Loading…
Find range and identify outliers in data sets
Learn step-by-step with practice exercises built right in.
How spread out is data? How do we identify unusual values? Understanding range and outliers helps us analyze data variation and spot anomalies!
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!
Range measures variability:
Example 1: 10, 12, 14, 16, 18 Range = 18 - 10 = 8
Find the range of: 5, 12, 8, 15, 3, 10
Step 1: Find the maximum (largest value). Max = 15
Step 2: Find the minimum (smallest value). Min = 3
Step 3: Calculate range. Range = Max - Min Range = 15 - 3 = 12
Answer: Range = 12
Data set: 22, 25, 23, 24, 26, 23. What is the range?
Avoid these 3 frequent errors
See how this math is used in the real world
Solve .
Review key concepts with our flashcard system
Explore more Pre-Algebra topics
Example 2: 10, 10, 10, 10, 10 Range = 10 - 10 = 0
Different spreads!
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 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!
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!
Range only uses two 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.
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:
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!
Simple rule: A value is an outlier if it's far from the rest
More specific:
For pre-algebra, visual inspection is usually enough!
Ask: "Does this value seem way different?"
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!
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!
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!
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!
Measurement error:
Natural variation:
Different population:
Must investigate to determine cause!
Option 1: Keep the outlier
Option 2: Remove the outlier
Always explain what you did!
Test scores: 85, 88, 90, 92, 87, 25
Range: 92 - 25 = 67
Analysis:
Without outlier: Range = 92 - 85 = 7 (much smaller!)
The outlier hides that most students did well!
Home prices on a street: 220K, 210K, $2.5M
Range: 200K = $2.3M
Analysis:
Median price (672K) inflated by mansion!
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 plots (box-and-whisker plots) show outliers!
Structure:
Visual way to identify outliers!
IQR is another measure of spread:
Detailed IQR typically in later courses!
For now: Know that range isn't the only spread measure!
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!
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!
Store A daily sales: 800 (Range = 600-50)
Analysis:
Range reveals different patterns!
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!
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!
❌ Mistake 1: Subtracting in wrong order
❌ Mistake 2: Forgetting to order data first
❌ Mistake 3: Assuming one unusual value is an error
❌ Mistake 4: Using range as only spread measure
❌ Mistake 5: Forgetting units
To find range:
To identify outliers:
To analyze spread:
Range:
Outlier:
Effects:
When to use:
Tip 1: Always look at your data
Tip 2: Order data first
Tip 3: Consider context
Tip 4: Report outliers
Tip 5: Use multiple measures
Range measures how spread out data is:
Formula: Range = Maximum - Minimum
Characteristics:
Outliers are unusual values:
Definition: Values much larger or smaller than others
Effects:
Key concepts:
Applications:
Problem-solving:
Understanding range and outliers is essential for data analysis and recognizing patterns and anomalies!
Step 1: Identify maximum. Max = 26
Step 2: Identify minimum. Min = 22
Step 3: Subtract. Range = 26 - 22 = 4
Answer: Range = 4
Identify any outliers in this data set: 15, 18, 17, 16, 19, 45, 18
Step 1: Look at all values. 15, 16, 17, 18, 18, 19 are clustered together (15-19) 45 is much larger
Step 2: Identify outliers. 45 is FAR from the cluster 45 - 19 = 26 (much bigger gap than within cluster)
Step 3: Conclusion. 45 is an outlier (unusually high value)
Answer: 45 is an outlier
Test scores: 82, 78, 85, 90, 88, 30, 84, 86. Find the range with and without the outlier, and explain the effect.
Step 1: Identify outlier. 30 is much lower than all others (which are 78-90) 30 is an outlier
Step 2: Range WITH outlier. Max = 90, Min = 30 Range = 90 - 30 = 60
Step 3: Range WITHOUT outlier. Max = 90, Min = 78 Range = 90 - 78 = 12
Step 4: Effect. Outlier GREATLY increases range from 12 to 60 Range is very sensitive to outliers
Answer: With outlier: Range = 60. Without outlier: Range = 12. The outlier dramatically increases the range.
Two classes took the same test. Class A scores: 70, 75, 80, 85, 90 (range = 20). Class B scores: 60, 70, 80, 90, 100 (range = 40). What does the range tell you about each class's performance?
Step 1: Analyze Class A. Range = 20 (smaller) Scores are 70-90, fairly clustered More CONSISTENT performance Less variation between students
Step 2: Analyze Class B. Range = 40 (larger) Scores are 60-100, more spread out Less CONSISTENT performance More variation between students
Step 3: Compare. Both have same median (80) Class A: students performed more similarly Class B: wider gap between top and bottom
Answer: Class A has more consistent scores (smaller range = less spread). Class B has more variation (larger range = more spread between highest and lowest).