Outliers in Data

🎯 Outliers in Data

Part 1 of 5 — What Is an Outlier?

Topics in This Part

🔑 Key Concept: An outlier is a data value that lies unusually far from the rest of the data. One stray value can quietly distort an average, stretch a graph, and lead to wrong conclusions — so learning to spot and handle outliers is a core data skill.

The Idea of an Outlier

Imagine seven students report how many minutes it took them to get to school:

$12,\; 14,\; 15,\; 16,\; 18,\; 19,\; 47$

Six of the values huddle between $12$ and $19$ minutes. Then there's a $47$ — more than double the next-largest value. That lonely $47$ is an outlier: a value that doesn't fit the pattern of the rest.

Outliers can be:

• Real and important — maybe that student walked from across town, or a sensor caught a genuine spike.
• Mistakes — a typo ($47$ instead of $17$), a broken instrument, or the wrong units.

💡 An outlier is not automatically "bad data." It is simply a value worth a second look. The job of statistics is to flag it; the job of a thoughtful person is to investigate why it's there.

Concept Check 🎯

Why a Single Value Can Mislead

Return to the commute times. Their mean (average) is:

$\frac{12+14+15+16+18+19+47}{7} = \frac{141}{7} \approx 20.1 \text{ minutes}$

But look — six of the seven students got to school in under $20$ minutes! The "average" of $20.1$ is larger than almost every actual value, all because of one $47$.

If we set the outlier aside, the other six average:

$\frac{12+14+15+16+18+19}{6} = \frac{94}{6} \approx 15.7 \text{ minutes}$

That $15.7$ is a far more honest summary of a typical commute.

⚠️ The mean is pulled toward outliers. A single extreme value can drag the average up or down until it no longer represents the group. We will study this "tug-of-war" carefully in Part 4.

See the Pull 🧮

A small shop records the ages of $5$ customers:

$19,\; 21,\; 22,\; 24,\; 64$

1) Mean of all five ages $= \,?$ (it divides evenly) 2) Mean of just the first four ages $\;19, 21, 22, 24\;$ (the outlier removed) $= \,?$

Spotting by Eye Is Not Enough

In the examples above, the outliers were obvious. But what about a value that is somewhat large — is $32$ an outlier in a data set that runs $10$ to $28$? Eyeballing it is risky and subjective.

We need an objective rule that any two people would apply the same way. There are two standard ones:

🔑 Both rules turn the fuzzy phrase "unusually far" into a precise calculation. Part 2 tackles the most common one — the $1.5 \times \text{IQR}$ rule.

First Ideas 🔽

Lock in the vocabulary from Part 1.

🎯 Outliers in Data

Part 2 of 5 — The $1.5 \times \text{IQR}$ Rule

🔑 The Idea: Measure the spread of the middle half of the data with the IQR, then build two fences. Any value beyond a fence is officially an outlier.

Step 1 — Find the Quartiles and the IQR

The interquartile range (IQR) is the spread of the middle $50\%$ of the data:

$\text{IQR} = Q_3 - Q_1$

where $Q_1$ (the first quartile) is the median of the lower half of the sorted data and $Q_3$ (the third quartile) is the median of the upper half.

Quick refresher

For sorted data $\;3,\, 7,\, 8,\, 12,\, 13,\, 14,\, 15,\, 16,\, 17,\, 38\;$ ($n = 10$):

• The median is the average of the $5^{\text{th}}$ and $6^{\text{th}}$ values: $\frac{13+14}{2} = 13.5$.
• Lower half $\{3, 7, 8, 12, 13\}$ → $Q_1 = 8$ (the middle one).
• Upper half $\{14, 15, 16, 17, 38\}$ → $Q_3 = 16$.

$\text{IQR} = Q_3 - Q_1 = 16 - 8 = 8$

💡 The IQR ignores the extremes on purpose, so it is a stable measure of spread — exactly what we want when hunting the extremes.

Step 2 — Build the Fences

Multiply the IQR by $1.5$ and step that far outside each quartile:

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

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

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

Finishing our example

With $Q_1 = 8$, $Q_3 = 16$, and $\text{IQR} = 8$:

$\text{Lower fence} = 8 - 1.5(8) = 8 - 12 = -4$ $\text{Upper fence} = 16 + 1.5(8) = 16 + 12 = 28$

Now scan the data $\;3, 7, 8, 12, 13, 14, 15, 16, 17, 38$. Is anything below $-4$ or above $28$? Only the $38$ clears the upper fence — $38$ is an outlier. ✓

⚠️ A value exactly on a fence is not beyond it, so it is not flagged. Only values strictly past a fence count.

Concept Check 🎯

Build the Fences 🧮

Monthly rainfall (cm), sorted, with the five-number summary min $=2$, $Q_1 = 6$, median $=9$, $Q_3 = 14$, max $=31$.

1) $\text{IQR} = Q_3 - Q_1 = \,?$ 2) Upper fence $= Q_3 + 1.5 \cdot \text{IQR} = \,?$ 3) Lower fence $= Q_1 - 1.5 \cdot \text{IQR} = \,?$

Putting the Fences to Work

Once you have both fences, classifying every value is a simple scan: walk through the data and ask, for each one, "Is it below the lower fence or above the upper fence?"

Remember the boundary detail from the worked example:

⚠️ Strictly beyond. A value has to be past a fence to count. A value sitting exactly on a fence is inside the gate, so it is not an outlier.

The drill below gives you the fences directly — your only job is the verdict.

Outlier or Not? 🔽

A data set has $Q_1 = 40$ and $Q_3 = 70$, so $\text{IQR} = 30$. The fences are lower $= 40 - 1.5(30) = -5$ and upper $= 70 + 1.5(30) = 115$. For each value, decide.

🎯 Outliers in Data

Part 3 of 5 — The Standard-Deviation Rule

🔑 A second test: Instead of quartiles, measure how many standard deviations a value sits from the mean. If it is more than about $2$ away, it is unusually far out.

Measuring Distance in Standard Deviations

The standard deviation $\sigma$ measures the typical distance of values from the mean $\bar{x}$. We can rewrite any value as a number of standard deviations away from the mean — called its $z$-score:

$z = \frac{x - \bar{x}}{\sigma}$

• $z = 0$ means the value sits exactly at the mean.
• $z = 1$ means it is one standard deviation above the mean.
• $z = -2$ means it is two standard deviations below the mean.

🔑 The rule: A common cutoff flags a value as an outlier when its $z$-score satisfies $|z| > 2$ (some textbooks use the stricter $|z| > 3$). The farther $|z|$ is from $0$, the more unusual the value.

Worked Example: Test Scores

A class has a mean score of $\bar{x} = 70$ with a standard deviation of $\sigma = 5$. Using the $|z| > 2$ rule, check three scores.

A score of $82$: $z = \frac{82 - 70}{5} = \frac{12}{5} = 2.4$ Since $|2.4| > 2$, the $82$ is an outlier (unusually high). ✓

A score of $78$: $z = \frac{78 - 70}{5} = \frac{8}{5} = 1.6$ Since $|1.6| < 2$, the $78$ is not an outlier — just a good score.

A score of $58$: $z = \frac{58 - 70}{5} = \frac{-12}{5} = -2.4$ Since $|-2.4| > 2$, the $58$ is an outlier (unusually low). ✓

💡 The $z$-score is a "universal ruler." A score of $82$ might be ordinary in one class and extraordinary in another — the $z$-score accounts for how spread out each class actually is.

Concept Check 🎯

Compute the $z$-score 🧮

A factory fills bottles with a mean of $\bar{x} = 500$ mL and a standard deviation of $\sigma = 8$ mL.

1) A bottle holds $516$ mL. Its $z$-score is $\,?$ 2) A bottle holds $480$ mL. Its $z$-score is $\,?$ 3) Using $|z| > 2$, how many of these two bottles are outliers? $\,?$

Two Rules, Side by Side

You now own two outlier tests. They usually agree, but they are built from different ingredients:

💡 Because the IQR rule leans on the median and quartiles, it is itself resistant to outliers — which is why it's the go-to for messy, skewed data. The next check makes sure you can tell the two rules apart.

Two Rules, One Goal 🔽

Match each phrase to the rule it describes.

🎯 Outliers in Data

Part 4 of 5 — How Outliers Affect Statistics

🔑 Big Idea: Some statistics get yanked around by outliers and some shrug them off. Knowing which is which tells you the right numbers to report when an outlier is present.

Resistant vs. Non-Resistant

A statistic is resistant (or robust) if a single outlier barely changes it. It is non-resistant if an outlier can move it a lot.

🔑 Rule of thumb: When a data set contains an outlier, report the median for center and the IQR for spread. They describe the typical value honestly.

Worked Example: Adding One Outlier

Start with five values:

$2,\; 4,\; 5,\; 6,\; 8$

• Mean $= \dfrac{2+4+5+6+8}{5} = \dfrac{25}{5} = 5$
• Median $= 5$ (the middle value)

Now toss in a single outlier, $50$:

$2,\; 4,\; 5,\; 6,\; 8,\; 50$

• Mean $= \dfrac{2+4+5+6+8+50}{6} = \dfrac{75}{6} = 12.5$
• Median $= \dfrac{5+6}{2} = 5.5$ (average of the two middle values)

⚠️ The mean leaped by $7.5$, while the median barely budged. This is exactly why a report that hides an outlier behind "the average" can be so misleading.

Concept Check 🎯

Measure the Damage 🧮

A study group's quiz scores:

$70,\; 72,\; 74,\; 76,\; 78$

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

Now a sixth member scores an $8$, giving $\;8,\, 70,\, 72,\, 74,\, 76,\, 78$.

3) New mean $= \,?$ (it divides evenly) 4) New median $= \,?$

Why Some Statistics Resist and Others Don't

The pattern in that drill is no accident. It comes down to what each statistic actually uses:

• The mean sums every value, so an extreme number changes the total — and the average shifts.
• The range is built from the very smallest and largest values, so an outlier is one of its inputs.
• The median depends only on the middle position, so a far-off value just sits at the end without moving it.
• The IQR uses the quartiles — the edges of the middle half — and ignores the extremes entirely.

🔑 Memory hook: statistics built from positions in the middle (median, IQR) resist outliers; statistics built from every value or the extremes (mean, range, standard deviation) do not.

Use the next check to sort each statistic into the right bucket.

Resistant or Not? 🔽

For each statistic, choose whether a single outlier changes it a lot or barely at all.

🎯 Outliers in Data

Part 5 of 5 — Causes, Decisions & Mastery Check

You can now (1) recognize an outlier, (2) flag one with the $1.5 \times \text{IQR}$ rule, (3) flag one with the $z$-score rule, and (4) describe how outliers warp the mean, range, and standard deviation. The last skill is judgment: what to do once you've found one.

Where Outliers Come From — and What to Do

⚠️ Never delete an outlier just because it is inconvenient. Removing real data to make a chart look tidy is a form of dishonesty. Investigate the cause first; only remove a value if you have good reason to believe it is an error.

💡 A good report often shows the analysis both ways — with and without the outlier — and explains the difference, so the reader can judge for themselves.

Concept Check 🎯

Full Analysis 🧮

Ages of people at a small workshop, sorted, with $n = 9$:

$12,\; 13,\; 14,\; 15,\; 16,\; 17,\; 18,\; 19,\; 52$

The five-number summary is min $=12$, $Q_1 = 13.5$, median $=16$, $Q_3 = 18.5$, max $=52$.

1) $\text{IQR} = Q_3 - Q_1 = \,?$ 2) Upper fence $= Q_3 + 1.5 \cdot \text{IQR} = \,?$ (decimal is fine) 3) How many values are outliers? $\,?$

Your Outlier Toolkit

Before the final practice, here is everything in one place:

⚠️ A value exactly on a fence (or with $|z|$ exactly $2$) is not an outlier — it has to be strictly beyond the cutoff.

Mixed Practice 🎯

One Last Word

Finding an outlier is the beginning of the story, not the end. A flagged value is an invitation to ask "why?" — a typo, a broken sensor, or a genuinely remarkable event. The math points at the unusual value; you decide what it means.

Three questions ahead to lock it all in.

Exit Quiz ✅

Answer all three to finish the lesson.