4, 0.6. Squares: 0.64, 0.04, 0, 0.16, 0.36. Sum = 1.20.
- s² = 1.20 ÷ (5 − 1) = 0.30, so s = √0.30 ≈ 0.55 h.
- Brand B deviations: −3, −1.5, 0, 1.5, 3. Squares: 9, 2.25, 0, 2.25, 9. Sum = 22.5.
- s² = 22.5 ÷ 4 = 5.625, so s = √5.625 ≈ 2.37 h.
- These are samples of each brand's production, so the n − 1 formula is correct.
Same mean, but Brand A's standard deviation (about 0.55 h) is about a quarter of Brand B's (about 2.37 h). Brand A is far more consistent, and is the better choice when you need to predict how long a battery will last.
Both sets have mean 8. Set B's values are spread farther from 8, so its standard deviation is about 3 times as large.Interpreting the standard deviation
- σ (or s) is roughly a typical distance from the mean. It is never negative.
- σ = 0 only when every value is the same.
- Like the mean, σ is not resistant: one extreme value can increase it a lot, because deviations are squared.
- For mound-shaped, roughly symmetric data, most values (about two thirds) lie within one standard deviation of the mean and almost all lie within two or three. Lesson 11 makes this precise with the normal model.
- Use mean with standard deviation for roughly symmetric data, and median with IQR for skewed data or data with outliers.
Calculator and spreadsheet names
On a graphing calculator's 1-Var Stats, σx is the population standard deviation and Sx is the sample one. In a spreadsheet, =STDEV.P(range) and =VAR.P(range) are the population versions, and =STDEV.S(range) and =VAR.S(range) are the sample versions. (The older =STDEV is the sample version.) Always decide first which one your situation needs.
Just for funWhy did the standard deviation break up with the variance?
It said, "You're just too square for me."
Hands-on activityThe ruler-drop reaction test
You need: a 30 cm ruler, a helper, paper, a pencil and a calculator.
- Your helper holds the ruler vertically by the 30 cm end. Hold your thumb and index finger open at the 0 cm mark, without touching the ruler.
- Without warning, the helper drops the ruler; catch it as fast as you can. Record the cm mark at the top of your thumb.
- Do 10 trials with your dominant hand and 10 trials with your other hand.
- For each hand, make a deviation table and calculate the mean and the sample standard deviation s (treat your trials as a sample of all the catches you could make). Check with a calculator or spreadsheet.
- Which hand is faster on average? Which is more consistent? Explain using the numbers.
What to look for: typical catch distances are often somewhere between about 5 cm and 25 cm. A smaller mean means faster reactions; a smaller s means more consistent reactions. One startled late catch can raise s a lot, which shows that the standard deviation is not resistant to outliers.
Your turn
1Find the range and the population standard deviation of 4, 4, 6, 8, 8.
Range = 8 − 4 = 4. μ = 30 ÷ 5 = 6. Deviations −2, −2, 0, 2, 2; squares 4, 4, 0, 4, 4; sum 16. σ² = 16 ÷ 5 = 3.2, σ ≈ 1.79.
2Find the sample standard deviation of 4, 4, 6, 8, 8.
s² = 16 ÷ (5 − 1) = 4, so s = 2.
3A data set has σ = 0. What do you know about it?
Every value is the same (all deviations are 0).
4Find σ for 10, 12, 14, 16, 18.
μ = 14. Deviations −4, −2, 0, 2, 4; squares 16, 4, 0, 4, 16; sum 40. σ² = 8, σ ≈ 2.83.
5A teacher has the marks of all 28 students in her class and wants their standard deviation. Population or sample formula?
Population: the class is the whole group she cares about.
6A researcher measures 50 randomly chosen trout to estimate the spread of lengths in a whole lake. Population or sample formula?
Sample (divide by n − 1): the 50 fish are used to estimate the lake's population.
7Why is the variance harder to interpret than the standard deviation?
The variance is in squared units (such as cm²), while the standard deviation is in the same units as the data.
More practice: warm-up
1What do the deviations from the mean always add to?
3Which spreadsheet function gives the sample standard deviation?
4Find the range of 12, 19, 7, 25, 16.
5Is the standard deviation resistant to outliers?
No. Squaring makes large deviations count heavily.
More practice: on your own
1Find x̄ and s for the sample 2, 5, 7, 7, 9.
x̄ = 30 ÷ 5 = 6. Deviations −4, −1, 1, 1, 3; squares 16, 1, 1, 1, 9; sum 28. s² = 28 ÷ 4 = 7, s ≈ 2.65.
2Two goalies' save percentages over 6 games: Goalie P: 0.900, 0.910, 0.905, 0.915, 0.895, 0.905. Goalie Q: 0.950, 0.860, 0.930, 0.880, 0.940, 0.870. Without full calculation, which has the smaller standard deviation? Then check Goalie P's population σ.
Goalie P: values stay within 0.895 to 0.915, while Q ranges from 0.860 to 0.950. For P: μ = 5.430 ÷ 6 = 0.905; deviations −0.005, 0.005, 0, 0.010, −0.010, 0; squares 0.000025, 0.000025, 0, 0.0001, 0.0001, 0; sum 0.00025. σ² = 0.00025 ÷ 6 ≈ 0.0000417, σ ≈ 0.0065.
3A data set of 4 values has mean 10 and three of its deviations are −3, 1 and 4. Find the fourth deviation and the fourth value.
Deviations add to 0: −3 + 1 + 4 + d = 0, so d = −2, and the value is 10 − 2 = 8.
4A sample has n = 10 and Σ(x − x̄)² = 90. Find s² and s. What would σ be if the 10 values were the whole population?
s² = 90 ÷ 9 = 10, s ≈ 3.16. σ² = 90 ÷ 10 = 9, so σ = 3.
5Which would change the standard deviation more: changing the median value of a data set by 1, or changing the maximum by 10? Explain.
Changing the maximum by 10. Its deviation grows by 10 and is then squared, so it raises Σ(x − μ)² a lot (the mean also shifts a little). Changing a middle value by 1 changes its small deviation by only 1.
Challenge
1Find a set of 3 whole numbers with mean 10 and population standard deviation √(2/3) (about 0.816). Then explain why no set of 3 whole numbers can have mean 10 and σ = 0.5.
σ² = 2/3 means Σ(x − 10)² = 3 × 2/3 = 2. Whole-number deviations that add to 0 and whose squares add to 2 must be −1, 0 and 1, so the set is 9, 10, 11 (in any order). For σ = 0.5 we would need Σ(x − 10)² = 3 × 0.25 = 0.75, but the squares of whole-number deviations are whole numbers, so their sum cannot be 0.75. Impossible.
2Show that the population variance can be written as σ² = Σx² ÷ N − μ². Use it to check Example 1 (3, 5, 6, 8, 13).
Σ(x − μ)² = Σ(x² − 2μx + μ²) = Σx² − 2μΣx + Nμ² = Σx² − 2μ(Nμ) + Nμ² = Σx² − Nμ². Dividing by N: σ² = Σx² ÷ N − μ². Check: Σx² = 9 + 25 + 36 + 64 + 169 = 303. 303 ÷ 5 − 7² = 60.6 − 49 = 11.6 ✓.
3For the batteries in Example 2, Brand B costs 30% less than Brand A. A hospital needs batteries for emergency lights that must last at least 10 hours. Which brand should it buy? Use the data to justify your answer, and name one limitation of the data.
Brand A. Both brands average 11.0 h, but Brand B's s ≈ 2.37 h means many batteries will fall well below 10 h (two of its five sample batteries lasted 8.0 h and 9.5 h). Brand A's s ≈ 0.55 h puts nearly all batteries near 11 h; its lowest sample value was 10.2 h. For a safety use, consistency matters more than price. Limitation: samples of only 5 batteries per brand are very small, so the estimates of s could be far off; a larger sample would be needed before a final decision.
Quiz
1. Which statement about the standard deviation is true?
2. For the population 1, 3, 5, what is σ²?
3. For the sample 1, 3, 5, what is s?
4. Why does the sample variance divide by n − 1?
5. Two classes have the same mean mark. Class X has σ = 4 and Class Y has σ = 11. Which is the best conclusion?
6. Which measure of spread uses every value in the data set?
Big idea (D1.1): the standard deviation measures a typical distance from the mean, using every value. Use σ (divide by N) for a whole population and s (divide by n − 1) for a sample used to estimate a population.
How to help: The ruler-drop activity is quick and fun for two people. Ask your teen to organize the work in a deviation table (x, x − x̄, (x − x̄)²); a neat table is the best defence against errors.
Common difficulties:
- Forgetting to square the deviations, or forgetting the final square root.
- Mixing up σ and s (the calculator shows both: σx and Sx).
- Rounding the mean too early; keep full precision until the end.
- Thinking a large standard deviation means a large mean. Spread and centre are separate ideas.