Range and Outliers

📏 Range and Outliers

Part 1 of 5 — Range: Measuring Spread

Topics in This Part

🔑 Key Concept: Two data sets can have the same average but feel completely different. The range is the simplest way to measure how spread out a data set is — that's where we start.

What Does "Spread" Mean?

Compare two basketball players who each average 20 points per game over five games:

Same average — but Wendy is far less consistent. Her scores are spread out; Sam's are bunched together.

A measure of spread captures this difference. The easiest one is the range.

💡 Center vs. Spread: The mean and median tell you where data is centered. The range tells you how far apart the values are. You need both to describe a data set well.

Maximum, Minimum, and Range

The range is the distance from the smallest value to the largest:

$\text{Range} = \text{Maximum} - \text{Minimum}$

To find it:

1. Find the maximum (largest value).
2. Find the minimum (smallest value).
3. Subtract: maximum $-$ minimum.

Worked Example

Data set: $7,\ 3,\ 9,\ 12,\ 5$

Sorting helps you spot the ends: $3,\ 5,\ 7,\ 9,\ 12$.

• Maximum $= 12$
• Minimum $= 3$

$\text{Range} = 12 - 3 = 9$

⚠️ The range is a single number, not an interval. Write the answer as $9$, not "$3$ to $12$."

Identify the Pieces 🔽

Sorted data set: $14,\ 22,\ 9,\ 31,\ 18,\ 6$ → sorted: $6,\ 9,\ 14,\ 18,\ 22,\ 31$.

Range in Context

Range works for any kind of numeric data — temperatures, prices, test scores, heights.

Example — Daily high temperatures (°F): $68,\ 72,\ 65,\ 80,\ 71$

• Maximum $= 80$, Minimum $= 65$
• $\text{Range} = 80 - 65 = 15$

So the temperature varied by 15 °F across the week.

💡 Notice the range is fast and intuitive, but it uses only two values — the highest and the lowest. Keep that fragility in mind as you practice.

Concept Check 🎯

Find the Range 🧮

Compute the range (maximum $-$ minimum) of each data set.

1) $5,\ 8,\ 2,\ 11,\ 7$ 2) $20,\ 15,\ 28,\ 12,\ 19,\ 25$ 3) $100,\ 45,\ 67,\ 89$

Looking Ahead

You can now measure spread with the range. But remember: because it depends on only the two extreme values, a single unusual data point can change it dramatically.

🔑 Takeaway: That fragile point is exactly what Part 2 is about — outliers, the values that lie far from the rest.

📏 Range and Outliers

Part 2 of 5 — Spotting Outliers

🔑 The Idea: An outlier is a value that lies far away from the rest of the data. Outliers can be real surprises, data-entry typos, or rare events — and they can quietly distort your statistics.

What Is an Outlier?

An outlier is a data value that is unusually far from the others.

Example — quiz scores (out of 20):

$12,\ 14,\ 15,\ 16,\ 18,\ 19,\ 75$

Wait — a quiz is out of 20, so $75$ is impossible. It's almost certainly a typo for $7.5$ or $17$. Even before any formula, your eye flags it: there's a big gap between $19$ and $75$.

When you sort data and one value sits far past a noticeable gap, that value is a likely outlier.

💡 Outliers aren't always errors. A genuinely tall student, a record-breaking sale, or an unusually cold day are real outliers worth keeping. The goal is to notice them and decide what they mean.

How One Outlier Wrecks the Range

Because the range uses only the max and min, a single extreme value can blow it up.

Using the quiz scores $12,\ 14,\ 15,\ 16,\ 18,\ 19,\ 75$:

One bad value changed the range from a sensible 7 to a misleading 63.

⚠️ This is why range alone can mislead. A statistic that is barely affected by outliers is called resistant — we'll meet resistant measures in Part 4.

Concept Check 🎯

Outlier vs. Range 🧮

For each set, find the range with the outlier, then the range without it (drop the bold value).

1) $22,\ 24,\ 25,\ 27,\ 29,\ 30,\ \mathbf{88}$    Range with $88 = \,?$    Range without $88 = \,?$

Read the Data 🔽

Sorted heights (cm) of a small team: $150,\ 152,\ 153,\ 155,\ 156,\ 190$.

📏 Range and Outliers

Part 3 of 5 — The 1.5 × IQR Rule

🔑 Why a rule? "It looks far away" isn't precise. Statisticians use the interquartile range (IQR) to draw exact boundaries — called fences — so an outlier is defined, not guessed.

Quartiles and the IQR

Quartiles split sorted data into four equal parts:

• $Q_1$ (first quartile) — the median of the lower half.
• $Q_2$ — the overall median.
• $Q_3$ (third quartile) — the median of the upper half.

The interquartile range is the spread of the middle 50% of the data:

$\text{IQR} = Q_3 - Q_1$

💡 Unlike the range, the IQR ignores the extreme ends and looks only at the middle — that's what makes it resistant to outliers and perfect for detecting them.

Concept Check 🎯

Worked Example — Finding the IQR

Data (already sorted, $8$ values): $4,\ 8,\ 11,\ 13,\ 15,\ 18,\ 21,\ 24$

Step 1 — Split into halves (even count, so split down the middle):

• Lower half: $4,\ 8,\ 11,\ 13$
• Upper half: $15,\ 18,\ 21,\ 24$

Step 2 — Median of each half:

$Q_1 = \frac{8 + 11}{2} = 9.5 \qquad Q_3 = \frac{18 + 21}{2} = 19.5$

Step 3 — Subtract:

$\text{IQR} = Q_3 - Q_1 = 19.5 - 9.5 = 10$

⚠️ Odd number of values? Leave the middle value out of both halves when finding $Q_1$ and $Q_3$. (E.g., for $2,5,6,9,12,14,21$ the median $9$ is skipped: $Q_1 = 5$, $Q_3 = 14$.)

The Fences: 1.5 × IQR

A value is an outlier if it falls outside the fences:

$\text{Lower fence} = Q_1 - 1.5 \times \text{IQR}$ $\text{Upper fence} = Q_3 + 1.5 \times \text{IQR}$

Any value below the lower fence or above the upper fence is an outlier.

Example — detecting an outlier

Data: $3,\ 7,\ 8,\ 9,\ 10,\ 11,\ 12,\ 40$ → $Q_1 = 7.5$, $Q_3 = 11.5$, so $\text{IQR} = 4$.

$\text{Upper fence} = 11.5 + 1.5 \times 4 = 11.5 + 6 = 17.5$ $\text{Lower fence} = 7.5 - 6 = 1.5$

Is $40$ an outlier? Since $40 > 17.5$, yes — $40$ lies above the upper fence. ✓

Build the Fences 🔽

For a data set, $Q_1 = 20$ and $Q_3 = 32$.

Find the IQR 🧮

For each sorted set, find $Q_1$, $Q_3$, and the IQR.

1) $3,\ 5,\ 7,\ 9,\ 11,\ 13,\ 15,\ 17$    $Q_1 = \,?$    $Q_3 = \,?$    $\text{IQR} = \,?$

Test the Fence 🧮

A data set has $Q_1 = 8$, $Q_3 = 13$ (so $\text{IQR} = 5$). The data includes the value $30$.

1) Upper fence $= Q_3 + 1.5 \times \text{IQR} = \,?$ 2) Is $30$ above the upper fence? Enter 1 for yes (outlier) or 0 for no.

📏 Range and Outliers

Part 4 of 5 — How Outliers Distort Statistics

🔑 Big Idea: Outliers pull some statistics hard and barely touch others. Knowing which measures resist outliers tells you which number to trust.

Resistant vs. Non-Resistant

A statistic is resistant (or robust) if outliers barely change it.

💡 Rule of thumb: When a data set has outliers, report the median and IQR. When it's roughly symmetric with no outliers, the mean and range are fine.

Mean vs. Median Under an Outlier

Start with five values: $10,\ 12,\ 13,\ 14,\ 15$.

$\text{Mean} = \frac{10+12+13+14+15}{5} = \frac{64}{5} = 12.8 \qquad \text{Median} = 13$

Now add one outlier, $100$, giving $10,\ 12,\ 13,\ 14,\ 15,\ 100$:

$\text{Mean} = \frac{164}{6} \approx 27.3 \qquad \text{Median} = \frac{13+14}{2} = 13.5$

The mean more than doubled; the median barely moved. The median is resistant.

⚠️ A single outlier can drag the mean to a value that no longer represents a "typical" data point. That's why news reports often quote the median home price or income.

Concept Check 🎯

Measure the Impact 🧮

Start with $4,\ 5,\ 6,\ 7,\ 8$. Then add the outlier $50$ to get $4,\ 5,\ 6,\ 7,\ 8,\ 50$.

1) Mean of the original five values $= \,?$ 2) Mean after adding $50$ $= \,?$ (round to 2 decimals) 3) Median after adding $50$ $= \,?$

Resistant or Not? 🔽

Classify each statistic by whether an outlier strongly affects it.

📏 Range and Outliers

Part 5 of 5 — Mixed Practice & Mastery Check

You can now (1) compute the range, (2) spot outliers by gaps, (3) use the 1.5 × IQR fences to define outliers exactly, and (4) judge which statistics resist outliers. Time to put it together.

Quick Reference

⚠️ Top mistakes to avoid: writing range as an interval instead of a single number; forgetting to sort before finding quartiles; and using the mean/range when an outlier is present (use median/IQR instead).

Mixed Practice 🎯

Full Analysis 🧮

Sorted data: $40,\ 42,\ 45,\ 48,\ 50$.

1) Range $= \,?$ 2) Median $= \,?$

Now a different set has $Q_1 = 15$ and $Q_3 = 22$.

3) Upper fence $= Q_3 + 1.5 \times \text{IQR} = \,?$

Choose the Right Tool 🔽

Exit Quiz ✅

Answer all three to finish the lesson.