Box And Whisker Plot Practice Problems

Let's dig into the world of box and whisker plots through practical exercises that will sharpen your understanding and application of this valuable statistical tool. Day to day, box and whisker plots, also known as boxplots, are visual representations that summarize the distribution of a dataset by showing the median, quartiles, and outliers. Mastering them equips you with a powerful skill for data analysis and interpretation.

Understanding the Anatomy of a Box and Whisker Plot

Before tackling practice problems, it's crucial to grasp the core components of a box and whisker plot:

• Minimum Value: The smallest data point in the set (excluding outliers).
• First Quartile (Q1): The median of the lower half of the data. 25% of the data falls below this value.
• Median (Q2): The middle value of the dataset. 50% of the data falls below this value.
• Third Quartile (Q3): The median of the upper half of the data. 75% of the data falls below this value.
• Maximum Value: The largest data point in the set (excluding outliers).
• Interquartile Range (IQR): The difference between the third and first quartiles (Q3 - Q1). It represents the spread of the middle 50% of the data.
• Whiskers: Lines extending from the box to the minimum and maximum values within a defined range. Typically, this range is 1.5 times the IQR beyond Q1 and Q3.
• Outliers: Data points that fall outside the whiskers. They are often represented as individual points or asterisks.

Practice Problems: Putting Knowledge into Action

Now, let's dive into some practice problems to solidify your understanding. Each problem will present a dataset, and your task is to create a box and whisker plot and interpret its key features.

Problem 1: Test Scores

A class of 20 students took a test. The scores are as follows:

65, 70, 72, 75, 78, 80, 82, 85, 85, 88, 90, 92, 92, 95, 95, 98, 98, 100, 100, 100

Solution:

1. Order the Data: The data is already ordered.
2. Find the Median (Q2): Since there are 20 data points (an even number), the median is the average of the 10th and 11th values: (88 + 90) / 2 = 89.
3. Find Q1: The median of the lower half (65 to 88) is the average of the 5th and 6th values: (78 + 80) / 2 = 79.
4. Find Q3: The median of the upper half (90 to 100) is the average of the 15th and 16th values: (95 + 98) / 2 = 96.5.
5. Identify Minimum and Maximum: Minimum = 65, Maximum = 100.
6. Calculate IQR: IQR = Q3 - Q1 = 96.5 - 79 = 17.5.
7. Calculate Whisker Limits:
   • Lower Limit: Q1 - 1.5 * IQR = 79 - 1.5 * 17.5 = 52.75. Since the minimum value (65) is greater than 52.75, the lower whisker extends to 65.
   • Upper Limit: Q3 + 1.5 * IQR = 96.5 + 1.5 * 17.5 = 122.75. Since the maximum value (100) is less than 122.75, the upper whisker extends to 100.
8. Identify Outliers: There are no outliers as all data points fall within the whisker limits.

Interpretation: The box and whisker plot shows a relatively concentrated distribution of test scores. The median is 89, indicating that half the class scored below this mark. The IQR of 17.5 suggests that the middle 50% of the scores are clustered relatively close together. No outliers are present, suggesting no exceptionally high or low scores.

Problem 2: Employee Salaries (in thousands of dollars)

A company has the following employee salaries:

40, 42, 45, 48, 50, 52, 55, 60, 65, 70, 75, 80, 90, 100, 200

Solution:

1. Order the Data: The data is already ordered.
2. Find the Median (Q2): Since there are 15 data points, the median is the 8th value: 60.
3. Find Q1: The median of the lower half (40 to 55) is the 4th value: 48.
4. Find Q3: The median of the upper half (65 to 200) is the 12th value: 80.
5. Identify Minimum and Maximum: Minimum = 40, Maximum = 200.
6. Calculate IQR: IQR = Q3 - Q1 = 80 - 48 = 32.
7. Calculate Whisker Limits:
   • Lower Limit: Q1 - 1.5 * IQR = 48 - 1.5 * 32 = 0. Since the minimum value (40) is greater than 0, the lower whisker extends to 40.
   • Upper Limit: Q3 + 1.5 * IQR = 80 + 1.5 * 32 = 128. The maximum value (200) is greater than 128, so it is an outlier. The upper whisker extends to the highest data point below 128, which is 100.
8. Identify Outliers: 200 is an outlier.

Interpretation: The box and whisker plot reveals a positive skew (rightward skew) in the salary distribution. The median salary is $60,000. The large outlier at $200,000 indicates a significant disparity in salaries, likely representing a high-level executive or owner. The IQR of $32,000 shows the spread of the middle 50% of salaries.

Problem 3: Waiting Times at a Restaurant (in minutes)

Customers' waiting times at a restaurant were recorded as follows:

10, 12, 15, 15, 18, 20, 22, 25, 28, 30, 35, 40

Solution:

1. Order the Data: The data is already ordered.
2. Find the Median (Q2): Since there are 12 data points, the median is the average of the 6th and 7th values: (20 + 22) / 2 = 21.
3. Find Q1: The median of the lower half (10 to 20) is the average of the 3rd and 4th values: (15 + 15) / 2 = 15.
4. Find Q3: The median of the upper half (22 to 40) is the average of the 9th and 10th values: (28 + 30) / 2 = 29.
5. Identify Minimum and Maximum: Minimum = 10, Maximum = 40.
6. Calculate IQR: IQR = Q3 - Q1 = 29 - 15 = 14.
7. Calculate Whisker Limits:
   • Lower Limit: Q1 - 1.5 * IQR = 15 - 1.5 * 14 = -6. Since the minimum value (10) is greater than -6, the lower whisker extends to 10.
   • Upper Limit: Q3 + 1.5 * IQR = 29 + 1.5 * 14 = 50. Since the maximum value (40) is less than 50, the upper whisker extends to 40.
8. Identify Outliers: There are no outliers.

Interpretation: The box and whisker plot suggests that waiting times are fairly consistent. The median waiting time is 21 minutes. The IQR of 14 minutes indicates a reasonable spread around the median. The absence of outliers suggests that no customers experienced exceptionally long or short waits.

Problem 4: Number of Books Read per Year

A group of people were surveyed about the number of books they read in a year. The results are:

2, 5, 5, 8, 10, 12, 15, 15, 20, 25, 30, 40, 50, 75

Solution:

1. Order the Data: The data is already ordered.
2. Find the Median (Q2): With 14 data points, the median is the average of the 7th and 8th values: (15 + 15) / 2 = 15.
3. Find Q1: The median of the lower half (2 to 15) is the 4th value: 8.
4. Find Q3: The median of the upper half (15 to 75) is the 11th value: 30.
5. Identify Minimum and Maximum: Minimum = 2, Maximum = 75.
6. Calculate IQR: IQR = Q3 - Q1 = 30 - 8 = 22.
7. Calculate Whisker Limits:
   • Lower Limit: Q1 - 1.5 * IQR = 8 - 1.5 * 22 = -25. Since the minimum value (2) is greater than -25, the lower whisker extends to 2.
   • Upper Limit: Q3 + 1.5 * IQR = 30 + 1.5 * 22 = 63. The maximum value (75) is greater than 63, so it is an outlier. The upper whisker extends to the highest value below 63, which is 50.
8. Identify Outliers: 75 is an outlier.

Interpretation: The box and whisker plot reveals a positive skew. The median number of books read is 15. The outlier at 75 suggests that one individual reads significantly more books than the rest of the group.

Problem 5: Plant Heights (in cm)

The heights of a sample of plants are:

5, 7, 8, 9, 10, 10, 11, 12, 13, 14, 15, 15, 20, 25

Solution:

1. Order the Data: The data is already ordered.
2. Find the Median (Q2): With 14 data points, the median is the average of the 7th and 8th values: (11 + 12) / 2 = 11.5
3. Find Q1: The median of the lower half (5 to 11) is the 4th value: 9.
4. Find Q3: The median of the upper half (12 to 25) is the 11th value: 15.
5. Identify Minimum and Maximum: Minimum = 5, Maximum = 25.
6. Calculate IQR: IQR = Q3 - Q1 = 15 - 9 = 6.
7. Calculate Whisker Limits:
   • Lower Limit: Q1 - 1.5 * IQR = 9 - 1.5 * 6 = 0. The lower whisker extends to 5.
   • Upper Limit: Q3 + 1.5 * IQR = 15 + 1.5 * 6 = 24. The maximum value (25) is an outlier. The upper whisker extends to the highest value below 24, which is 20.
8. Identify Outliers: 25 is an outlier.

Interpretation: The boxplot indicates that plant heights are mostly concentrated within a smaller range, with a couple of plants significantly taller than the rest. The median height is 11.5 cm. The presence of an outlier at 25 cm suggests a plant that grew exceptionally tall.

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Problem 6: Website Load Times (in seconds)

A website's load times were recorded over a period of time:

1.2, 1.5, 1.8, 2.0, 2.2, 2.5, 2.7, 2.8, 3.0, 3.3, 3.5, 4.0, 5.0, 8.0

Solution:

1. Order the Data: The data is already ordered.
2. Find the Median (Q2): With 14 data points, the median is the average of the 7th and 8th values: (2.7 + 2.8) / 2 = 2.75
3. Find Q1: The median of the lower half (1.2 to 2.7) is the 4th value: 2.0.
4. Find Q3: The median of the upper half (2.8 to 8.0) is the 11th value: 3.5.
5. Identify Minimum and Maximum: Minimum = 1.2, Maximum = 8.0.
6. Calculate IQR: IQR = Q3 - Q1 = 3.5 - 2.0 = 1.5.
7. Calculate Whisker Limits:
   • Lower Limit: Q1 - 1.5 * IQR = 2.0 - 1.5 * 1.5 = -0.25. The lower whisker extends to 1.2.
   • Upper Limit: Q3 + 1.5 * IQR = 3.5 + 1.5 * 1.5 = 5.75. The maximum value (8.0) is an outlier. The upper whisker extends to the highest value below 5.75, which is 5.0.
8. Identify Outliers: 8.0 is an outlier.

Interpretation: Most load times are relatively quick, but there's at least one instance where the load time was significantly higher. The median load time is 2.75 seconds. The outlier at 8.0 seconds suggests a potential performance issue that needs investigation.

Problem 7: Customer Satisfaction Scores (1-10)

A survey asked customers to rate their satisfaction on a scale of 1 to 10. The results are:

3, 4, 5, 5, 6, 6, 7, 7, 8, 8, 9, 9, 10, 10

Solution:

1. Order the Data: The data is already ordered.
2. Find the Median (Q2): With 14 data points, the median is the average of the 7th and 8th values: (7 + 7) / 2 = 7.
3. Find Q1: The median of the lower half (3 to 7) is the 4th value: 5.
4. Find Q3: The median of the upper half (7 to 10) is the 11th value: 9.
5. Identify Minimum and Maximum: Minimum = 3, Maximum = 10.
6. Calculate IQR: IQR = Q3 - Q1 = 9 - 5 = 4.
7. Calculate Whisker Limits:
   • Lower Limit: Q1 - 1.5 * IQR = 5 - 1.5 * 4 = -1. The lower whisker extends to 3.
   • Upper Limit: Q3 + 1.5 * IQR = 9 + 1.5 * 4 = 15. The upper whisker extends to 10.
8. Identify Outliers: There are no outliers.

Interpretation: Customer satisfaction is generally high. The median satisfaction score is 7. The IQR of 4 indicates a reasonable spread in satisfaction levels. The absence of outliers suggests consistent satisfaction among customers.

Problem 8: Number of Emails Received per Day

The number of emails a person received each day for two weeks:

15, 20, 22, 25, 25, 28, 30, 32, 35, 35, 38, 40, 42, 60

Solution:

1. Order the Data: The data is already ordered.
2. Find the Median (Q2): With 14 data points, the median is the average of the 7th and 8th values: (30 + 32) / 2 = 31.
3. Find Q1: The median of the lower half (15 to 30) is the 4th value: 25.
4. Find Q3: The median of the upper half (32 to 60) is the 11th value: 38.
5. Identify Minimum and Maximum: Minimum = 15, Maximum = 60.
6. Calculate IQR: IQR = Q3 - Q1 = 38 - 25 = 13.
7. Calculate Whisker Limits:
   • Lower Limit: Q1 - 1.5 * IQR = 25 - 1.5 * 13 = 5.5. The lower whisker extends to 15.
   • Upper Limit: Q3 + 1.5 * IQR = 38 + 1.5 * 13 = 57.5. The maximum value (60) is an outlier. The upper whisker extends to the highest value below 57.5, which is 42.
8. Identify Outliers: 60 is an outlier.

Interpretation: The number of emails received is fairly consistent most days, with one day having a significantly higher volume. The median number of emails is 31. The outlier at 60 suggests a day with an unusually high email load.

Problem 9: Commute Times (in minutes)

A person's commute times for a month are:

10, 15, 15, 18, 20, 20, 22, 25, 25, 25, 28, 30, 30, 45

Solution:

1. Order the Data: The data is already ordered.
2. Find the Median (Q2): With 14 data points, the median is the average of the 7th and 8th values: (22 + 25) / 2 = 23.5
3. Find Q1: The median of the lower half (10 to 22) is the 4th value: 18.
4. Find Q3: The median of the upper half (25 to 45) is the 11th value: 28.
5. Identify Minimum and Maximum: Minimum = 10, Maximum = 45.
6. Calculate IQR: IQR = Q3 - Q1 = 28 - 18 = 10.
7. Calculate Whisker Limits:
   • Lower Limit: Q1 - 1.5 * IQR = 18 - 1.5 * 10 = 3. The lower whisker extends to 10.
   • Upper Limit: Q3 + 1.5 * IQR = 28 + 1.5 * 10 = 43. The maximum value (45) is an outlier. The upper whisker extends to the highest value below 43, which is 30.
8. Identify Outliers: 45 is an outlier.

Interpretation: Commute times are typically around 20-30 minutes, with one day having a significantly longer commute. The median commute time is 23.5 minutes. The outlier at 45 minutes suggests a day with unusual traffic or other delays.

Problem 10: Ages of Participants in a Study

The ages of participants in a research study:

18, 20, 22, 25, 28, 30, 32, 35, 38, 40, 45, 50, 60, 75

Solution:

1. Order the Data: The data is already ordered.
2. Find the Median (Q2): With 14 data points, the median is the average of the 7th and 8th values: (32 + 35) / 2 = 33.5
3. Find Q1: The median of the lower half (18 to 32) is the 4th value: 25.
4. Find Q3: The median of the upper half (35 to 75) is the 11th value: 45.
5. Identify Minimum and Maximum: Minimum = 18, Maximum = 75.
6. Calculate IQR: IQR = Q3 - Q1 = 45 - 25 = 20.
7. Calculate Whisker Limits:
   • Lower Limit: Q1 - 1.5 * IQR = 25 - 1.5 * 20 = -5. The lower whisker extends to 18.
   • Upper Limit: Q3 + 1.5 * IQR = 45 + 1.5 * 20 = 75. The upper whisker extends to 75.
8. Identify Outliers: There are no outliers.

Interpretation: The ages of the participants are quite varied, ranging from young adults to senior citizens. The median age is 33.5 years old. The range is pretty wide.

Advanced Considerations: Skewness and Comparisons

Beyond the basic calculations and interpretations, box and whisker plots offer insights into the skewness of the data and enable comparisons between different datasets.

• Skewness: The position of the median within the box and the relative lengths of the whiskers reveal the skewness of the data.

  • Symmetrical Distribution: The median is centered within the box, and the whiskers are roughly equal in length.
  • Positive Skew (Right Skew): The median is closer to the bottom of the box, and the right whisker is longer. This indicates a tail of higher values.
  • Negative Skew (Left Skew): The median is closer to the top of the box, and the left whisker is longer. This indicates a tail of lower values.

• Comparisons: Box and whisker plots are excellent for comparing the distributions of multiple datasets side-by-side. You can visually assess differences in medians, spreads (IQRs), and the presence of outliers.

Common Mistakes to Avoid

• Confusing Median and Mean: The box and whisker plot displays the median, not the mean.
• Misinterpreting Outliers: Outliers aren't necessarily errors. They represent genuine data points that are significantly different from the rest.
• Ignoring Skewness: Pay attention to the skewness of the data, as it can significantly influence the interpretation.
• Using Boxplots for Small Datasets: Boxplots are most effective with larger datasets (generally n > 10) to provide a stable representation of the distribution.

Conclusion

Mastering box and whisker plots requires practice and a thorough understanding of their components. Still, by working through these problems and considering the advanced concepts of skewness and comparisons, you will be well-equipped to analyze and interpret data effectively using this powerful visualization tool. Practically speaking, remember to pay attention to the context of the data and use your insights to draw meaningful conclusions. The box and whisker plot is more than just a visual aid; it's a window into the underlying characteristics of your data.