Key takeaways
- The five number summary (min Q1 median Q3 max) is a fast way to describe a dataset.
- IQR = Q3 − Q1 tells the spread of the middle 50% (the “box” in a box plot).
- If you can find the five numbers, you can draw a box plot and compare two groups quickly.
The five number summary (min Q1 median Q3 max) is a topic that feels harder than it is.
You’ll learn a simple method, see worked examples, and practice like it’s a quiz.
You’re not behind. You just need a clear method.
Quick answer
The five-number summary is min, Q1, median, Q3, max.
Step 1: Sort the data.
Step 2: Find the median (middle value).
Step 3: Find Q1 (median of the lower half) and Q3 (median of the upper half).
Step 4: Take the min and max from the sorted list.
What each part means (with simple definitions)
- Minimum (min) (smallest value).
- Maximum (max) (largest value).
- Median (middle value after sorting).
- Quartiles (cut points that split sorted data into 4 parts).
- Q1 (25% of values are at or below this point).
- Q3 (75% of values are at or below this point).
- IQR (interquartile range = Q3 − Q1; spread of the middle 50%).
When the five-number summary is useful
Use it when you need to:
- Describe a dataset fast (center + spread + extremes).
- Make or read a box plot (a box plot is built from these five values).
- Compare two groups (which group has the higher median? bigger spread?).
- Talk about outliers using IQR (common in many classes).
Step-by-step method
Step 1: Sort the data
Write the numbers from smallest to largest.
Step 2: Find the median
- If n is odd, the median is the middle number.
- If n is even, the median is the average of the two middle numbers.
Step 3: Split into two halves
- Lower half = values below the median
- Upper half = values above the median
Rule many classes use: If n is odd, do not include the median in either half. (This is a common classroom method.)
Step 4: Find Q1 and Q3
- Q1 = median of the lower half
- Q3 = median of the upper half
Step 5: Record min and max
Min is the first number in the sorted list.
Max is the last number.
What to write on your paper (mini-checklist)
- Sorted data: ________
- n = ________
- min = ________
- median = ________
- lower half = ________ → Q1 = ________
- upper half = ________ → Q3 = ________
- max = ________
- (optional) IQR = Q3 − Q1 = ________
Quick self-check (fast)
- Q1 ≤ median ≤ Q3 should be true.
- min ≤ Q1 and Q3 ≤ max should be true.
- If your Q1 or Q3 is outside the min/max, you split wrong.
Worked examples (step-by-step)
Example 1 (very easy)
Data: 2, 5, 1, 4, 3
- Sort: 1, 2, 3, 4, 5
- Median = 3 (middle)
- Lower half: 1, 2 → Q1 = (1+2)/2 = 1.5
- Upper half: 4, 5 → Q3 = (4+5)/2 = 4.5
- min = 1, max = 5
Five-number summary: (1, 1.5, 3, 4.5, 5)
Example 2 (typical homework level, even n)
Data: 7, 2, 9, 4, 10, 6
- Sort: 2, 4, 6, 7, 9, 10
- Median = (6+7)/2 = 6.5
- Lower half: 2, 4, 6 → Q1 = 4
- Upper half: 7, 9, 10 → Q3 = 9
- min = 2, max = 10
Five-number summary: (2, 4, 6.5, 9, 10)
IQR: 9 − 4 = 5
Example 3 (exam-style “trap”: repeats + one big value)
Data: 5, 5, 5, 6, 6, 7, 30
- Sort: 5, 5, 5, 6, 6, 7, 30
- Median = 6
- Lower half (exclude median): 5, 5, 5 → Q1 = 5
- Upper half: 6, 7, 30 → Q3 = 7
- min = 5, max = 30
Five-number summary: (5, 5, 6, 7, 30)
Interpretation: Most values are near 5–7, but the max is far away (possible outlier idea).
How it connects to a box plot (and why teachers love it)
A box plot (box-and-whisker plot) is made from the five-number summary:
- whiskers go to min and max
- the box goes from Q1 to Q3
- a line inside the box is the median
That’s why your teacher often teaches these together.
Practice set (with answer key)
Directions: For each dataset, find the five-number summary (min, Q1, median, Q3, max).
- 3, 1, 2
- 8, 4, 6, 10, 12
- 2, 2, 2, 2, 2
- 9, 1, 7, 3, 5, 11
- 4, 6, 6, 6, 7, 8, 20 (trap: big value)
- 15, 18, 10, 12, 12, 14, 16, 20
- 5, 7, 8, 9, 10, 10, 11, 30
Answer key
- (1, 1, 2, 3, 3)
- (4, 5, 8, 11, 12)
- (2, 2, 2, 2, 2)
- (1, 3, 6, 9, 11)
- (4, 6, 6, 8, 20)
- (10, 12, 14, 18, 20)
- (5, 8, 9.5, 10.5, 30)
Short solutions (2 items only):
- #4 Sort: 1,3,5,7,9,11 → median = (5+7)/2=6 → Q1=3 → Q3=9 → summary (1,3,6,9,11)
- #7 Sort: 5,7,8,9,10,10,11,30 → median=(9+10)/2=9.5 → lower half 5,7,8,9 → Q1=(7+8)/2=7.5? (Stop!)
Use the median-of-halves method consistently: lower half is 5,7,8,9 → Q1=(7+8)/2=7.5 and upper half 10,10,11,30 → Q3=(10+11)/2=10.5.
(Note: Some classes use a different quartile rule. Follow your teacher’s rule.)
Common mistakes
| Mistake | Why it happens | Do this instead |
|---|---|---|
| Not sorting the data first | You try to “see” the median in messy order | Always sort from least to greatest first |
| Using the wrong median rule for even n | You forget to average the two middle values | If n is even, median = average of the two middle numbers |
| Including the median in both halves (when n is odd) | You mix quartile methods | Use one class rule consistently (many classes exclude the median) |
| Q1 or Q3 ends up outside min/max | The split into halves was wrong | Re-check your halves and re-find the medians |
| Thinking IQR is max − min | You confuse IQR with range | IQR = Q3 − Q1 |
If you get stuck…
- “I don’t know what to do first.”
Start with one action: sort the data. - “My median is not in the list.”
That can happen with even n. You are allowed to average the two middle values. - “My Q1 and Q3 don’t match my friend’s.”
Some tools use different quartile methods. Use your teacher’s method. - “My calculator gives a different answer.”
Check if it uses a different quartile rule, or if you typed the data wrong. - “I keep mixing up IQR and range.”
Range uses extremes: max − min.
IQR uses quartiles: Q3 − Q1.
If you’re still stuck, ask your teacher for one checked example and use it as your template (that’s normal).
Next steps
- Statistics Calculators and Study Guides — find the right tool fast for homework and quizzes.
- Descriptive Statistics Calculators and Study Guides — tools focused on quartiles, IQR, and summaries.
- Descriptive Statistics Calculator (All-in-One) — paste your data and get min/Q1/median/Q3/max plus IQR.
- Summary Statistics Calculator — a simple “what to include” checklist for assignments.
- When to Use Mean Median or Mode — helps when questions ask for “best center.”
- Best High-Quality Notebooks for Math Notes — if your work gets messy, a structured notebook can help.
How we know
- Box plots are built from min, Q1, median, Q3, max in standard classroom texts.
- The definition and reading of box plots match widely used learning resources.
- IQR is defined as Q3 − Q1 and describes the middle 50% spread.
- We used consistent, worked examples to test each step (sort → median → halves → Q1/Q3).
Use this the right way
Use this guide to learn the method and check your understanding.
If this is for an assignment or project, write your own steps in your own words.
If you include definitions in a report, cite your sources and avoid copy-paste.
Study tools that can help
- If you get distracted easily: use one clean page per problem and box each step (sorted list, median, halves, Q1/Q3).
- If your work gets messy: a structured notebook may help you keep the five numbers clear: Best High-Quality Notebooks for Math Notes.
- If you’re time-crunched: compute one example by hand, then check with the Descriptive Statistics Calculator (All-in-One).
References
Khan Academy. (n.d.). Identifying outliers with the 1.5×IQR rule. https://www.khanacademy.org/math/statistics-probability/summarizing-quantitative-data/box-whisker-plots/a/identifying-outliers-iqr-rule
OpenStax. (2023). Introductory Statistics 2e: 2.4 Box plots. https://openstax.org/books/introductory-statistics-2e/pages/2-4-box-plots
OpenStax. (2023). Introductory Statistics 2e: 2.3 Measures of the location of the data. https://openstax.org/books/introductory-statistics-2e/pages/2-3-measures-of-the-location-of-the-data
