Box Plots

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Concept Review

The 5-Number Summary: Finding the Story Hidden in Data

Imagine you're the coach of a basketball team, and 15 players just finished running sprints. Their times are all over the place: some blazingly fast, others struggling to finish. How do you quickly understand what happened? The 5-number summary tells you the complete story in just five key numbers.

Every dataset has five crucial landmarks that reveal its shape and spread. Think of these like the most important stops on a road trip — they tell you where the journey begins, where it pauses, and where it ends.

The Five Essential Numbers

Let's use our basketball sprint times (in seconds): 12, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 25, 27, 30, 35

🏁

Minimum

Fastest time

🥉

25th percentile

🎯

Median

Middle value

🥈

75th percentile

🐌

Maximum

Slowest time

Finding the Quartiles

The quartiles (Q1 and Q3) split your data into four equal groups. With 15 players, the median is the 8th value (20 seconds). Q1 is the median of the bottom half (positions 1-7), which is the 4th value: 16 seconds. Q3 is the median of the top half (positions 9-15), which is the 12th value: 25 seconds.

🔑 Key Insight

The 5-number summary doesn't care about every single data point — it focuses on position, not specific values. You could change the slowest time from 35 to 100 seconds, and four of your five summary numbers would stay exactly the same! This makes it incredibly stable and reliable for comparing datasets.

Reading the Story

Our summary (12, 16, 20, 25, 35) reveals that most players clustered between 16-25 seconds, with one standout fast runner and one player who struggled significantly. The gaps between numbers show us where the data spreads out or bunches together.

Key Takeaway: Just like a coach can instantly understand team performance from those five sprint times, the 5-number summary transforms any messy dataset into a clear, meaningful story. These five landmarks become the foundation for creating box plots and comparing different groups of data.

Sample questions

1. What are the five numbers in a 5-number summary?

○ Mean, Median, Mode, Range, MAD

○ 1, 2, 3, 4, 5

✓ Min, Q1, Median, Q3, Max

○ Min, Max, Mean, Q1, Q2

Answer: Min, Q1, Median, Q3, Max — These five points divide the data into four equal-sized groups.

2. How do you find Q1?

✓ It is the median of the lower half of the data

○ It is the smallest number

○ It is the mean of the data

○ It is the first number in the list

Answer: It is the median of the lower half of the data — Q1 is the "median of the bottom half".

3. Find the 5-number summary for: 2, 4, 6, 8, 10, 12, 14.

○ 2, 4, 8, 12, 14

✓ 2, 4, 8, 12, 14? Min=2, Median=8, Q1=4, Q3=12, Max=14

○ 2, 4, 6, 8, 14

○ 2, 6, 8, 10, 14

Answer: 2, 4, 8, 12, 14? Min=2, Median=8, Q1=4, Q3=12, Max=14 — Correctly identifies all five boundary points.

Skills in this topic

Find the 5-number summary of a data set (Minimum, Q1, Median, Q3, Maximum)
Create a box plot (box-and-whisker plot) on a number line
Read and interpret quartiles from a given box plot
Compare two different sets of data using side-by-side box plots
Connect the visual length of the "box" to the Interquartile Range (IQR)

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