# Measures of central tendency

> IB Mathematics AI SL · Statistics and Probability
> Source: https://www.owlsprep.com/study/ib-math-ai-sl-u4-measures-of-central-tendency/

This module introduces the three core measures of central tendency: mean, median, and mode. You will learn to calculate each for raw and grouped data, and select the most appropriate measure for different data contexts.

**Prerequisites:** [Types of data (categorical, discrete, continuous)](https://www.owlsprep.com/study/ib-math-ai-sl-u4-types-of-data/)

## Learning objectives

- Calculate mean, median and mode for raw discrete and continuous data
- Estimate the mean of grouped continuous data
- Select the appropriate measure of central tendency for different data types and distributions
- Predict changes to central tendency after linear data transformations

## Mean: Calculation for Raw and Grouped Data

**Arithmetic Mean** — The sum of all data values divided by the number of values. For grouped data, it is an estimate calculated using midpoints of class intervals.

*Notation:* \bar{x}

*Example:* For data $[2, 4, 6]$, $\bar{x} = 12/3 = 4$

**Worked example:** The number of text messages 5 students sent in one hour are $[12, 15, 8, 10, 15]$. Calculate the mean.

1. Sum all values in the data set:
2. $$12 + 15 + 8 + 10 + 15 = 60$$
3. Divide the sum by the number of observations $n=5$:
4. $$\bar{x} = \frac{60}{5} = 12$$

**Worked example:** The table below shows heights of 30 students. Estimate the mean height.

1. First, calculate midpoints and $f \times x$ for each class:
2. | Class interval | Midpoint $(x)$ | Frequency $(f)$ | $f \times x$ |
| --- | --- | --- | --- |
| $140 \leq h < 150$ | 145 | 4 | 580 |
| $150 \leq h < 160$ | 155 | 10 | 1550 |
| $160 \leq h < 170$ | 165 | 11 | 1815 |
| $170 \leq h < 180$ | 175 | 5 | 875 |
3. Sum frequencies and $f \times x$:
4. $$\sum f = 30, \quad \sum fx = 4820$$
5. Calculate the estimated mean:
6. $$\bar{x} = \frac{\sum fx}{\sum f} = \frac{4820}{30} \approx 160.7 \text{ cm}$$

> **Exam tip:** Always remember the mean from grouped data is only an estimate, not an exact value.

## Median and Mode: Definition and Calculation

**Median** — The middle value of an ordered data set. For $n$ observations: if $n$ is odd, it is the $\frac{n+1}{2}$th value; if even, average of the $\frac{n}{2}$th and $\frac{n}{2}+1$th values.

**Mode** — The most frequently occurring value in a data set. For grouped data, the modal class is the class interval with the highest frequency.

**Worked example:** For the ordered text message data $[8, 10, 12, 15, 15]$, find the median and mode.

1. We have $n=5$ (odd), so the median is the $\frac{5+1}{2} = 3$rd value. The 3rd value is 12, so median $= 12$.
2. The mode is the most frequent value. 15 occurs twice, so mode $= 15$.

**Worked example:** Add a 6th data point $7$ to the set above. Find the new median.

1. Reorder the data: $[7, 8, 10, 12, 15, 15]$, $n=6$ (even).
2. Average the 3rd and 4th values: $\frac{10 + 12}{2} = 11$. New median $= 11$.

## Selecting the Appropriate Measure of Central Tendency

The choice of measure depends on the type of data and shape of the distribution. Outliers and skewness affect which measure is most representative of the 'typical' value.

- **Mode**: Only appropriate for nominal categorical data; can also be used for discrete numerical data.
- **Median**: Appropriate for ordinal data, skewed numerical data, and data with outliers, as it is not affected by extreme values.
- **Mean**: Appropriate for symmetric numerical data without outliers, it uses all data values but is sensitive to extreme values.

> **tip**
>
> If an exam question asks for 'average' without specifying, always check the context to select the most appropriate measure.

**Worked example:** A business has 4 employees earning \$2000 per week, and an owner earning \$12000 per week. Which measure best describes the typical weekly wage? Justify.

1. Ordered wages: $[2000, 2000, 2000, 2000, 12000]$. Mode = 2000, median = 2000, mean = $\frac{4(2000) + 12000}{5} = 4000$.
2. The owner's wage is an outlier that right-skews the distribution. The mean is pulled up to \$4000, which is higher than 4 of 5 wages.
3. Conclusion: The median (or mode) is best, as it is not affected by the outlier and reflects the typical wage.

## Effects of Linear Data Transformations

When all values in a data set are transformed by adding or multiplying by a constant, all three measures of central tendency change in predictable ways:

- Adding a constant $c$ to every value: mean, median and mode all increase by $c$.
- Multiplying every value by a constant $k$: mean, median and mode are all multiplied by $k$.

**Worked example:** The mean height of a class is 161 cm, measured in centimeters. What is the new mean when converted to meters?

1. Converting cm to meters requires multiplying every value by $k = 0.01$.
2. All measures of central tendency are scaled by the same constant, so new mean = $161 \times 0.01 = 1.61$ m.

## Common pitfalls

- **Wrong:** Forgetting to order the data set before calculating the median
  - Why it fails: The median is only defined as the middle value of an ordered set; unordered data gives an incorrect result
  - Correct: Always sort your data from smallest to largest before finding the median
- **Wrong:** Treating the estimated mean for grouped data as an exact value
  - Why it fails: We use midpoints to approximate the original raw data, which is not available for grouped sets
  - Correct: Always remember the mean from grouped data is an estimate, and round it appropriately
- **Wrong:** Using the mean for skewed data with outliers
  - Why it fails: Extreme values pull the mean away from the center, leading to a misleading measure of typical value
  - Correct: Use the median for skewed data or data with extreme outliers
- **Wrong:** Calculating a single mode for continuous raw data with all unique values
  - Why it fails: Continuous data rarely has repeated values, so a single mode is meaningless for raw continuous data
  - Correct: Use the median or mean for raw continuous data, or the modal class for grouped continuous data
- **Wrong:** Expecting the median to always be a value from the original data set
  - Why it fails: For even-sized data sets, the median is the average of two middle values, which may not exist in the original data
  - Correct: When $n$ is even, average the two middle values even if the result is not in the original data

## Cheatsheet

| Measure | When to use | Formula/Rule |
| --- | --- | --- |
| Mode | Nominal categorical data | Most frequent value / modal class |
| Median | Ordinal data, skewed data, data with outliers | Middle value of ordered data |
| Mean (raw) | Symmetric numerical data, no outliers | $\bar{x} = \frac{\sum x_i}{n}$ |
| Mean (grouped) | Estimate for grouped continuous data | $\bar{x} = \frac{\sum f_i x_i}{\sum f_i}$ |

## What's next

Measures of central tendency are the foundation of descriptive statistics, and they are almost always paired with measures of spread to fully describe the shape and center of a data distribution. Understanding when to use each measure helps you avoid common mistakes in interpretation when analyzing real-world data, which is a core skill for IB AI SL statistics questions. You will often be asked to compare measures of central tendency between two or more data sets in exam questions, so building a solid understanding here will support all future statistical work from correlation to hypothesis testing.

- [Measures of spread: variance, standard deviation](https://www.owlsprep.com/study/ib-math-ai-sl-u4-measures-of-spread-variance-standard/)
- [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/)

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