Measures of central tendency

IB Mathematics AI SL· 10 min read

1. Mean: Calculation for Raw and Grouped Data★★☆☆☆⏱ 10 min

📘 Definition

Arithmetic Mean

\bar{x}

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.

Example:

For data $[2, 4, 6]$ , $\overset{x}{ˉ} = 12/3 = 4$

📐 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
$\bar{x} = \frac{\sum fx}{\sum f} = \frac{4820}{30} \approx 160.7 \text{ cm}$

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

Exam tip:

Always remember the mean from grouped data is only an estimate, not an exact value.

2. Median and Mode: Definition and Calculation★★☆☆☆⏱ 8 min

📘 Definition

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.

📘 Definition

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 $= 11$ .

3. Selecting the Appropriate Measure of Central Tendency★★★☆☆⏱ 10 min

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.

📐 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.

4. Effects of Linear Data Transformations★★★☆☆⏱ 7 min

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.

5. Common Pitfalls

Wrong move:

Forgetting to order the data set before calculating the median

Why:

The median is only defined as the middle value of an ordered set; unordered data gives an incorrect result

Correct move:

Always sort your data from smallest to largest before finding the median

Wrong move:

Treating the estimated mean for grouped data as an exact value

Why:

We use midpoints to approximate the original raw data, which is not available for grouped sets

Correct move:

Always remember the mean from grouped data is an estimate, and round it appropriately

Wrong move:

Using the mean for skewed data with outliers

Why:

Extreme values pull the mean away from the center, leading to a misleading measure of typical value

Correct move:

Use the median for skewed data or data with extreme outliers

Wrong move:

Calculating a single mode for continuous raw data with all unique values

Why:

Continuous data rarely has repeated values, so a single mode is meaningless for raw continuous data

Correct move:

Use the median or mean for raw continuous data, or the modal class for grouped continuous data

Wrong move:

Expecting the median to always be a value from the original data set

Why:

For even-sized data sets, the median is the average of two middle values, which may not exist in the original data

Correct move:

When $n$ is even, average the two middle values even if the result is not in the original data

6. Quick Reference 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	$\overset{x}{ˉ} = \frac{\sum x _{i}}{n}$
Mean (grouped)	Estimate for grouped continuous data	$\overset{x}{ˉ} = \frac{\sum f _{i} x _{i}}{\sum f _{i}}$

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.