Solved Exercises on Box Plots in Grade 7

Master box plots: quartiles, median, interquartile range, outliers, and data interpretation through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Basic Box Plot Construction
Exercise 1
The test scores of 11 students are: 45, 52, 58, 63, 67, 71, 75, 78, 82, 85, 90. Find the five-number summary and construct a box plot.
Definition:

Box Plot: A graphical representation showing the five-number summary of a dataset: minimum, Q1, median (Q2), Q3, and maximum.

Quartiles: Values that divide ordered data into four equal parts (Q1 = 25%, Q2 = 50%, Q3 = 75%).

Box plot construction method:
  1. Arrange data in ascending order
  2. Find the median (Q2)
  3. Find Q1 (median of lower half)
  4. Find Q3 (median of upper half)
  5. Identify minimum and maximum values
  6. Draw the box plot with whiskers
Step 1: Organize the data

Sorted data: 45, 52, 58, 63, 67, 71, 75, 78, 82, 85, 90

Step 2: Find the median (Q2)

With 11 values, median is the 6th value: 71

Step 3: Find Q1 (lower quartile)

Lower half: 45, 52, 58, 63, 67. Q1 = median = 58

Step 4: Find Q3 (upper quartile)

Upper half: 75, 78, 82, 85, 90. Q3 = median = 82

Step 5: Identify extremes

Minimum = 45, Maximum = 90

Step 6: Create five-number summary
StatisticValue
Minimum45
Q158
Median (Q2)71
Q382
Maximum90
Final answer:

The box plot has: Minimum = 45, Q1 = 58, Median = 71, Q3 = 82, Maximum = 90.

Applied rules:

• Quartile calculation: Q1 = median of lower half, Q3 = median of upper half

• Five-number summary: Includes min, Q1, median, Q3, max

• Box boundaries: Box spans from Q1 to Q3

Min: 45 Q1: 58 Med: 71 Q3: 82 Max: 90 Test Score Distribution
2 Interquartile Range and Outliers
Exercise 2
The ages of 12 participants in a study are: 15, 18, 20, 22, 25, 27, 30, 32, 35, 38, 40, 55. Calculate the interquartile range (IQR) and identify any outliers using the 1.5×IQR rule.
Definition:

Interquartile Range (IQR): Q3 - Q1, representing the middle 50% of data.

Outliers: Data points beyond Q1 - 1.5×IQR or Q3 + 1.5×IQR.

Step 1: Organize the data

Sorted data: 15, 18, 20, 22, 25, 27, 30, 32, 35, 38, 40, 55

Step 2: Find quartiles

With 12 values: Q1 = average of 3rd and 4th = (20+22)/2 = 21

Median = average of 6th and 7th = (27+30)/2 = 28.5

Q3 = average of 9th and 10th = (35+38)/2 = 36.5

Step 3: Calculate IQR

IQR = Q3 - Q1 = 36.5 - 21 = 15.5

Step 4: Find outlier boundaries

Lower boundary = Q1 - 1.5×IQR = 21 - 1.5×15.5 = 21 - 23.25 = -2.25

Upper boundary = Q3 + 1.5×IQR = 36.5 + 1.5×15.5 = 36.5 + 23.25 = 59.75

Step 5: Identify outliers

All values between -2.25 and 59.75 are within bounds. Since all data points fall within these limits, there are no outliers.

Final answer:

IQR = 15.5. There are no outliers in this dataset.

Applied rules:

• IQR formula: IQR = Q3 - Q1

• Outlier detection: Values beyond Q1 - 1.5×IQR or Q3 + 1.5×IQR

• Boundaries: Lower = Q1 - 1.5×IQR, Upper = Q3 + 1.5×IQR

Min: 15 Q1: 21 Med: 28.5 Q3: 36.5 Max: 55 IQR = 15.5 Age Distribution with IQR
3 Data Interpretation
Exercise 3
A box plot shows the daily temperatures (°F) in a city. Min=32°F, Q1=45°F, Median=58°F, Q3=72°F, Max=85°F. What percentage of days had temperatures between Q1 and Q3? What is the range of the middle 50% of temperatures?
Definition:

Interquartile Range (IQR): The range containing the middle 50% of data, from Q1 to Q3.

Percentiles: Q1 represents 25th percentile, Q3 represents 75th percentile.

Step 1: Understand the given data

Five-number summary: Min=32, Q1=45, Median=58, Q3=72, Max=85

Step 2: Find percentage between Q1 and Q3

By definition, 50% of data lies between Q1 and Q3

Step 3: Calculate IQR

IQR = Q3 - Q1 = 72 - 45 = 27°F

Step 4: Interpret the results

The middle 50% of temperatures range from 45°F to 72°F

Final answer:

50% of days had temperatures between 45°F and 72°F. The range of the middle 50% is 27°F.

Applied rules:

• Quartile interpretation: Q1 to Q3 contains exactly 50% of data

• IQR significance: Measures spread of central data

• Data distribution: Understanding what each part of the box plot represents

Min: 32°F Q1: 45°F Med: 58°F Q3: 72°F Max: 85°F 50% of data here Temperature Distribution
Box Plot Concepts, Rules and Methods
IQR = Q3 - Q1
Interquartile Range Formula
Quartile 1 (Q1)
25th Percentile
Median of lower half
Quartile 2 (Q2)
50th Percentile
Median of data
Quartile 3 (Q3)
75th Percentile
Median of upper half
Key definitions:

Box Plot: A graphical representation of the five-number summary showing the distribution of a dataset.

Five-Number Summary: Minimum, Q1, Median, Q3, Maximum values of a dataset.

Quartiles: Three values that divide ordered data into four equal parts.

Interquartile Range (IQR): The difference between Q3 and Q1, representing the middle 50% of data.

Box plot construction steps:
  1. Sort data: Arrange values in ascending order
  2. Find median (Q2): Middle value of the dataset
  3. Find Q1: Median of the lower half (excluding median if odd)
  4. Find Q3: Median of the upper half (excluding median if odd)
  5. Identify extremes: Find minimum and maximum values
  6. Draw plot: Create box from Q1 to Q3 with median line, add whiskers to extremes
Tip 1: The box represents the middle 50% of the data.
Tip 2: Longer boxes indicate greater variability in the middle 50%.
Tip 3: Whiskers show the range of non-outlier data points.
Tip 4: Outliers are plotted as individual points beyond whiskers.
Key characteristics: Shows spread, central tendency, skewness, and outliers in a compact form.
Common applications: Quality control, statistical analysis, comparing distributions, outlier detection.
Solution: Exercises 4 to 5
4 Finding Missing Values
Exercise 4
A dataset has 9 values arranged in order. The median is 35, Q1 is 25, Q3 is 45, minimum is 15, and maximum is 60. If the IQR is 20, verify if these values are consistent and construct the box plot.
Definition:

Consistency Check: Verifying that calculated values match expected relationships.

Step 1: Verify IQR calculation

Given: Q3 = 45, Q1 = 25, so IQR = 45 - 25 = 20 ✓

Step 2: Check median position

With 9 values, median should be the 5th value, which is 35 ✓

Step 3: Verify quartile positions

Lower half: 1st to 4th values, median of which should be Q1 = 25 ✓

Upper half: 6th to 9th values, median of which should be Q3 = 45 ✓

Step 4: Confirm extremes

Minimum = 15, Maximum = 60 are the smallest and largest values ✓

Final answer:

All values are consistent. The box plot shows: Min=15, Q1=25, Median=35, Q3=45, Max=60.

Applied rules:

• IQR relationship: IQR must equal Q3 - Q1

• Position verification: Check that values are in correct positions

• Consistency check: Ensure all relationships hold true

Min: 15 Q1: 25 Med: 35 Q3: 45 Max: 60 ✓ Consistent Values IQR = 20 = Q3-Q1 Consistent Data Distribution
5 Comparing Distributions
Exercise 5
Compare two datasets using box plots. Dataset A: Min=10, Q1=20, Median=30, Q3=40, Max=50. Dataset B: Min=15, Q1=25, Median=35, Q3=45, Max=55. Which dataset has greater variability? Which is shifted higher?
Definition:

Variability: How spread out the data values are.

Central Tendency: Where the center of the data is located.

Step 1: Calculate IQR for both datasets

Dataset A: IQR = 40 - 20 = 20

Dataset B: IQR = 45 - 25 = 20

Step 2: Compare medians

Dataset A median = 30, Dataset B median = 35

Dataset B is shifted higher

Step 3: Compare ranges

Dataset A range = 50 - 10 = 40

Dataset B range = 55 - 15 = 40

Step 4: Compare overall positions

Each value in Dataset B is exactly 5 units higher than Dataset A

Final answer:

Both datasets have the same variability (IQR = 20). Dataset B is shifted 5 units higher than Dataset A.

Applied rules:

• Comparative analysis: Compare IQR for variability

• Position comparison: Compare medians and overall ranges

• Pattern recognition: Identifying shifts in distribution

A: 10 20 30 40 50 B: 15 25 35 45 55 A B Comparison: Dataset A vs Dataset B
Box Plot Theory: Laws, Methods, Definitions, and Formulas
IQR = Q3 - Q1
Interquartile Range
Key definitions:

Box Plot (Box-and-Whisker Plot): A standardized way of displaying the distribution of data based on a five-number summary.

Quartiles: Three values that divide sorted data into four equal parts: Q1 (25th percentile), Q2 (50th percentile/median), Q3 (75th percentile).

Interquartile Range (IQR): The range between Q1 and Q3, representing the middle 50% of the data.

Outliers: Data points that fall significantly outside the overall pattern of the data.

Box plot creation methodology:
  1. Data Collection: Gather all data points to be represented
  2. Sorting: Arrange data in ascending order
  3. Quartile Calculation: Find Q1, Q2 (median), and Q3
  4. Extremes Identification: Determine minimum and maximum values
  5. Outlier Detection: Identify values beyond Q1-1.5×IQR or Q3+1.5×IQR
  6. Plot Construction: Draw box from Q1 to Q3 with median line, add whiskers to non-outlier extremes
Tip 1: The length of the box represents the IQR and indicates data variability.
Tip 2: A median line closer to Q1 suggests right skewness; closer to Q3 suggests left skewness.
Tip 3: Equal-sized whiskers suggest symmetric distribution around the median.
Tip 4: Multiple outliers may indicate unusual data patterns or measurement errors.
Key characteristics: Shows central tendency, spread, skewness, and outliers in a single compact view.
Common applications: Statistical analysis, quality control, research, comparing multiple datasets.
Essential formulas and rules:

• Quartile positions: Q1 = (n+1)/4th value, Q3 = 3(n+1)/4th value

• Interquartile Range: IQR = Q3 - Q1

• Outlier boundaries: Lower = Q1 - 1.5×IQR, Upper = Q3 + 1.5×IQR

• Five-number summary: Min, Q1, Median, Q3, Max

• Percentile interpretation: 25% below Q1, 50% below median, 75% below Q3

Questions & Answers

Question: I'm confused about how to find Q1 and Q3 when there's an even number of data points. Can you explain the process?

Answer: Great question! Here's how to find Q1 and Q3 with even numbers of data points:

  • For Q1: Take the median of the lower half of the data (the first half when arranged in order)
  • For Q3: Take the median of the upper half of the data (the second half when arranged in order)

If the halves have an even number of values, average the two middle values. If they have an odd number, take the middle value.

Example with 10 data points: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10

Lower half (5 values): 1, 2, 3, 4, 5 → Q1 = 3 (middle value)

Upper half (5 values): 6, 7, 8, 9, 10 → Q3 = 8 (middle value)

Example with 12 data points: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12

Lower half (6 values): 1, 2, 3, 4, 5, 6 → Q1 = (3+4)/2 = 3.5

Upper half (6 values): 7, 8, 9, 10, 11, 12 → Q3 = (9+10)/2 = 9.5

Question: How do I know if a data point is an outlier? What makes a value qualify as an outlier?

Answer: An outlier is identified using the 1.5×IQR rule:

  1. Calculate the Interquartile Range (IQR) = Q3 - Q1
  2. Calculate the lower boundary: Q1 - 1.5×IQR
  3. Calculate the upper boundary: Q3 + 1.5×IQR
  4. Any data point below the lower boundary OR above the upper boundary is an outlier

Example: If Q1 = 20, Q3 = 40, then IQR = 20

Lower boundary = 20 - 1.5×20 = 20 - 30 = -10

Upper boundary = 40 + 1.5×20 = 40 + 30 = 70

Any value less than -10 or greater than 70 would be considered an outlier.

Note: This is a statistical convention, not an absolute rule. Context matters when interpreting outliers.

Question: What does it mean when a box plot is skewed to the left or right? How can I tell?

Answer: Skewness in a box plot indicates the direction of data concentration:

  • Right-skewed (positive skew): The right whisker is longer than the left; the median line is closer to Q1; tail extends to the right
  • Left-skewed (negative skew): The left whisker is longer than the right; the median line is closer to Q3; tail extends to the left
  • Symmetric: Whiskers are roughly equal; median is centered in the box

To identify skewness, look at the position of the median line within the box and the relative lengths of the whiskers.

Example: Income data is often right-skewed because most people earn moderate incomes, but a few earn extremely high incomes, creating a long right tail.

Question: How can I use box plots to compare different datasets? What should I look for?

Answer: Box plots are excellent for comparing datasets. When comparing:

  • Center: Compare the positions of the median lines
  • Variability: Compare the lengths of the boxes (IQRs) and overall ranges
  • Shape: Compare skewness patterns in each box plot
  • Outliers: Note differences in outlier presence and location

A higher median indicates a higher central tendency. A longer box indicates greater variability. Different outlier patterns may indicate different data characteristics.

For example, comparing test scores between two classes: if one box plot is shifted higher and has a shorter box, that class performed better with less variation.

Question: What does the interquartile range (IQR) tell me about my data that the range doesn't?

Answer: The IQR and range provide different insights about your data:

  • Range: Difference between maximum and minimum values; sensitive to outliers
  • IQR: Difference between Q3 and Q1; measures spread of the middle 50% of data; resistant to outliers

The IQR gives you a better sense of typical variability because it excludes the extreme values that might distort the range.

Example: In a dataset of salaries where one CEO earns $1,000,000 while others earn $50,000-$70,000, the range would be huge ($950,000+) but the IQR would show the actual salary variation among typical employees.

Use the IQR when you want to measure central variability, and the range when you want to see the full extent of data spread.