Data presentation and summary statistics

IB Mathematics AA HL· 4.2· 15 min read

1. Graphical Displays for Different Data Types★★☆☆☆⏱ 5 min

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📘 Definition

Univariate data

Data that records a single characteristic from each sampling unit, used for descriptive and inferential analysis

Example:

Heights of 50 12th grade students is univariate; height + weight is bivariate

Different data types require different displays to clearly show patterns. Categorical data uses bar charts or pie charts, discrete quantitative data uses bar charts, and continuous data uses histograms, frequency polygons, or cumulative frequency curves.

📐 Worked Example

A teacher records absences per student in a 30-person class: | Absences | 0 | 1 | 2 | 3 | 4 | | Frequency | 12 | 10 | 5 | 2 | 1 | What display is most appropriate, and how should it be structured?

1
1. Identify data type: Number of absences is discrete quantitative data.
2
1. Match to display: Discrete data uses a separated bar chart (not a histogram, which implies continuity).
3
1. Axes setup: X-axis labeled "Number of absences" with ticks at 0–4, Y-axis labeled "Frequency" ranging 0–12.
4
1. Draw separated bars with height equal to the frequency for each absence value.

2. Measures of Center and Spread★★★☆☆⏱ 6 min

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📘 Definition

Interquartile Range (IQR)

IQR = Q_3 - Q_1

The spread of the middle 50% of ordered data, a measure of spread that is robust to outliers

Example:

For [2, 3, 5, 7, 11], Q1=3, Q3=7, IQR = 4

Measures of center describe the typical value of a distribution, while measures of spread describe how much data varies. Common pairs used are mean and standard deviation, or median and IQR.

Mean = average of all data, affected by extreme outliers
Median = middle value of ordered data, not affected by outliers
Standard deviation = average distance of points from the mean, same units as original data

📐 Worked Example

Find the mean, median, and population standard deviation for [12, 15, 18, 22, 25], where this is the full population.

1
Calculate the mean:
2
$\bar{x} = \frac{12 + 15 + 18 + 22 + 25}{5} = 18.4$
3
Find the median: ordered data, the 3rd value is the median: Median = 18
4
Calculate sum of squared deviations from the mean:
5
$(12-18.4)^2 + ... + (25-18.4)^2 = 109.2$
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Population standard deviation is the square root of average squared deviation:
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$\sigma = \sqrt{\frac{109.2}{5}} \approx 4.67$

3. Outliers and Box Plots★★★☆☆⏱ 5 min

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📘 Definition

Outlier (1.5 × IQR rule)

An observation is an outlier if it is less than $Q_{1} - 1.5 \times I QR$ or greater than $Q_{3} + 1.5 \times I QR$

Example:

Q1=10, Q3=20, IQR=10: any value < -5 or > 35 is an outlier

Box plots (box-and-whisker plots) display the five-number summary (minimum, Q1, median, Q3, maximum) and explicitly mark outliers. They are ideal for comparing two or more distributions side-by-side.

📐 Worked Example

Identify any outliers in [4, 6, 7, 8, 10, 12, 35] using the 1.5×IQR rule.

1
Data is ordered, n=7. Median = 4th term = 8. Split into lower half [4,6,7] and upper half [10,12,35], excluding the median.
2
Q1 = median of lower half = 6, Q3 = median of upper half = 12
3
Calculate outlier boundaries:
4
$IQR = 12 - 6 = 6 \\ Lower = 6 - 1.5(6) = -3 \\ Upper = 12 + 1.5(6) = 21$
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35 is greater than 21, so 35 is classified as an outlier.

4. Cumulative Frequency Graphs★★★★☆⏱ 7 min

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A cumulative frequency graph plots cumulative frequency (total observations up to a given value) against the upper bound of class intervals. It is used to estimate medians, quartiles, and percentiles from grouped continuous data.

📐 Worked Example

For 100 plant heights, cumulative frequency is: 10 at 10cm, 40 at 20cm, 75 at 30cm, 100 at 40cm. Estimate the median and IQR.

1
Median is the height at cumulative frequency $\frac{100}{2} = 50$ .
2
Interpolate between (20cm, 40) and (30cm, 75): $20 + \frac{50 - 40}{75 - 40} \times 10 \approx 22.9$ cm
3
Q1 is at cumulative frequency 25, Q3 at 75:
4
Q1 $\approx 10 + \frac{25 - 10}{40 - 10} \times 10 = 15$ cm, Q3 = 30 cm (given)
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Calculate IQR:
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$IQR = 30 - 15 = 15 \text{ cm}$

5. Common Pitfalls

Wrong move:

Using a histogram for discrete categorical data

Why:

Connected histogram bars imply the variable is continuous, which is misleading and loses marks

Correct move:

Use separated bar charts for discrete or categorical data

Wrong move:

Using the wrong standard deviation (sample vs population) from a GDC

Why:

IB expects you to select the correct value for the context, using the wrong one gives an incorrect answer

Correct move:

Use $σ$ for full population data, $s$ for sample data from a larger population

Wrong move:

Using mean instead of median for skewed distributions with outliers

Why:

The mean is pulled towards the tail of skewed distributions and extreme outliers, so it does not reflect the typical value

Correct move:

Report median as the measure of center for skewed distributions or datasets with outliers

Wrong move:

Including the overall median when calculating Q1/Q3 for odd n

Why:

This leads to incorrect quartile values and wrong outlier classification

Correct move:

Exclude the overall median from both the lower and upper halves when n is odd before calculating Q1/Q3

6. Quick Reference Cheatsheet

Measure Type	Name	Key Rule	Robust to Outliers?
Center	Mean	$\overset{x}{ˉ} = \frac{\sum x _{i}}{n}$	No
Center	Median	Middle value of ordered data	Yes
Spread	Range	Max - Min	No
Spread	IQR	Q3 - Q1	Yes
Spread	Population SD	$σ = \frac{\sum ( x _{i} - x ˉ ) ^{2}}{n}$	No
Spread	Sample SD	$s = \frac{\sum ( x _{i} - x ˉ ) ^{2}}{n - 1}$	No
Outlier Rule	1.5×IQR	Outlier if < Q1 - 1.5IQR or > Q3 + 1.5IQR	N/A

7. Frequently Asked

Do I need to distinguish between population and sample standard deviation?

Yes, IB exams regularly test this distinction. Use population standard deviation ( $σ$ ) when you have data for the entire group of interest, and sample standard deviation ( $s$ ) when your data is a subset of a larger group.

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 · 1
Outlier calculation and box plot
2023 · 2
Compare two distributions
2021 · 1
Cumulative frequency interpretation

Going deeper

syllabusIB AA HL Official Topic 4 SyllabusOfficial syllabus requirements

What's Next

Data presentation and summary statistics is the foundation for all statistical work in IB AA HL. These descriptive tools help you explore patterns in raw data before conducting formal inferential tests, and you will use summary measures throughout all remaining statistics topics. You will rely on the properties of mean and standard deviation when working with probability distributions, use box plots to compare groups for hypothesis testing, and interpret summary statistics for regression analysis.