Data Representation & Summary Stats — SL 4.1 Data representation and summary statistics (mean, median, quartiles, IQR) - IB Mathematics Analysis and Approaches

SECTION 1

Historical Context & Motivation

Long before spreadsheets and calculators, people needed ways to make sense of large collections of numbers. Ancient tax records, census counts, and astronomical observations all posed the same challenge: how do you condense hundreds or thousands of individual values into a handful of meaningful numbers? The branch of mathematics we now call statistics grew directly from this practical need. Understanding its origins helps us appreciate why the tools in SL 4.1—mean, median, quartiles, and IQR—are so powerful and widely used today.

1663 • John Graunt's Bills of Mortality

English statistician John Graunt published analyses of London death records, pioneering the idea of summarizing raw data into tables and averages to reveal patterns in public health.

1774 • Laplace and the Arithmetic Mean

Pierre-Simon Laplace formally justified the arithmetic mean as the best single estimate of a measured quantity, giving the familiar "add them up and divide" rule a rigorous mathematical foundation.

1882 • Galton's Percentile Methods

Sir Francis Galton introduced percentile ranks and the concept of dividing data into quartiles, providing a way to describe spread and position beyond just an average.

1977 • Tukey's Box-and-Whisker Plot

Mathematician John Tukey popularized the box-and-whisker plot, combining five-number summaries (minimum, Q₁, median, Q₃, maximum) into a single visual that reveals center, spread, and outliers at a glance.

The central question has remained the same across centuries: How can we represent an entire data set with just a few numbers and a clear picture? SL 4.1 gives you the essential toolkit to answer that question—measures of central tendency (mean, median), measures of spread (quartiles, IQR), and graphical representations that bring data to life.

SECTION 2

Core Principles & Definitions

Before diving into calculations, it helps to understand the big ideas behind summary statistics. Every data set tells a story, and that story has two main characters: center (where the data clusters) and spread (how far the data stretches). The concepts below form the foundation for everything in this topic.

Mean (x̄)

The arithmetic average—add every value and divide by how many there are. It uses all data points but is sensitive to extreme values (outliers).

Median

The middle value when data are ordered from smallest to largest. It splits the data into two equal halves and resists the pull of outliers.

Quartiles (Q₁ and Q₃)

Q₁ is the median of the lower half; Q₃ is the median of the upper half. Together with the median, they divide data into four equal-frequency groups.

Interquartile Range (IQR)

IQR = Q₃ − Q₁. It measures the spread of the central 50% of the data, making it a robust alternative to the range.

Five-Number Summary

Minimum, Q₁, Median, Q₃, Maximum—these five values provide a compact snapshot of a data set's shape, center, and spread.

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KEY TAKEAWAY

Think of the mean as the balance point of a seesaw—every data point exerts a pull, so one very heavy kid on one end (an outlier) shifts the balance. The median is like finding the middle seat in a movie theater row—no matter who sits at the edges, the middle seat stays the same. When you want to describe how widely the audience is spread out, the IQR tells you the width that covers the central half of the row.

SECTION 3

Visual Explanation — The Box-and-Whisker Plot

A box-and-whisker plot (also called a box plot) is the standard graphical representation for a five-number summary. It instantly shows you the center, spread, and symmetry (or skewness) of a data set. The diagram below uses the ordered data set: 3, 5, 7, 8, 10, 12, 14, 16, 20.

The box spans from Q₁ (6) to Q₃ (14), capturing the middle 50 % of the data. The vertical line inside the box marks the median (10). The whiskers extend to the minimum (3) and maximum (20). The IQR of 8 is the width of the box.

Notice how the right whisker is longer than the left whisker. This tells you the data is positively skewed (stretched toward higher values). When the whiskers are roughly equal in length and the median line sits near the center of the box, the data is approximately symmetric. Being able to read these visual cues is a key skill on IB exams.

SECTION 4

Mathematical Framework

Now let's formalize the calculations. Each formula below is defined for a data set of n values, listed in ascending order as x₁, x₂, …, xₙ.

MEAN (ARITHMETIC AVERAGE)

x̄ = (x₁ + x₂ + ⋯ + xₙ) / n = (Σxᵢ) / n

x̄ (read "x-bar") is the mean. Σxᵢ denotes the sum of all n data values. Every value in the set contributes equally.

MEDIAN

Median = x₍ₙ₊₁₎⁄₂ if n is odd; Median = (xₙ⁄₂ + xₙ⁄₂₊₁) / 2 if n is even

First sort the data in ascending order. If there is an odd number of values, the median is the single middle value. If even, average the two central values.

QUARTILES

Q₁ = Median of the lower half; Q₃ = Median of the upper half

Split the ordered data at the median. Q₁ is the median of the values below the overall median; Q₃ is the median of the values above it. If n is odd, the overall median is excluded from both halves.

INTERQUARTILE RANGE

IQR = Q₃ − Q₁

The IQR measures the spread of the central 50 % of the data. A small IQR means data points cluster tightly around the median; a large IQR means greater variability.

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IB Exam Tip

Your GDC (graphing calculator) can compute all of these statistics from a list. On the TI-84, enter data into L1, then press STAT → CALC → 1-Var Stats. On the TI-Nspire, use Menu → Statistics → Stat Calculations → One-Variable Statistics. Knowing the formulas helps you check calculator output and solve problems without technology.

SECTION 5

Detailed Breakdown — Choosing the Right Measure

Not every data set is best described by the same statistic. The mean is excellent when data is roughly symmetric, but it can be misleading when a few extreme values pull it away from the bulk of the data. The diagram below compares a symmetric data set with a skewed one to show how the mean and median behave differently.

Left: In a symmetric data set (2, 4, 5, 6, 8), the mean and median both equal 5. Right: Replacing 8 with an outlier of 25 drags the mean to 8.4 while the median remains at 5.

Scenario	Best Measure of Center	Why?
Data is symmetric, no outliers	Mean	Uses all values; gives the most precise summary
Data is skewed or has outliers	Median	Resistant to extreme values; stays near the bulk of data
You want to describe spread	IQR	Captures the middle 50 %, ignoring extremes at both tails
Quick overall range needed	Range (Max − Min)	Simple to compute, but heavily affected by a single outlier

Choosing the appropriate summary statistic

SECTION 6

Worked Example

A class of 12 students recorded the number of hours they studied for a math test. The raw data (in hours) are: 4, 7, 5, 10, 3, 8, 6, 12, 5, 9, 7, 6. Find the mean, median, Q₁, Q₃, and IQR.

Finding the Five-Number Summary

1
Step 1 — Order the Data

Arrange the 12 values from smallest to largest: 3, 4, 5, 5, 6, 6, 7, 7, 8, 9, 10, 12. Always order first—quartile calculations depend on position.

Ordered: 3, 4, 5, 5, 6, 6, 7, 7, 8, 9, 10, 12
2
Step 2 — Calculate the Mean

Sum all values: 3 + 4 + 5 + 5 + 6 + 6 + 7 + 7 + 8 + 9 + 10 + 12 = 82. Divide by n = 12: x̄ = 82 / 12 ≈ 6.83.

Mean ≈ 6.83 hours
3
Step 3 — Find the Median

With n = 12 (even), the median is the average of the 6th and 7th values. The 6th value is 6 and the 7th value is 7, so the median = (6 + 7) / 2 = 6.5.

Median = 6.5 hours
4
Step 4 — Find Q₁ and Q₃

Split the data at the median into two halves of 6 values each. Lower half: 3, 4, 5, 5, 6, 6. Q₁ is the median of this group: the average of the 3rd and 4th values = (5 + 5) / 2 = 5. Upper half: 7, 7, 8, 9, 10, 12. Q₃ is the median of this group: the average of the 3rd and 4th values = (8 + 9) / 2 = 8.5.

Q₁ = 5 hours; Q₃ = 8.5 hours
5
Step 5 — Calculate the IQR

IQR = Q₃ − Q₁ = 8.5 − 5 = 3.5. This means the middle 50 % of students studied within a 3.5-hour window.

IQR = 3.5 hours

ℹ

Quick Check

The five-number summary is: Min = 3, Q₁ = 5, Median = 6.5, Q₃ = 8.5, Max = 12. You could now draw a box plot from these five values to visualize the data's center and spread.

SECTION 7

Strengths & Limitations of Each Measure

Each summary statistic has trade-offs. Knowing when to rely on the mean versus the median—and when the IQR tells a more honest story than the range—is a critical thinking skill tested on the IB exam. The table below puts these trade-offs side by side.

Measure	Strengths	Limitations
Mean	Uses every data point; algebraically convenient; required for standard deviation and further analyses	Sensitive to outliers; can be misleading for highly skewed data
Median	Resistant to outliers; always represents the "middle" of the data; easy to interpret	Ignores extreme values entirely; less useful for algebraic operations
Range	Very simple to compute; gives a quick sense of total spread	Entirely determined by two extreme values; one outlier can inflate it dramatically
IQR	Describes the spread of the central 50 %; resistant to outliers; used to identify outliers	Ignores the tails of the distribution; less precise than standard deviation for symmetric data

Comparing summary statistics

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KEY TAKEAWAY

Think of choosing a summary statistic like choosing a camera filter for a photo. The mean is like a wide-angle lens—it captures everything, including distracting objects in the background. The median is like a portrait mode that focuses on the center subject and blurs out the extremes. The IQR is like cropping the photo to just the important middle section. Each serves a purpose; the best choice depends on the shape of your data.

SECTION 8

Connection to Advanced Topics

The summary statistics in SL 4.1 are the building blocks for more sophisticated analysis later in the course and at higher levels. Understanding how these concepts extend will help you see the bigger picture of statistics.

SL 4.1 Concept	Extends To …	Why It Matters
Mean (x̄)	Standard deviation (σ) and variance (σ²) in SL 4.2	Variance measures how far data points deviate from the mean on average, giving a more precise picture of spread
Median & Quartiles	Cumulative frequency curves and percentiles	Reading Q₁, Q₂, Q₃ from a cumulative frequency graph is a common IB exam technique
IQR	Outlier detection rule: values beyond Q₁ − 1.5 × IQR or Q₃ + 1.5 × IQR	The IQR gives you a formal method to decide whether an extreme value is genuinely unusual
Box plots	Histograms, frequency polygons, and normal distribution curves	More detailed representations become necessary as data sets grow larger and analysis becomes more complex

How SL 4.1 connects to later topics

As you progress through HL topics or pursue statistics at university, you'll encounter concepts like standard deviation, z-scores, and regression analysis. Every one of these builds on the foundational idea that data has a center and a spread—exactly what you've learned here in SL 4.1.

SECTION 9

Practice Problems

Question 1: CONCEPTUAL

A data set has a mean of 50 and a median of 42. What does this tell you about the shape of the distribution? Explain your reasoning.

Show answer

Because the mean (50) is greater than the median (42), the distribution is positively skewed (skewed to the right). This means there are some high values that pull the mean above the median. In a perfectly symmetric distribution, the mean and median would be equal. The larger the gap between the two, the more pronounced the skew.

Question 2: BASIC CALCULATION

Find the mean and median of the data set: 12, 15, 18, 22, 25, 30.

Show answer

The data is already in order. Mean = (12 + 15 + 18 + 22 + 25 + 30) / 6 = 122 / 6 ≈ 20.33. Since n = 6 (even), the median is the average of the 3rd and 4th values: (18 + 22) / 2 = 20. The mean and median are very close, suggesting the data is roughly symmetric.

Question 3: INTERMEDIATE

The ages (in years) of participants in a workshop are: 19, 21, 22, 24, 25, 27, 29, 31, 35, 40. Find Q₁, Q₃, and the IQR. Determine whether the value 40 is an outlier using the 1.5 × IQR rule.

Show answer

The data is already ordered with n = 10. Lower half: 19, 21, 22, 24, 25 → Q₁ = 22 (middle value). Upper half: 27, 29, 31, 35, 40 → Q₃ = 31. IQR = 31 − 22 = 9. Upper fence = Q₃ + 1.5 × IQR = 31 + 13.5 = 44.5. Since 40 < 44.5, the value 40 is not an outlier by this rule, even though it is the largest value in the set.

Question 4: APPLIED

A small company has 7 employees with the following annual salaries (in thousands of dollars): 32, 35, 38, 40, 42, 45, 150. The CEO (earning 150k) argues the average salary shows employees are well-paid. The employees disagree. Calculate both the mean and median salary, and explain which measure better represents the typical employee's pay.

Show answer

Mean = (32 + 35 + 38 + 40 + 42 + 45 + 150) / 7 = 382 / 7 ≈ $54,570. Median (4th value out of 7) = $40,000. The CEO's salary of $150k is an outlier that inflates the mean. The median of $40,000 is far more representative of what a typical employee earns. This is a classic example of why economists often report median household income rather than mean income—it avoids distortion from extreme values at the top.

Question 5: CRITICAL THINKING

Two classes, A and B, both scored a median of 72 on a test. Class A has an IQR of 8 and Class B has an IQR of 24. Without seeing any individual scores, compare the performance of the two classes. Which class would you prefer to teach if your goal were to ensure every student is prepared for the next unit? Justify your answer using the concepts of center and spread.

Show answer

Both classes share the same median, so their "typical" student scored equally well. However, Class A's IQR of 8 means the middle 50 % of students scored within an 8-point band around the median (roughly 68–76), indicating consistent performance. Class B's IQR of 24 means the middle 50 % spans a 24-point range (roughly 60–84), indicating much greater variability. If the goal is ensuring every student is ready for the next unit, Class A is preferable because students' understanding is more uniform. Class B likely contains students who excelled alongside students who struggled significantly, requiring more differentiated instruction. This problem illustrates that center alone never tells the full story—spread is essential context.

SUMMARY

Summary

SL 4.1 equips you with the essential toolkit for describing any data set. The mean (x̄ = Σxᵢ / n) gives the arithmetic average and uses every value, while the median identifies the exact middle of ordered data and resists the influence of outliers. Together, these two measures of center reveal both the typical value and the data's symmetry. When mean > median, the distribution is positively skewed; when mean < median, it is negatively skewed.

To describe spread, the quartiles Q₁ and Q₃ divide the data into four equal-frequency groups, and the interquartile range (IQR = Q₃ − Q₁) captures the width of the central 50 %. The five-number summary (Min, Q₁, Median, Q₃, Max) provides all the information needed to construct a box-and-whisker plot, which visually displays center, spread, and skewness in a single diagram. Always choose the summary statistic that matches your data's shape: the mean for symmetric distributions, the median and IQR when outliers or skewness are present.