# Measures of spread: variance, standard deviation

> IB Mathematics: Applications and Interpretation SL · Statistics and Probability
> Source: https://www.owlsprep.com/study/ib-math-ai-sl-u4-measures-of-spread-variance-standard/

This sub-topic covers how to quantify the dispersion of data values around the mean using variance and standard deviation, the most widely used measures of spread for numerical data. You will learn hand calculation, GDC use, and interpretation for exam questions.

**Prerequisites:** [Measures of central tendency: mean, median, mode](https://www.owlsprep.com/study/ib-math-ai-sl-u4-measures-of-central-tendency/)

## Learning objectives

- Distinguish between population and sample variance and standard deviation
- Calculate variance and standard deviation by hand and using a GDC
- Interpret what standard deviation tells you about the spread of data
- Use variance/standard deviation to compare different data sets

## Key Definitions and Basic Hand Calculation

**Variance** — 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.

*Notation:* Population variance: $\sigma^2$, Sample variance: $s^2$

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

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

*Notation:* Population: $\sigma$, Sample: $s$

**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 $\mu$:
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. $$\sigma = \sqrt{1.84} \approx 1.36$$

## Population vs Sample Variance

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.

**Sample Variance** — 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.

*Notation:* $s^2$

**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.

> **tip**
>
> If no population/sample is specified, use population standard deviation for the full data set given in the question, and sample standard deviation when the data is explicitly a sample.

## GDC Calculation and Interpretation

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 $\sigma \approx 2.67$ °C, sample standard deviation $s \approx 2.88$ °C.
3. Since we have all 7 days of data (the full population), we use $\sigma$.
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.

**Check your understanding**

Check you can select the correct output from your GDC:

1. A question asks for the standard deviation of all 30 test scores from your class. Your GDC outputs $\sigma = 5.2$ and $s = 5.4$. Which do you use?

   - 5.2
   - 5.4
   - Either is acceptable

   *Answer:* 5.2

   *Why:* Correct! The entire class is the full population, so we use $\sigma = 5.2$.

2. You have a sample of 15 trees from a forest, and need to estimate the population variance. Which value do you use?

   - $\sigma^2$
   - $s^2$
   - Either

   *Answer:* $s^2$

   *Why:* Correct! $s^2$ is the unbiased estimate of population variance from a sample.

*Calculator:* allowed

## Common pitfalls

- **Wrong:** Dividing by $n$ instead of $n-1$ when calculating sample variance
  - Why it fails: This gives a biased, lower-than-correct value for the estimated population variance, costing easy marks.
  - Correct: Always confirm if data is a full population or a sample: divide by $n$ for population, $n-1$ for sample.
- **Wrong:** Forgetting to take the square root of variance to get standard deviation
  - Why it fails: Students often output the variance value from their GDC when asked for standard deviation, leading to lost marks.
  - Correct: Double check what the question asks for: give $\sigma^2/s^2$ for variance, $\sigma/s$ for standard deviation.
- **Wrong:** Selecting the wrong output value from the GDC
  - Why it fails: GDCs always output both population and sample standard deviation, so it is easy to pick the wrong one by accident.
  - Correct: Always read the question carefully to confirm if you need population or sample statistics before writing your answer.
- **Wrong:** Interpreting variance instead of standard deviation for context questions
  - Why it fails: Variance is measured in squared units, which are not meaningful for real-world interpretation.
  - Correct: Always use standard deviation to interpret spread in the original units of the data.

## Cheatsheet

| Measure | Notation | Formula | Use Case |
| --- | --- | --- | --- |
| Population Variance | $\sigma^2$ | $\frac{\sum (x_i - \mu)^2}{n}$ | Full data set / entire population |
| Population Std Dev | $\sigma$ | $\sqrt{\sigma^2}$ | Interpreting spread for full population |
| Sample Variance | $s^2$ | $\frac{\sum (x_i - \bar{x})^2}{n-1}$ | Estimating population variance from sample |
| Sample Std Dev | $s$ | $\sqrt{s^2}$ | Interpreting spread for a sample |

## 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.

- [Correlation and linear regression](https://www.owlsprep.com/study/ib-math-ai-sl-u4-correlation-and-linear-regression/)
- [Basic probability concepts and combined events](https://www.owlsprep.com/study/ib-math-ai-sl-u4-basic-probability-concepts-and-combined/)
- [Conditional probability and independence](https://www.owlsprep.com/study/ib-math-ai-sl-u4-conditional-probability-and-independence/)

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