# Measures of central tendency and dispersion

> IB Mathematics: Analysis and Approaches SL · Unit 4: Statistics & Probability
> Source: https://www.owlsprep.com/study/ib-math-aa-sl-u4-measures-of-central-tendency-and/

This module covers how to summarize numerical data using measures that describe the center (central tendency) and spread (dispersion), for both ungrouped and grouped discrete and continuous data, including calculation and contextual interpretation.

**Prerequisites:** [Organizing and representing data](https://www.owlsprep.com/study/ib-math-aa-sl-u4-organizing-data/)

## Learning objectives

- Calculate mean, median and mode for ungrouped and grouped numerical data
- Compute common measures of dispersion: range, interquartile range, variance and standard deviation
- Select and justify appropriate measures for different data types and distributions
- Interpret these measures in real-world contexts

## Measures of Central Tendency

Measures of central tendency describe the typical or center value of a data set. There are three main measures, each suited to different scenarios.

**Mean** — The sum of all data values divided by the number of values. For grouped data, we use midpoints of class intervals to calculate an estimated mean.

*Notation:* $\bar{x}$ (sample), $\mu$ (population)

**Median** — The middle value of an ordered data set. For even $n$, it is the average of the two middle values.

**Mode** — The most frequently occurring value in a data set. Can be used for categorical data.

**Worked example:** Find the mean, median and mode for the ungrouped data set: $[2, 3, 5, 5, 7, 8, 10]$

1. Calculate the mean by summing values and dividing by $n=7$:
2. $$\text{Mean} = \frac{2+3+5+5+7+8+10}{7} = \frac{40}{7} \approx 5.71$$
3. Data is already ordered, $n=7$ (odd) so the median is the $\frac{7+1}{2} = 4^{\text{th}}$ value:
4. Median = $5$
5. The most frequent value is $5$, so the mode is $5$.

**Worked example:** Calculate the estimated mean for this grouped frequency table: $0 < x \leq 10, f=4$; $10 < x \leq 20, f=8$; $20 < x \leq 30, f=3$

1. Find the midpoint $x$ of each class interval: $5, 15, 25$
2. Calculate $\sum fx$ and $\sum f$:
3. $$\sum fx = (4 \times 5) + (8 \times 15) + (3 \times 25) = 20 + 120 + 75 = 215$$
4. $$\sum f = 4 + 8 + 3 = 15$$
5. Calculate estimated mean:
6. $$\bar{x} = \frac{215}{15} \approx 14.33$$

> **Exam tip:** Always state that the mean for grouped data is an estimate, not an exact value, you may lose a mark if you forget this.

## Measures of Dispersion

Measures of dispersion describe how spread out the data values are around the central tendency. Common measures include range, interquartile range (IQR), variance and standard deviation.

**Standard Deviation** — The average distance of data values from the mean, measured in the same units as the original data. Variance is the square of standard deviation.

> **note**
>
> IB requires you to distinguish between sample and population standard deviation. On your GDC, $s_x$ is sample standard deviation (divides by $n-1$) and $\sigma_x$ is population standard deviation (divides by $n$).

**Worked example:** Calculate the sample variance and standard deviation for the data set $[2, 4, 6]$

1. First calculate the sample mean:
2. $$\bar{x} = \frac{2+4+6}{3} = 4$$
3. Calculate the sum of squared deviations from the mean:
4. $$\sum (x - \bar{x})^2 = (2-4)^2 + (4-4)^2 + (6-4)^2 = 4 + 0 + 4 = 8$$
5. For sample variance, divide by $n-1 = 2$:
6. $$s^2 = \frac{8}{2} = 4$$
7. Sample standard deviation is the square root of variance:
8. $$s = \sqrt{4} = 2$$

*Calculator:* allowed

## Choosing Appropriate Measures

The choice of measure depends on the shape of the distribution and presence of outliers. The mean is affected by extreme values, while the median and IQR are not.

**Worked example:** Hourly wages at a small store: \$12, \$12, \$15, \$16, \$18, \$22, \$75 (manager's wage). Which measure of central tendency best represents the typical wage? Justify your answer.

1. Calculate all three measures: Mean ≈ \$24.29, Median = \$16, Mode = \$12
2. Identify that \$75 is an outlier, much higher than all other wages.
3. The outlier skews the mean, pulling it far above the wage of most employees. The mode is too low, and not representative.
4. Conclusion: The median is the best measure, as it is not affected by the extreme outlier and reflects the typical wage.

- Use mean/standard deviation for symmetric, outlier-free data
- Use median/IQR for skewed data or data with outliers
- Use mode only for categorical data or discrete data with repeated values

> **Exam tip:** If a question asks you to justify your choice, you must explicitly mention outliers or skewness to get full marks.

## Five Number Summary

The five number summary combines minimum, first quartile ($Q_1$), median, third quartile ($Q_3$) and maximum, used to construct box plots and identify outliers.

**Interquartile Range (IQR)** — The spread of the middle 50% of the data, calculated as $IQR = Q_3 - Q_1$

**Worked example:** Find the five number summary for test scores: $[45, 52, 60, 65, 68, 72, 75, 80, 85, 90]$

1. Minimum = 45, Maximum = 90, $n=10$ even.
2. Median = average of 5th and 6th terms: $\frac{68 + 72}{2} = 70$
3. $Q_1$ = median of lower half (first 5 terms): 3rd term = 60
4. $Q_3$ = median of upper half (last 5 terms): 3rd term of upper half = 80
5. Five number summary: $[45, 60, 70, 80, 90]$, $IQR = 20$

## Common pitfalls

- **Wrong:** Reporting an exact mean for grouped data
  - Why it fails: Grouped data does not retain individual raw values, so midpoints are only an approximation
  - Correct: Explicitly state that your answer is an estimated mean for grouped data
- **Wrong:** Confusing sample and population standard deviation from GDC output
  - Why it fails: IB examiners award full marks only for the value requested in the question
  - Correct: Always read the question carefully: use $s_x$ for sample, $\sigma_x$ for population
- **Wrong:** Using mean for skewed data with outliers
  - Why it fails: Outliers pull the mean away from the center, making it unrepresentative
  - Correct: Use the median as the measure of central tendency for skewed data
- **Wrong:** Forgetting to sort data before finding median or quartiles
  - Why it fails: Unordered data gives incorrect position values for median and quartiles
  - Correct: Always sort data from smallest to largest before calculating any positional measures
- **Wrong:** Calculating IQR as $\frac{Q_3 - Q_1}{2}$
  - Why it fails: IQR is the full difference between upper and lower quartile, not the half difference
  - Correct: Calculate IQR directly as $IQR = Q_3 - Q_1$

## Cheatsheet

| Measure | Type | Key Notes |
| --- | --- | --- |
| Mean (ungrouped) | Central Tendency | $\frac{\sum x}{n}$, affected by outliers |
| Est. Mean (grouped) | Central Tendency | $\frac{\sum fx}{\sum f}$, uses midpoints, estimate only |
| Median | Central Tendency | Middle ordered value, not affected by outliers |
| Mode | Central Tendency | Most frequent value, for categorical/discrete data |
| Range | Dispersion | $\text{Max} - \text{Min}$, affected by outliers |
| IQR | Dispersion | $Q_3 - Q_1$, spread of middle 50%, robust to outliers |
| Sample Variance | Dispersion | $\frac{\sum (x-\bar{x})^2}{n-1}$, for sample data |
| Population Variance | Dispersion | $\frac{\sum (x-\mu)^2}{N}$, for full population |

## What's next

Measures of central tendency and dispersion are the foundation of all statistical analysis in IB AA SL. These concepts appear regularly in both Paper 1 and Paper 2, often combined with data representation, box plots, or cumulative frequency questions. You will use these skills to compare distributions, analyze bivariate relationships, and describe probability distributions in later topics. Mastery of these calculations and interpretation is essential for higher scoring statistics questions.

- [Cumulative distributions and percentiles](https://www.owlsprep.com/study/ib-math-aa-sl-u4-cumulative-distributions-and-percentiles/)
- [Probability, conditional probability and independent events](https://www.owlsprep.com/study/ib-math-aa-sl-u4-probability-conditional-probability-and-independent/)
- [Discrete probability distributions, expectation and variance](https://www.owlsprep.com/study/ib-math-aa-sl-u4-discrete-probability-distributions-expectation-and/)

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