Solved Exercises on Range in Grade 7

Master range calculations: finding differences between maximum and minimum values, interpreting data spread, and comparing data sets through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Basic Range Calculation
Exercise 1
Find the range of the following data set: 12, 18, 15, 21, 10, 16, 19.
Definition:

Range: The difference between the maximum and minimum values in a data set.

Formula: Range = Maximum - Minimum

Range Calculation Method:
  1. Identify all values in the data set
  2. Find the maximum value (largest number)
  3. Find the minimum value (smallest number)
  4. Subtract the minimum from the maximum
  5. Range = Maximum - Minimum
Data Set
12, 18, 15, 21, 10, 16, 19
Maximum
21
Minimum
10
Range
11
Step 1: List all values

Data: 12, 18, 15, 21, 10, 16, 19

Step 2: Identify the maximum value

Looking at all values: 12, 18, 15, 21, 10, 16, 19

The largest value is 21

Step 3: Identify the minimum value

Looking at all values: 12, 18, 15, 21, 10, 16, 19

The smallest value is 10

Step 4: Calculate the range

Range = Maximum - Minimum

Range = 21 - 10 = 11

Range = 11
Final Answer:

The range of the data set is 11.

Applied Rules:

• Range Formula: Range = Maximum - Minimum

• Value Identification: Correctly identify max and min values

• Subtraction: Perform the subtraction accurately

2 Comparing Ranges
Exercise 2
Two classes took the same math test. Class A scores: 75, 80, 85, 90, 95. Class B scores: 60, 70, 80, 90, 100. Which class has a greater range of scores? What does this tell you about the data?
Definition:

Range Comparison: Using range to compare the spread or variability of different data sets.

Data Spread: How spread out the values are in a data set.

Class A
Max=95, Min=75, Range=20
Class B
Max=100, Min=60, Range=40
Greater Range
Class B
Step 1: Calculate range for Class A

Class A scores: 75, 80, 85, 90, 95

Maximum: 95, Minimum: 75

Range = 95 - 75 = 20

Step 2: Calculate range for Class B

Class B scores: 60, 70, 80, 90, 100

Maximum: 100, Minimum: 60

Range = 100 - 60 = 40

Step 3: Compare the ranges

Class A range: 20

Class B range: 40

Class B has a greater range

Step 4: Interpret the results

Class B has more variability in scores

Class A scores are more closely grouped together

Class B shows more diversity in performance levels

Class A Range = 20
Class B Range = 40
Class B has greater range
Final Answer:

Class B has a greater range (40) than Class A (20). This indicates that Class B has more variability in test scores, with scores spread over a wider range.

Applied Rules:

• Range Comparison: Higher range indicates greater data spread

• Variability Measure: Range shows how spread out values are

• Data Interpretation: Greater range suggests more diversity in data

3 Effect of Outliers on Range
Exercise 3
Calculate the range with and without the outlier for the following data set: 15, 18, 20, 17, 16, 19, 18, 100. Explain how the outlier affects the range.
Definition:

Outlier: A value that is significantly different from other values in the data set.

Range Sensitivity: Range is highly sensitive to outliers since it only considers maximum and minimum values.

With Outlier
Range = 85
Without Outlier
Range = 5
Effect
Increases by 80
Step 1: Identify the outlier

Data set: 15, 18, 20, 17, 16, 19, 18, 100

Value 100 is significantly larger than other values (15-20 range)

This is the outlier.

Step 2: Calculate range with outlier

With outlier: 15, 18, 20, 17, 16, 19, 18, 100

Maximum: 100, Minimum: 15

Range = 100 - 15 = 85

Step 3: Calculate range without outlier

Without outlier: 15, 18, 20, 17, 16, 19, 18

Maximum: 20, Minimum: 15

Range = 20 - 15 = 5

Step 4: Compare the results

Range with outlier: 85

Range without outlier: 5

Difference: 85 - 5 = 80

Step 5: Explain the effect

The outlier (100) dramatically increased the range from 5 to 85.

This shows that range is very sensitive to extreme values.

With outlier: Range = 85
Without outlier: Range = 5
Final Answer:

With outlier: Range = 85. Without outlier: Range = 5. The outlier increased the range by 80, demonstrating that range is highly sensitive to extreme values.

Applied Rules:

• Outlier Sensitivity: Range is highly sensitive to outliers

• Extreme Values: Outliers directly affect maximum or minimum

• Data Interpretation: Consider outliers when interpreting range

Rules and methods, laws,...
\(\text{Range} = \text{Maximum} - \text{Minimum}\)
Range Formula
Range Formula
\(\text{Range} = \text{Max} - \text{Min}\)
Difference between extremes
Sensitivity
Affected by outliers
Only considers max/min
Interpretation
Measures data spread
Higher = more variability
Key Definitions:

Range: The difference between the maximum and minimum values in a data set

Maximum: The largest value in the data set

Minimum: The smallest value in the data set

Outlier: A value that is significantly different from other values in the data set

Data Spread: How spread out the values are in a data set

Variability: The degree to which data points differ from each other

Dispersion: A measure of how spread out values are in a data set

Complete Methodology:
  1. Value Identification: Identify all values in the data set
  2. Maximum Finding: Find the largest value
  3. Minimum Finding: Find the smallest value
  4. Subtraction: Calculate maximum - minimum
  5. Verification: Ensure the result is positive
  6. Interpretation: Understand what the range tells you about the data
Tip 1: Always arrange data in order to easily identify maximum and minimum values.
Tip 2: Range is sensitive to outliers, so consider the context when interpreting.
Tip 3: Range only considers two values, so it doesn't reflect the distribution of other values.
Tip 4: Range is always positive since maximum is always ≥ minimum.
Tip 5: Use range to compare spread between different data sets.
Common Errors: Confusing maximum and minimum, calculation mistakes, forgetting that range is always positive, not considering outliers.
Exam Preparation: Practice with various data sets, memorize the formula, understand interpretation, know the sensitivity to outliers.
Solution: Exercises 4 to 5
4 Real-World Application
Exercise 4
A company recorded daily sales for a week: Monday $240, Tuesday $180, Wednesday $320, Thursday $260, Friday $200, Saturday $450, Sunday $380. What is the range of daily sales? If the company wants to reduce the range by $50, what would be the new maximum sales value if the minimum remains the same?
Definition:

Real-World Application: Applying statistical concepts to practical business situations.

Range Reduction: Adjusting data to achieve a desired range.

Sales Data
$180-$450
Current Range
$270
Target Range
$220
Step 1: Identify all sales values

Sales: $240, $180, $320, $260, $200, $450, $380

Step 2: Find the maximum and minimum values

Maximum: $450 (Saturday)

Minimum: $180 (Tuesday)

Step 3: Calculate the current range

Range = Maximum - Minimum

Range = $450 - $180 = $270

Step 4: Calculate the target range

Target range = Current range - $50

Target range = $270 - $50 = $220

Step 5: Find the new maximum with same minimum

New range = New maximum - Minimum

$220 = New maximum - $180

New maximum = $220 + $180 = $400

Step 6: Verify the result

New range = $400 - $180 = $220 ✓

Current Range = $270
New Maximum = $400
Final Answer:

The current range of daily sales is $270. To reduce the range by $50 while keeping the minimum the same, the new maximum sales value would be $400.

Applied Rules:

• Real-World Context: Apply statistical concepts to practical situations

• Range Formula: Range = Maximum - Minimum

• Algebraic Thinking: Rearrange formulas to solve for unknowns

5 Range in Context
Exercise 5
For each situation, explain whether range is an appropriate measure of variability and why:
a) Heights of students in a class
b) Test scores with several identical high scores
c) Ages of people in a retirement community
Definition:

Appropriate Measure: The measure that best represents the variability in a data set.

Context Consideration: Evaluating whether a statistical measure is suitable for the given situation.

Situation a
Appropriate
Situation b
Less appropriate
Situation c
Appropriate
Step 1: Analyze Situation a - Student Heights

Heights are continuous numerical data with natural variation

Range shows the difference between tallest and shortest students

This is meaningful and useful information

Step 2: Analyze Situation b - Test Scores

With identical high scores, range only captures the difference to lowest score

Range doesn't reflect the clustering of high scores

Other measures (standard deviation) might be more informative

Step 3: Analyze Situation c - Retirement Community Ages

Ages are continuous numerical data

Range shows the age span in the community

This provides meaningful information about demographic spread

Step 4: Consider limitations of range

Range only considers two values (max and min)

Range is sensitive to outliers

Range doesn't reflect distribution of other values

Step 5: Provide recommendations

a) Yes - range is appropriate for height data

b) Less appropriate - range doesn't capture clustering

c) Yes - range is appropriate for age data

a) Yes, b) Less appropriate, c) Yes
Final Answer:

a) Yes - Range is appropriate for student heights as it shows the spread from shortest to tallest
b) Less appropriate - Range doesn't capture clustering of high scores, other measures might be better
c) Yes - Range is appropriate for ages as it shows the age span in the community

Applied Rules:

• Context Appropriateness: Consider the nature of the data

• Limitation Awareness: Understand range's limitations

• Alternative Measures: Know when other measures might be better

Detailed Summary: Range Fundamentals
\(\text{Range} = \text{Maximum} - \text{Minimum}\)
Range Formula
Key definitions:

Range: The difference between the maximum and minimum values in a data set, calculated as Maximum - Minimum

Maximum: The largest value in the data set

Minimum: The smallest value in the data set

Data Spread: How spread out the values are in a data set

Variability: The degree to which data points differ from each other

Dispersion: A measure of how spread out values are in a data set

Outlier: A value that is significantly different from other values in the data set

Complete methodology:
  1. Data Organization: List all values in the data set
  2. Maximum Identification: Find the largest value in the data set
  3. Minimum Identification: Find the smallest value in the data set
  4. Subtraction: Calculate Maximum - Minimum
  5. Verification: Ensure the result is positive and reasonable
  6. Interpretation: Understand what the range tells you about data spread
Tip 1: Arrange data in ascending order to easily identify maximum and minimum values.
Tip 2: Range is always positive since maximum is always greater than or equal to minimum.
Tip 3: Range is highly sensitive to outliers since it only considers maximum and minimum values.
Tip 4: Range doesn't reflect how other values are distributed between max and min.
Tip 5: Use range to compare variability between different data sets.

Common errors: Confusing maximum and minimum, calculation mistakes, not considering the effect of outliers, forgetting that range is always positive.
Exam preparation: Practice with various data sets, memorize the formula, understand interpretation, know how outliers affect range.
Formulas to know by heart:

• Range Formula: Range = Maximum - Minimum

• Value Identification: Always identify max and min first

• Positive Result: Range is always ≥ 0

• Outlier Sensitivity: Range is highly sensitive to extreme values

• Comparison Tool: Use range to compare data spread across sets

Exercise with Visualization: Range Comparison
Exercise 6: Comparing Data Spreads
Compare the ranges of the following data sets:
Set A: 10, 15, 20, 25, 30 (range = 20)
Set B: 5, 15, 20, 25, 35 (range = 30)
Set C: 18, 19, 20, 21, 22 (range = 4)

Analysis: The visualization shows how different data sets have different ranges.

  • Set A: Moderate spread (range = 20)
  • Set B: Wide spread (range = 30)
  • Set C: Tight cluster (range = 4)

Questions & Answers

Question: How do I make sure I don't mix up the maximum and minimum values when calculating range?

Answer: Here are effective strategies to avoid mixing up max and min:

  • Sort the Data: Arrange all values in ascending order first
  • Label Clearly: Write "Maximum =" and "Minimum =" before each value
  • Circle Values: Visually mark the largest and smallest values
  • Check Reasonableness: Ensure max ≥ min before calculating

Example: For [12, 18, 15, 21, 10, 16, 19]

Sorted: [10, 12, 15, 16, 18, 19, 21]

Now clearly: Maximum = 21, Minimum = 10

Range = 21 - 10 = 11

This systematic approach prevents errors and builds good habits!

Question: Why is range so sensitive to outliers? How does this affect interpretation?

Answer: Range is sensitive to outliers because:

  • Only Uses Two Values: Range only considers the maximum and minimum
  • Extreme Impact: An outlier automatically becomes the max or min
  • Complete Influence: One extreme value can drastically change the range

Example: For [15, 18, 20, 17, 16, 19, 18], range = 20-15 = 5

Adding outlier 100: [15, 18, 20, 17, 16, 19, 18, 100], range = 100-15 = 85

The range increased from 5 to 85 due to just one outlier!

This sensitivity means range might not accurately represent the typical spread of most data points when outliers are present.

Question: Can range ever be negative? What happens if the minimum is greater than the maximum?

Answer: No, range cannot be negative! Here's why:

  • Definition Requirement: By definition, maximum ≥ minimum
  • Formula Structure: Range = Maximum - Minimum
  • Natural Outcome: Since max ≥ min, (max - min) ≥ 0

If you calculate a negative range, you made an error:

  • You might have mixed up max and min values
  • You might have subtracted in the wrong order
  • You might have misidentified the maximum or minimum

Example: For [10, 15, 20], Maximum = 20, Minimum = 10

Range = 20 - 10 = 10 (positive)

If you calculated 10 - 20 = -10, you switched the values!

Question: How can I check if my range calculation is correct?

Answer: Here are several verification methods:

  1. Reasonableness Check: Ensure range is positive and makes sense for your data
  2. Re-identification: Find max and min again to confirm values
  3. Addition Check: Minimum + Range should equal Maximum
  4. Visual Check: Look at your sorted data to confirm max and min

Example: For [12, 18, 15, 21, 10, 16, 19]

  • Calculated: Max = 21, Min = 10, Range = 11
  • Verification: 10 + 11 = 21 ✓
  • Also: 21 - 11 = 10 ✓

These checks help catch calculation errors and ensure accuracy!

Question: When would I encounter range calculations in real life? How is this useful?

Answer: Range calculations are common in everyday life:

  • Weather: Daily temperature range (high - low)
  • Finance: Stock price range during trading day
  • Business: Sales range across different periods
  • Health: Blood pressure range, heart rate range
  • Education: Test score ranges, grade ranges
  • Sports: Score ranges, performance ranges

Range helps you understand the spread or variability in data. For example, knowing the temperature range helps you prepare for the day's weather variations.

Understanding range helps you interpret data and make informed decisions based on the spread of values in both academic and real-world contexts!