Measures of spread: variance, standard deviation

IB Mathematics: Applications and Interpretation SL· 45 min read

1. Key Definitions and Basic Hand Calculation★★☆☆☆⏱ 10 min

📘 Definition

Variance

Population variance: $σ^{2}$ , Sample variance: $s^{2}$

The average of the squared deviations of all data values from the mean. It quantifies how far spread out data is from the central mean.

Example:

For data $[1, 2, 3]$ with mean $2$ , population variance is $\frac{( 1 - 2 ) ^{2} + ( 2 - 2 ) ^{2} + ( 3 - 2 ) ^{2}}{3} = \frac{2}{3} \approx 0.67$

📘 Definition

Standard Deviation

Population: $σ$ , Sample: $s$

The square root of the variance. It is measured in the same units as the original data, making it easier to interpret than variance.

📐 Worked Example

A small population of 5 students has test scores out of 10: $[6, 7, 7, 8, 10]$ . Calculate population variance and standard deviation.

1
First calculate the population mean $μ$ :
2
$\mu = \frac{6 + 7 + 7 + 8 + 10}{5} = 7.6$
3
Calculate each squared deviation from the mean:
4
$(6-7.6)^2 = 2.56,\ (7-7.6)^2 = 0.36,\ (7-7.6)^2 = 0.36,\ (8-7.6)^2 = 0.16,\ (10-7.6)^2 = 5.76$
5
Sum the squared deviations and divide by the population size $n = 5$ :
6
$\sigma^2 = \frac{2.56 + 0.36 + 0.36 + 0.16 + 5.76}{5} = \frac{9.2}{5} = 1.84$
7
Take the square root to get population standard deviation:
8
$\sigma = \sqrt{1.84} \approx 1.36$

2. Population vs Sample Variance★★★☆☆⏱ 15 min

When you collect a sample from a larger population, you need an unbiased estimate of the true population variance. To correct for bias introduced by sampling, we divide the sum of squared deviations by $n - 1$ instead of $n$ , where $n$ is the sample size.

📘 Definition

Sample Variance

$s^{2}$

An unbiased estimator of the population variance, calculated from a sample, that adjusts for sampling bias by using $n - 1$ instead of $n$ in the denominator.

📐 Worked Example

Treat the same 5 test scores $[6, 7, 7, 8, 10]$ as a sample from a larger population. Calculate sample variance and standard deviation.

1
The mean is still $7.6$ , and the sum of squared deviations is still $9.2$ , same as the population example.
2
For sample variance, divide the sum of squared deviations by $n - 1 = 5 - 1 = 4$ :
3
$s^2 = \frac{9.2}{4} = 2.3$
4
Take the square root to get sample standard deviation:
5
$s = \sqrt{2.3} \approx 1.52$
6
Sample variance is larger than population variance, which accounts for the uncertainty of sampling.

3. GDC Calculation and Interpretation★★★☆☆⏱ 20 min

✓ Calculator OK

In IB exams, you will almost always use your GDC to calculate variance and standard deviation for any data set larger than 5 values. A low standard deviation means data is clustered close to the mean; a high standard deviation means data is widely spread out.

📐 Worked Example

Daily high temperatures (°C) in City A over 7 days: $[12, 14, 15, 15, 16, 18, 21]$ . Find the population standard deviation using your GDC and interpret the result.

1
Enter the data into a list on your GDC, then run the 1-variable statistics function.
2
Your GDC will output both values: population standard deviation $σ \approx 2.67$ °C, sample standard deviation $s \approx 2.88$ °C.
3
Since we have all 7 days of data (the full population), we use $σ$ .
4
Interpretation: The mean temperature is 15.86 °C. Most daily temperatures are within ~2.7 °C of the mean, so temperatures are not highly variable over this week.

✓ Quick check

Check you can select the correct output from your GDC:

A question asks for the standard deviation of all 30 test scores from your class. Your GDC outputs $σ = 5.2$ and $s = 5.4$ . Which do you use?
- 5.2
- 5.4
- Either is acceptable
You have a sample of 15 trees from a forest, and need to estimate the population variance. Which value do you use?
- $σ^{2}$
- $s^{2}$
- Either
Reveal answer
1 —
Correct! $s^{2}$ is the unbiased estimate of population variance from a sample.

4. Common Pitfalls

Wrong move:

Dividing by $n$ instead of $n - 1$ when calculating sample variance

Why:

This gives a biased, lower-than-correct value for the estimated population variance, costing easy marks.

Correct move:

Always confirm if data is a full population or a sample: divide by $n$ for population, $n - 1$ for sample.

Wrong move:

Forgetting to take the square root of variance to get standard deviation

Why:

Students often output the variance value from their GDC when asked for standard deviation, leading to lost marks.

Correct move:

Double check what the question asks for: give $σ^{2} / s^{2}$ for variance, $σ / s$ for standard deviation.

Wrong move:

Selecting the wrong output value from the GDC

Why:

GDCs always output both population and sample standard deviation, so it is easy to pick the wrong one by accident.

Correct move:

Always read the question carefully to confirm if you need population or sample statistics before writing your answer.

Wrong move:

Interpreting variance instead of standard deviation for context questions

Why:

Variance is measured in squared units, which are not meaningful for real-world interpretation.

Correct move:

Always use standard deviation to interpret spread in the original units of the data.

5. Quick Reference Cheatsheet

Measure	Notation	Formula	Use Case
Population Variance	$σ^{2}$	$\frac{\sum ( x _{i} - μ ) ^{2}}{n}$	Full data set / entire population
Population Std Dev	$σ$	$σ^{2}$	Interpreting spread for full population
Sample Variance	$s^{2}$	$\frac{\sum ( x _{i} - x ˉ ) ^{2}}{n - 1}$	Estimating population variance from sample
Sample Std Dev	$s$	$s^{2}$	Interpreting spread for a sample

When this came up on past exams

AI-estimated based on syllabus patterns — cross-check with official past papers for accuracy. Use only as revision-focus signals.

2025 · Paper 1
2 mark standard deviation calculation
2024 · Paper 2
Compare spreads of two data sets
2023 · Paper 1
Interpret standard deviation value

What's Next

Now you have mastered variance and standard deviation, you can build on this knowledge for more advanced statistics topics in the IB AI SL curriculum. Standard deviation is a foundational concept for normal distributions, hypothesis testing, and confidence intervals, all of which are heavily tested on IB exams. You will now also be able to compare data sets by combining measures of central tendency and measures of spread to draw evidence-based conclusions in context-based questions, which appear on every Paper 1 and Paper 2.