Box And Whisker Plot Example Problems

Let's dive into the world of box and whisker plots, also known as box plots. These are powerful visual tools for summarizing and comparing data sets, revealing key statistics at a glance. In this article, we'll explore the fundamentals of box plots, walk through several example problems, and equip you with the knowledge to interpret and create these insightful diagrams.

Understanding Box and Whisker Plots

A box and whisker plot is a standardized way of displaying the distribution of data based on a five-number summary:

• Minimum: The smallest value in the dataset.
• First Quartile (Q1): The value below which 25% of the data falls.
• Median (Q2): The middle value of the dataset, dividing it into two equal halves.
• Third Quartile (Q3): The value below which 75% of the data falls.
• Maximum: The largest value in the dataset.

The "box" in the plot represents the interquartile range (IQR), which is the range between Q1 and Q3. The median is marked by a line within the box. The "whiskers" extend from the box to the minimum and maximum values, unless there are outliers. Outliers, which are data points significantly distant from the rest of the data, are often displayed as individual points beyond the whiskers.

Why Use Box and Whisker Plots?

• Summarize Data: They provide a concise summary of a dataset's central tendency, spread, and skewness.
• Identify Outliers: They quickly highlight potential outliers, which may warrant further investigation.
• Compare Distributions: They allow for easy comparison of the distributions of multiple datasets.
• Visual Representation: They offer a visually appealing and easily understandable representation of data.

Constructing a Box and Whisker Plot: Step-by-Step

Before we tackle example problems, let's outline the steps involved in creating a box and whisker plot:

1. Order the Data: Arrange the data in ascending order.
2. Find the Median (Q2): Determine the middle value of the dataset. If the dataset has an even number of values, the median is the average of the two middle values.
3. Find the First Quartile (Q1): Find the median of the lower half of the data (excluding the overall median if the dataset has an odd number of values).
4. Find the Third Quartile (Q3): Find the median of the upper half of the data (excluding the overall median if the dataset has an odd number of values).
5. Determine the Minimum and Maximum: Identify the smallest and largest values in the dataset.
6. Calculate the IQR: Subtract Q1 from Q3 (IQR = Q3 - Q1).
7. Identify Outliers:
   • Calculate the lower bound: Q1 - 1.5 * IQR
   • Calculate the upper bound: Q3 + 1.5 * IQR
   • Any data points below the lower bound or above the upper bound are considered outliers.
8. Draw the Plot:
   • Draw a number line that spans the range of the data.
   • Draw a box from Q1 to Q3.
   • Draw a vertical line inside the box at the median (Q2).
   • Draw whiskers extending from the box to the smallest and largest values within the outlier bounds.
   • Plot any outliers as individual points beyond the whiskers.

Box and Whisker Plot Example Problems

Now, let's put our knowledge into practice with several example problems.

Example Problem 1: Test Scores

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

65, 70, 72, 75, 78, 80, 82, 84, 85, 85, 88, 90, 92, 92, 94, 95, 96, 98, 99, 100

Solution:

1. Order the Data: The data is already ordered.
2. Find the Median (Q2): Since there are 20 data points, the median is the average of the 10th and 11th values: (85 + 88) / 2 = 86.5
3. Find the First Quartile (Q1): The median of the lower half (65 to 85) is the average of the 5th and 6th values: (78 + 80) / 2 = 79
4. Find the Third Quartile (Q3): The median of the upper half (88 to 100) is the average of the 15th and 16th values: (94 + 95) / 2 = 94.5
5. Determine the Minimum and Maximum: Minimum = 65, Maximum = 100
6. Calculate the IQR: IQR = Q3 - Q1 = 94.5 - 79 = 15.5
7. Identify Outliers:
   • Lower Bound: Q1 - 1.5 * IQR = 79 - 1.5 * 15.5 = 55.75
   • Upper Bound: Q3 + 1.5 * IQR = 94.5 + 1.5 * 15.5 = 117.75
   • There are no outliers in this dataset.
8. Draw the Plot: Draw a number line from 60 to 100 (or slightly beyond). Mark Q1 (79), Median (86.5), and Q3 (94.5) to create the box. Extend the whiskers to the minimum (65) and maximum (100).

Interpretation: The box plot shows that the scores are fairly spread out. The median is slightly above the middle of the box, suggesting a slight positive skew (more scores clustered towards the lower end). There are no outliers.

Example Problem 2: Plant Heights

The heights (in cm) of 15 plants are:

10, 12, 15, 18, 20, 22, 25, 28, 30, 32, 35, 40, 45, 50, 75

Solution:

1. Order the Data: The data is already ordered.
2. Find the Median (Q2): The middle value (8th value) is 28.
3. Find the First Quartile (Q1): The median of the lower half (10 to 25) is the 4th value: 18.
4. Find the Third Quartile (Q3): The median of the upper half (30 to 75) is the 12th value: 40.
5. Determine the Minimum and Maximum: Minimum = 10, Maximum = 75
6. Calculate the IQR: IQR = Q3 - Q1 = 40 - 18 = 22
7. Identify Outliers:
   • Lower Bound: Q1 - 1.5 * IQR = 18 - 1.5 * 22 = -15
   • Upper Bound: Q3 + 1.5 * IQR = 40 + 1.5 * 22 = 73
   • The value 75 is an outlier because it's greater than 73.
8. Draw the Plot: Draw a number line from 0 to 80 (or slightly beyond). Mark Q1 (18), Median (28), and Q3 (40) to create the box. Extend the left whisker to the minimum (10). The right whisker extends to the largest value within the outlier bound, which is 50. Mark the outlier (75) as a separate point.

Interpretation: The box plot reveals a wide spread of plant heights. The presence of an outlier (75 cm) indicates a plant that is significantly taller than the rest. The relatively long whisker on the right side suggests a positive skew in the data.

Example Problem 3: Reaction Times

The reaction times (in seconds) of 10 participants in a study are:

0.2, 0.3, 0.25, 0.4, 0.35, 0.28, 0.32, 0.45, 0.22, 0.38

Solution:

1. Order the Data: 0.2, 0.22, 0.25, 0.28, 0.3, 0.32, 0.35, 0.38, 0.4, 0.45
2. Find the Median (Q2): The average of the 5th and 6th values: (0.3 + 0.32) / 2 = 0.31
3. Find the First Quartile (Q1): The average of the 2nd and 3rd values in the lower half: (0.22 + 0.25) / 2 = 0.235
4. Find the Third Quartile (Q3): The average of the 2nd and 3rd values in the upper half: (0.35 + 0.38) / 2 = 0.365
5. Determine the Minimum and Maximum: Minimum = 0.2, Maximum = 0.45
6. Calculate the IQR: IQR = Q3 - Q1 = 0.365 - 0.235 = 0.13
7. Identify Outliers:
   • Lower Bound: Q1 - 1.5 * IQR = 0.235 - 1.5 * 0.13 = 0.04
   • Upper Bound: Q3 + 1.5 * IQR = 0.365 + 1.5 * 0.13 = 0.56
   • There are no outliers.
8. Draw the Plot: Draw a number line from 0.15 to 0.5 (or slightly beyond). Mark Q1 (0.235), Median (0.31), and Q3 (0.365) to create the box. Extend the whiskers to the minimum (0.2) and maximum (0.45).

Interpretation: This box plot indicates a relatively small range of reaction times. The median is close to the center of the box, suggesting a symmetrical distribution.

Example Problem 4: Comparing Two Datasets

Two groups of students took the same exam. The scores for Group A are:

70, 75, 80, 85, 90, 95, 100

The scores for Group B are:

60, 65, 70, 75, 80, 85, 90

Solution:

We need to create separate box plots for each group and then compare them.

Group A:

1. Order the Data: Already ordered.
2. Median (Q2): 85
3. Q1: 75
4. Q3: 95
5. Minimum: 70
6. Maximum: 100
7. IQR: 20
8. Outliers: None

Group B:

1. Order the Data: Already ordered.
2. Median (Q2): 75
3. Q1: 65
4. Q3: 85
5. Minimum: 60
6. Maximum: 90
7. IQR: 20
8. Outliers: None

Draw the Plots: Draw two box plots on the same number line for easy comparison.

Interpretation:

• Group A has a higher median score (85) than Group B (75), indicating that Group A performed better overall.
• The IQR for both groups is the same (20), suggesting similar variability in scores.
• The minimum and maximum scores are higher for Group A than Group B, further supporting the conclusion that Group A performed better.

Example Problem 5: Customer Spending

The amount spent by 12 customers at a store is:

$15, $20, $25, $30, $35, $40, $45, $50, $55, $60, $65, $100

Solution:

1. Order the Data: The data is already ordered.
2. Median (Q2): ($40 + $45) / 2 = $42.5
3. Q1: ($25 + $30) / 2 = $27.5
4. Q3: ($55 + $60) / 2 = $57.5
5. Minimum: $15
6. Maximum: $100
7. IQR: $57.5 - $27.5 = $30
8. Outliers:
   • Lower Bound: $27.5 - 1.5 * $30 = -$17.5
   • Upper Bound: $57.5 + 1.5 * $30 = $102.5
   • No outliers below, but $100 is close. It's good practice to note any points approaching the outlier boundary. While technically not an outlier, it's a value to keep in mind.

Interpretation: The box plot shows a generally increasing trend in customer spending. The maximum value ($100) is relatively high compared to the rest of the data, suggesting that one customer spent significantly more than the others.

Example Problem 6: Employee Salaries

Salaries (in thousands of dollars) of 15 employees at a company are as follows:

40, 45, 50, 55, 60, 65, 70, 75, 80, 85, 90, 95, 100, 105, 200

Solution:

1. Order the Data: The data is already ordered.
2. Median (Q2): 75
3. Q1: 55
4. Q3: 95
5. Minimum: 40
6. Maximum: 200
7. IQR: 95 - 55 = 40
8. Outliers:
   • Lower Bound: 55 - 1.5 * 40 = -5
   • Upper Bound: 95 + 1.5 * 40 = 155
   • 200 is an outlier

Interpretation: The boxplot highlights a significant outlier representing a very high salary compared to other employees. The median is 75k, which indicates that half of the employees earn below and the other half earns above this salary.

Advanced Considerations

• Modified Box Plots: These plots adjust the whisker length to a certain percentile (e.g., 9th and 91st percentiles) instead of the minimum and maximum values, reducing the impact of extreme outliers on the whisker length.
• Variable Width Box Plots: The width of the box can be proportional to the square root of the sample size, allowing for a visual comparison of sample sizes across different groups.
• Notched Box Plots: Notches around the median provide a rough visual assessment of whether the medians of two groups are significantly different. If the notches of two box plots do not overlap, this suggests a statistically significant difference between the medians.

Conclusion

Box and whisker plots are invaluable tools for data analysis and visualization. By understanding the five-number summary and the steps involved in constructing these plots, you can effectively summarize data, identify outliers, and compare distributions across multiple datasets. The example problems presented here offer a practical foundation for applying this knowledge in various real-world scenarios. Remember to practice creating and interpreting box plots to further enhance your data analysis skills. They help take potentially confusing datasets and make them significantly easier to digest and understand, allowing for quicker decision making and insights.