Range and Outliers

How spread out is data? How do we identify unusual values? Understanding range and outliers helps us analyze data variation and spot anomalies!

---

What Is Range?

Range measures how spread out the data is.

Range = Maximum value - Minimum value

Formula: Range = Max - Min

Range tells you the span of the data!

Example: Data: 5, 8, 12, 15, 20

Max = 20, Min = 5 Range = 20 - 5 = 15

The data spans 15 units!

---

Understanding Range

Range measures variability:

• Large range: Data is spread out
• Small range: Data is clustered together
• Range of 0: All values are the same

Example 1: 10, 12, 14, 16, 18 Range = 18 - 10 = 8

Example 2: 10, 10, 10, 10, 10 Range = 10 - 10 = 0

Different spreads!

---

Finding the Range

Step-by-step:

Step 1: Find the largest value (maximum) Step 2: Find the smallest value (minimum) Step 3: Subtract: Max - Min

Example: Test scores: 78, 85, 92, 88, 95, 81

Max = 95 Min = 78 Range = 95 - 78 = 17

Scores range across 17 points!

---

Range Units

Range has the SAME units as the data.

Temperature: 65°F, 70°F, 80°F, 85°F Range = 85 - 65 = 20°F (degrees)

Height: 150 cm, 165 cm, 172 cm Range = 172 - 150 = 22 cm

Always include units with your answer!

---

Comparing Ranges

Class A scores: 70, 75, 80, 85, 90 Range = 90 - 70 = 20

Class B scores: 78, 79, 80, 81, 82 Range = 82 - 78 = 4

Class A has more variability (larger range)! Class B is more consistent (smaller range)!

Range helps compare data sets!

---

Limitation of Range

Range only uses two values:

• Only considers max and min
• Ignores everything in between
• Sensitive to extreme values

Example: 10, 11, 12, 13, 50 Range = 50 - 10 = 40

One extreme value (50) makes range large! Doesn't show that most values are 10-13.

---

What Is an Outlier?

An outlier is a value that is much larger or much smaller than the other values.

Outlier: Unusually far from the rest of the data

Think: A value that "stands out" or seems unusual

Examples:

• Most ages are 10-12, but one is 45 (outlier!)
• Most test scores are 70-90, but one is 15 (outlier!)

---

Identifying Outliers Visually

Look at the data:

Data: 12, 15, 14, 13, 16, 45

Most values: 12-16 (clustered) One value: 45 (far away)

45 is an outlier!

Graph or number line makes this obvious!

---

Common Rule for Outliers

Simple rule: A value is an outlier if it's far from the rest

More specific:

• Much more than 1.5 times the range from the median
• Outside the "typical" cluster

For pre-algebra, visual inspection is usually enough!

Ask: "Does this value seem way different?"

---

Outliers Can Be High or Low

High outlier: Much larger than others

Example: 10, 12, 11, 13, 98 98 is a high outlier

Low outlier: Much smaller than others

Example: 85, 90, 88, 92, 15 15 is a low outlier

Outliers can be on either end!

---

Effect of Outliers on Range

Outliers increase the range dramatically!

Without outlier: 10, 12, 14, 16, 18 Range = 18 - 10 = 8

With outlier: 10, 12, 14, 16, 18, 50 Range = 50 - 10 = 40

Outlier multiplied the range by 5!

Range is very sensitive to outliers!

---

Effect of Outliers on Mean

Outliers pull the mean toward them!

Example: 20, 22, 24, 26, 100

Without 100: Mean = (20+22+24+26) ÷ 4 = 23

With 100: Mean = (20+22+24+26+100) ÷ 5 = 38.4

The outlier (100) increased mean from 23 to 38.4!

Mean is sensitive to outliers!

---

Effect of Outliers on Median

Median is resistant to outliers!

Example: 20, 22, 24, 26, 100

Median: 20, 22, 24, 26, 100 = 24

If we change 100 to 1000: Median: 20, 22, 24, 26, 1000 = 24 (same!)

Outlier doesn't affect median much!

This is why median is often better than mean!

---

Causes of Outliers

Measurement error:

• Typo entering data (18 typed as 180)
• Broken measuring tool
• Recording mistake

Natural variation:

• Actually an unusual event
• Rare but real occurrence

Different population:

• Adult in group of children
• Professional among amateurs

Must investigate to determine cause!

---

What to Do with Outliers

Option 1: Keep the outlier

• If it's a real value
• Part of the story
• Shows the full picture

Option 2: Remove the outlier

• If it's a measurement error
• When analyzing typical cases
• Report separately

Always explain what you did!

---

Real-World Example: Test Scores

Test scores: 85, 88, 90, 92, 87, 25

Range: 92 - 25 = 67

Analysis:

• Most scores: 85-92 (doing well!)
• One score: 25 (outlier - maybe student was sick?)

Without outlier: Range = 92 - 85 = 7 (much smaller!)

The outlier hides that most students did well!

---

Real-World Example: Home Prices

Home prices on a street: $200K,$220K, $230K,$210K, $2.5M

Range: $2.5M -$200K = $2.3M

Analysis:

• Most homes: $200K-$230K (similar)
• One home: $2.5M (mansion - outlier!)

Median price ($220K) better represents typical home!** **Mean price ($672K) inflated by mansion!

---

Multiple Outliers

Data can have more than one outlier!

Example: 5, 7, 8, 9, 10, 11, 50, 52

Two high outliers: 50 and 52 Main cluster: 5-11

Multiple outliers possible on same side or both sides!

---

Box Plot and Outliers

Box plots (box-and-whisker plots) show outliers!

Structure:

• Box shows middle 50% of data
• Whiskers extend to min/max (within limits)
• Outliers shown as individual points beyond whiskers

Visual way to identify outliers!

---

Interquartile Range (IQR) Preview

IQR is another measure of spread:

• More resistant to outliers than range
• Uses middle 50% of data
• IQR = Q3 - Q1 (quartiles)

Detailed IQR typically in later courses!

For now: Know that range isn't the only spread measure!

---

Range in Sports

Basketball scores in a season: Lowest: 82 points Highest: 115 points

Range: 115 - 82 = 33 points

Interpretation: Team's scoring varies by 33 points game to game. Shows consistency or inconsistency!

---

Range in Weather

Daily high temperatures (°F): Monday: 65°, Tuesday: 70°, Wednesday: 68°, Thursday: 72°, Friday: 67°

Range: 72 - 65 = 7°F

Small range = stable weather! Large range = changing conditions!

---

Comparing Data Sets Using Range

Store A daily sales: $500-$800 (Range = $300) **Store B daily sales:**$600-$650 (Range =$50)

Analysis:

• Store A: More variable sales
• Store B: More consistent sales

Range reveals different patterns!

---

Finding Missing Value Given Range

Problem: Range is 20. Values are 5, 10, 15, and x. Find possible values for x.

Solution:

If x is maximum: x - 5 = 20 x = 25

If x is minimum: 15 - x = 20 x = -5

x could be 25, -5, or anything outside 5-15 that gives range 20!

---

Data Distribution and Spread

Three data sets with same range:

Set A: 0, 25, 50 Set B: 0, 10, 20, 30, 40, 50 Set C: 0, 0, 0, 50, 50, 50

All have range = 50!

But distributed very differently!

Range doesn't tell the whole story!

---

Common Mistakes to Avoid

❌ Mistake 1: Subtracting in wrong order

• Range = Max - Min (not Min - Max)
• Range is always positive or zero

❌ Mistake 2: Forgetting to order data first

• Easy to miss true max or min
• Order data to be sure!

❌ Mistake 3: Assuming one unusual value is an error

• Could be real!
• Investigate before removing

❌ Mistake 4: Using range as only spread measure

• Range very sensitive to outliers
• Consider other measures too

❌ Mistake 5: Forgetting units

• Range has same units as data
• Include in answer!

---

Problem-Solving Strategy

To find range:

1. Order data (optional but helpful)
2. Identify maximum value
3. Identify minimum value
4. Subtract: Max - Min
5. Include units

To identify outliers:

1. Look at data distribution
2. Find values far from cluster
3. Consider context (is it reasonable?)
4. Investigate cause if possible
5. Decide whether to keep or remove

To analyze spread:

1. Calculate range
2. Look for outliers
3. Consider other measures (IQR, standard deviation in later courses)
4. Interpret in context

---

Quick Reference

Range:

• Measures spread or variability
• Range = Max - Min
• Same units as data
• Sensitive to outliers

Outlier:

• Value much larger or smaller than others
• "Stands out" from the rest
• Can be high or low
• Affects mean and range, not median

Effects:

• Outliers increase range
• Outliers pull mean
• Outliers don't affect median much

When to use:

• Range: Quick measure of spread
• With median: More complete picture
• Identify outliers: Understand data better

---

Practice Tips

Tip 1: Always look at your data

• Plot on number line or graph
• Visual inspection helps spot outliers

Tip 2: Order data first

• Makes finding max/min easier
• Helps spot outliers

Tip 3: Consider context

• Is outlier reasonable?
• What might have caused it?

Tip 4: Report outliers

• Don't hide them
• Explain what you did with them

Tip 5: Use multiple measures

• Range alone doesn't tell whole story
• Combine with mean, median, mode

---

Summary

Range measures how spread out data is:

Formula: Range = Maximum - Minimum

Characteristics:

• Simple measure of variability
• Uses only two values (max and min)
• Same units as original data
• Very sensitive to outliers

Outliers are unusual values:

Definition: Values much larger or smaller than others

Effects:

• Dramatically increase range
• Pull mean toward them
• Little effect on median
• Important to identify and investigate

Key concepts:

• Large range = spread out data
• Small range = clustered data
• Outliers can be measurement errors or real unusual values
• Different measures needed for complete picture

Applications:

• Comparing consistency of data sets
• Identifying unusual events
• Quality control
• Understanding variability

Problem-solving:

• Order data to find max/min easily
• Look for values far from cluster
• Consider context and investigate outliers
• Use range with other measures for better understanding

Understanding range and outliers is essential for data analysis and recognizing patterns and anomalies!