Box Plots

📦 Box Plots

Part 1 of 5 — The Five-Number Summary

Topics in This Part

🔑 Key Concept: A box plot is a picture of five numbers: the minimum, $Q_1$, the median, $Q_3$, and the maximum. Master those five numbers in Part 1 and the picture practically draws itself in Part 2.

Why Box Plots Exist

Suppose a class earns these quiz scores:

$4,\; 7,\; 8,\; 11,\; 13,\; 16,\; 19$

You could stare at the raw list — but a box plot instead summarizes the whole data set with a handful of landmark values, so you can see the center, the spread, and any lopsidedness at a glance.

To build one, we first split the data into four equal groups using three cut-points:

• the median splits the data into a lower half and an upper half,
• the first quartile $Q_1$ is the median of the lower half,
• the third quartile $Q_3$ is the median of the upper half.

Together with the min and max, these are the five-number summary.

💡 "Quartile" comes from quarter. The three quartiles ($Q_1$, the median $Q_2$, and $Q_3$) chop the sorted data into four slices, each holding about $25\%$ of the values.

Step 1 — Sort, Then Find the Median

Always sort the data from smallest to largest first. Our scores are already sorted:

$4,\; 7,\; 8,\; \underbrace{11}_{\text{median}},\; 13,\; 16,\; 19$

There are $n = 7$ values (odd), so the median is the single middle value — the $4^{\text{th}}$ one: $11$.

The minimum is $4$ and the maximum is $19$.

What if there's an even count?

For $\;3,\; 5,\; 8,\; 9,\; 12,\; 14\;$ ($n=6$), there is no single middle value. Average the two middle numbers ($8$ and $9$):

$\text{median} = \frac{8 + 9}{2} = 8.5$

⚠️ The median is not always a number that appears in the data. With an even count it can land between two values.

Concept Check 🎯

Step 2 — Find the Quartiles

Go back to the scores $\;4,\, 7,\, 8,\, \mathbf{11},\, 13,\, 16,\, 19\;$ with median $11$.

When $n$ is odd, the median is one specific data value, so we leave it out when splitting into halves:

$\underbrace{4,\; 7,\; 8}_{\text{lower half}}\quad\Big|\quad \mathbf{11} \quad\Big|\quad \underbrace{13,\; 16,\; 19}_{\text{upper half}}$

• $Q_1$ = median of the lower half $\;\{4, 7, 8\}$ = $\mathbf{7}$
• $Q_3$ = median of the upper half $\;\{13, 16, 19\}$ = $\mathbf{16}$

🔑 The rule: $Q_1$ is the median of all values below the overall median; $Q_3$ is the median of all values above it. If $n$ is odd, exclude the median itself from both halves. If $n$ is even, the two halves are simply the lower $\frac{n}{2}$ and upper $\frac{n}{2}$ values.

Match Each Landmark 🔽

For the sorted data $\;2,\; 4,\; 6,\; 8,\; 10\;$ ($n = 5$), choose the correct value for each statistic.

The Five-Number Summary

Putting it all together for the quiz scores $\;4,\,7,\,8,\,11,\,13,\,16,\,19$:

These five numbers — and only these five — are everything a box plot shows. One more bit of practice, then we'll turn them into the picture.

💡 Always list a five-number summary in increasing order. If yours isn't increasing (e.g. $Q_1 >$ median), you made an arithmetic slip — go back and recheck.

Find the Quartiles 🧮

Daily high temperatures (°F), already sorted:

$64,\; 66,\; 68,\; 70,\; 72,\; 74,\; 76$

1) Median $= \,?$ 2) $Q_1 = \,?$ 3) $Q_3 = \,?$

📦 Box Plots

Part 2 of 5 — Building the Box & Whiskers

🔑 The Idea: Draw a number line, mark the five summary values, build a box from $Q_1$ to $Q_3$ with a line at the median, then extend whiskers out to the min and max.

Anatomy of a Box Plot

Every box plot is built from the five-number summary like this:

A rough sketch on a number line:

$\overset{\text{min}}{|}\!\!-\!\!-\!\!-\!\!\overset{Q_1}{[}\;\overset{\text{med}}{|}\;\overset{Q_3}{]}\!\!-\!\!-\!\!-\!\!\overset{\text{max}}{|}$

• The box spans $Q_1$ to $Q_3$ and holds the middle $50\%$ of the data.
• The whiskers reach out to the smallest and largest values.

💡 The width of the box is the most important visual: a wide box means the middle half of the data is spread out; a narrow box means it's tightly clustered.

The Interquartile Range (IQR)

The length of the box has a name — the interquartile range:

$\text{IQR} = Q_3 - Q_1$

It measures the spread of the middle $50\%$ of the data and ignores the extremes entirely.

Worked Example

A data set has the five-number summary

$4,\; 7,\; 12,\; 16,\; 20 \qquad (\text{min},\, Q_1,\, \text{med},\, Q_3,\, \text{max}).$

$\text{IQR} = Q_3 - Q_1 = 16 - 7 = 9$

$\text{Range} = \text{max} - \text{min} = 20 - 4 = 16$

⚠️ Don't confuse range (max $-$ min, the whole spread) with IQR (the spread of just the middle half). The IQR is much less affected by extreme values.

Concept Check 🎯

From Summary to IQR

So the full recipe for the box's width is: find $Q_1$ and $Q_3$ exactly as in Part 1, then subtract. That single number, the IQR, is the headline measure of spread for a box plot.

Time to run the whole pipeline yourself — sort is done for you, so go straight to the median, the quartiles, and the IQR.

Build the Summary 🧮

These reaction times (ms), already sorted:

$4,\; 6,\; 7,\; 9,\; 10,\; 12,\; 13,\; 15,\; 16,\; 18,\; 20$

There are $n = 11$ values. Find each value for the box plot.

1) Median $= \,?$ 2) $Q_1 = \,?$ 3) $Q_3 = \,?$ 4) IQR $= \,?$

Putting the Picture Together

With those five numbers, the box plot for the reaction times draws itself: the box runs from $Q_1 = 7$ to $Q_3 = 16$ (a width of $9$, the IQR), a line marks the median at $12$, and the whiskers reach left to $4$ and right to $20$.

💡 Notice the median line ($12$) is not centered in the box — it sits a little right of middle. That tells you the lower quarter of the box is more spread out than the upper quarter. We'll read clues like this in Part 4.

Before moving on, make sure you can name every part of the picture.

Label the Picture 🔽

A box plot is drawn on a number line. Match each feature of the plot to what it represents.

📦 Box Plots

Part 3 of 5 — Outliers & the 1.5 × IQR Rule

🔑 Why it matters: A single freakishly large or small value can stretch a whisker and distort the whole picture. The $1.5 \times \text{IQR}$ rule gives an objective test for flagging these outliers.

The Fences

An outlier is a value that lies unusually far from the middle of the data. To decide how far is "too far," we build two fences:

$\text{Lower fence} = Q_1 - 1.5 \cdot \text{IQR}$

$\text{Upper fence} = Q_3 + 1.5 \cdot \text{IQR}$

🔑 The rule: Any data value below the lower fence or above the upper fence is an outlier. Everything between the fences is considered ordinary.

The fences themselves are not drawn on the box plot — they're just the calculation that decides which points get flagged.

Worked Example: Morning Commute Times

A worker records commute times (minutes), sorted:

$15,\; 18,\; 20,\; 22,\; 25,\; 28,\; 30,\; 55$

Step 1 — Five-number summary. With $n=8$ (even), the median is $\frac{22+25}{2} = 23.5$. The lower half is $\{15,18,20,22\}$ so $Q_1 = \frac{18+20}{2} = 19$; the upper half is $\{25,28,30,55\}$ so $Q_3 = \frac{28+30}{2} = 29$.

Step 2 — IQR. $\text{IQR} = Q_3 - Q_1 = 29 - 19 = 10$

Step 3 — Fences. $\text{Lower} = 19 - 1.5(10) = 19 - 15 = 4$ $\text{Upper} = 29 + 1.5(10) = 29 + 15 = 44$

Step 4 — Flag outliers. Is any value below $4$ or above $44$? The $55$-minute commute is above $44$, so $55$ is an outlier. ✓

💡 When a data set has an outlier, the whisker stops at the most extreme value that is still inside the fence (here, $30$), and the outlier is drawn as a separate dot.

Concept Check 🎯

Your Outlier Checklist

For any data set, hunting outliers is always the same four moves:

1. Find $Q_1$ and $Q_3$, then $\text{IQR} = Q_3 - Q_1$.
2. Lower fence $= Q_1 - 1.5\cdot\text{IQR}$.
3. Upper fence $= Q_3 + 1.5\cdot\text{IQR}$.
4. Any value beyond either fence is an outlier.

Now apply all four steps to a real data set in the drill below.

Hunt the Outlier 🧮

Test scores, sorted, with $n = 9$:

$8,\; 10,\; 11,\; 13,\; 14,\; 15,\; 16,\; 18,\; 40$

The five-number summary is min $=8$, $Q_1 = 10.5$, median $=14$, $Q_3 = 17$, max $=40$.

1) IQR $= \,?$ 2) Upper fence $= Q_3 + 1.5 \cdot \text{IQR} = \,?$ (decimal is fine) 3) How many values are outliers? $\,?$

A Value On the Fence

The fences draw a hard line: a value counts as an outlier only if it is strictly beyond a fence — farther out than the fence itself.

• A value greater than the upper fence → outlier.
• A value less than the lower fence → outlier.
• A value equal to a fence → not an outlier (it's right on the boundary, still inside).

⚠️ A common trap is to flag a point that lands exactly on a fence. Don't — equality is not "beyond." Keep this in mind for the next check.

Decide: Outlier or Not? 🔽

A data set has $Q_1 = 30$ and $Q_3 = 50$ (so IQR $= 20$). The fences are: lower $= 30 - 1.5(20) = 0$ and upper $= 50 + 1.5(20) = 80$. For each value, choose whether it's an outlier.

📦 Box Plots

Part 4 of 5 — Reading & Comparing Box Plots

🔑 Big Payoff: Box plots really shine when you stack two of them on the same number line to compare groups — which has the higher center, which is more spread out, which is skewed.

Reading a Single Box Plot

You can recover the entire five-number summary just by reading positions off the axis:

Spotting Skew

The median line is rarely centered in the box. Its position tells you about skew:

• If the right side of the box and whisker is longer (median sits left of center), the data is skewed right (a few large values stretch it).
• If the left side is longer, the data is skewed left.
• If both sides look balanced, the data is roughly symmetric.

💡 The box plot hides how many points sit at each spot — it only shows the five landmarks. So two very different data sets can share the same box plot. It's a summary, not a full picture.

Concept Check 🎯

Comparing Two Box Plots

Two classes take the same test. Their five-number summaries:

How to compare them fairly:

• Center (median): Class B's median ($78$) is higher than Class A's ($75$) — B's typical score is a bit higher.
• Spread (IQR): Class A's IQR is $85 - 65 = 20$; Class B's is $90 - 60 = 30$. Class B's middle half is more spread out (less consistent).
• Overall range: A spans $90 - 60 = 30$; B spans $95 - 55 = 40$. Again B is wider.

⚠️ Always compare center with center (median vs. median) and spread with spread (IQR vs. IQR). Don't compare one group's median to another group's maximum — that's not a fair comparison.

Compare the Classes 🧮

Use the Class A and Class B summaries from above.

1) Class A's IQR $= \,?$ 2) Class B's IQR $= \,?$ 3) Class B's median minus Class A's median $= \,?$

Turning Numbers into Conclusions

The calculations only matter once you translate them into plain-language claims about the two groups:

• Higher median → higher typical score. Class B's median ($78$) tops Class A's ($75$), so a typical Class B student scored a bit higher.
• Smaller IQR → more consistent. Class A's IQR ($20$) is smaller than Class B's ($30$), so Class A's scores were more tightly clustered.

💡 A group can be better on average yet less consistent — exactly what we see here. Class B is higher and more spread out. Center and spread are two separate questions, so always answer both.

Interpret the Comparison 🔽

Still using Class A (median $75$, IQR $20$) and Class B (median $78$, IQR $30$). Choose the best conclusion in each case.

📦 Box Plots

Part 5 of 5 — Mixed Practice & Mastery Check

You can now (1) find a five-number summary, (2) build a box plot and its IQR, (3) flag outliers with the $1.5 \times \text{IQR}$ rule, and (4) read and compare plots. Let's put it all together.

Quick Reference

⚠️ Remember: sort first, exclude the median from the halves when $n$ is odd, and a value exactly on a fence is not an outlier.

Mixed Practice 🎯

The Full Method, Start to Finish

Here is the complete workflow for any data set, in order:

1. Sort the values from smallest to largest.
2. Find the median, then $Q_1$ and $Q_3$ (medians of the lower and upper halves).
3. Record the min and max for the whole five-number summary.
4. Compute $\text{IQR} = Q_3 - Q_1$.
5. Build fences at $Q_1 - 1.5\cdot\text{IQR}$ and $Q_3 + 1.5\cdot\text{IQR}$ and flag any outliers.

The next drill walks you through every one of these steps on a fresh data set.

Full Analysis 🧮

Hours of sleep recorded for $10$ students, sorted:

$4,\; 6,\; 7,\; 7,\; 8,\; 10,\; 12,\; 14,\; 15,\; 21$

The five-number summary is min $=4$, $Q_1 = 7$, median $=9$, $Q_3 = 14$, max $=21$.

1) IQR $= \,?$ 2) Upper fence $= Q_3 + 1.5 \cdot \text{IQR} = \,?$ (decimal is fine) 3) Is the maximum value $21$ an outlier? Enter yes or no.

You're Ready

That last drill used every skill in this lesson at once — five-number summary, IQR, fences, and an outlier decision. If it felt routine, you've got box plots down.

🔑 One-line summary: A box plot is the five-number summary made visible — a box of the middle $50\%$, whiskers to the extremes, and dots for any value past the $1.5\times\text{IQR}$ fences.

Finish with the exit quiz below.

Exit Quiz ✅

Answer all three to finish the lesson.