What Is A Box And Whisker Plot And How To Use It?

A box and whisker plot visually summarizes data distribution through quartiles, showcasing the median, spread, and potential outliers; WHAT.EDU.VN offers comprehensive explanations and resources for mastering this statistical tool. By understanding its components, you can quickly interpret data sets and compare distributions, gaining valuable insights into data variability and central tendencies, including using interquartile range and data visualization.

1. What Is A Box And Whisker Plot?

A box and whisker plot, often simply called a boxplot, is a standardized way of displaying the distribution of data based on a five-number summary (minimum, first quartile (Q1), median, third quartile (Q3), and maximum). It can tell you about your outliers and what their values are, and also if your data is symmetrical, how tightly your data is grouped, and if and how your data is skewed. This visualization is particularly useful for comparing distributions between different groups or datasets, helping to identify differences in central tendencies, variability, and skewness.

1.1. Key Components of a Box and Whisker Plot

Understanding the different parts of a boxplot is crucial for accurate interpretation:

• Minimum: The smallest data point in the dataset, excluding any outliers.
• First Quartile (Q1): Represents the 25th percentile of the data; 25% of the data falls below this value.
• Median (Q2): The middle value of the dataset, dividing it into two equal halves.
• Third Quartile (Q3): Represents the 75th percentile; 75% of the data falls below this value.
• Maximum: The largest data point in the dataset, excluding any outliers.
• Box: Drawn from Q1 to Q3, it contains the middle 50% of the data. The length of the box represents the interquartile range (IQR).
• Whiskers: Lines extending from each end of the box to the minimum and maximum values, showing the range of the remaining data.
• Outliers: Data points that fall outside the whiskers, typically defined as points below Q1 – 1.5(IQR) or above Q3 + 1.5(IQR). These are often plotted as individual points.

Alt Text: A visual representation showing the components of a box and whisker plot, including the minimum, first quartile (Q1), median, third quartile (Q3), maximum, outliers, box, and whiskers.

1.2. Why Use Box and Whisker Plots?

Box and whisker plots are used because they provide a clear and concise way to visualize and compare the distribution of data across different groups. Unlike histograms or density plots, which show the detailed shape of the distribution, boxplots offer a summary view, highlighting key statistics that are easy to interpret. Here are some specific advantages:

• Summarizes Data: A boxplot quickly summarizes a dataset, showing the median, quartiles, and range.
• Identifies Outliers: It easily identifies outliers, which can be important for data cleaning or further analysis.
• Compares Distributions: Boxplots make it easy to compare the distributions of different datasets side by side.
• Detects Skewness: The position of the median within the box and the length of the whiskers can indicate whether the data is skewed.

1.3. Applications in Various Fields

Box and whisker plots are widely used across various disciplines due to their simplicity and effectiveness in summarizing and comparing data:

• Education: Comparing test scores across different classes or schools.
• Healthcare: Analyzing patient data, such as blood pressure or cholesterol levels, across different treatment groups.
• Finance: Assessing the distribution of stock prices or investment returns.
• Environmental Science: Comparing pollution levels at different locations or times.
• Manufacturing: Monitoring product quality and identifying process variations.

2. How to Read and Interpret a Box and Whisker Plot?

Interpreting a box and whisker plot involves understanding what each component represents and how they collectively provide insights into the data distribution. Here’s a step-by-step guide:

2.1. Identifying the Median

The median is represented by the line inside the box. Its position within the box indicates the central tendency of the data:

• Median in the Middle: If the median is in the middle of the box, the data is likely symmetrically distributed.
• Median Closer to Q1: If the median is closer to the first quartile (Q1), the data is positively skewed (skewed to the right). This means there are more lower values and a few high values pulling the mean upwards.
• Median Closer to Q3: If the median is closer to the third quartile (Q3), the data is negatively skewed (skewed to the left). This means there are more higher values and a few low values pulling the mean downwards.

2.2. Understanding the Quartiles

The edges of the box represent the first quartile (Q1) and the third quartile (Q3). The distance between Q1 and Q3 is the interquartile range (IQR), which measures the spread of the middle 50% of the data.

• Small IQR: A small IQR indicates that the middle 50% of the data is tightly clustered around the median.
• Large IQR: A large IQR indicates that the middle 50% of the data is more spread out.

2.3. Analyzing the Whiskers

The whiskers extend from the box to the minimum and maximum values, excluding outliers. They show the range of the remaining data and can provide insights into the data’s variability:

• Equal Length Whiskers: If the whiskers are approximately equal in length, the data is likely symmetrically distributed.
• Unequal Length Whiskers: If one whisker is longer than the other, the data is skewed in the direction of the longer whisker.

2.4. Spotting Outliers

Outliers are plotted as individual points beyond the whiskers. They represent data points that are significantly different from the rest of the data:

• No Outliers: The absence of outliers suggests that the data is relatively consistent.
• Few Outliers: A few outliers may indicate unusual but valid data points.
• Many Outliers: A large number of outliers may suggest errors in data collection or the presence of a different underlying process.

2.5. Interpreting Skewness

Skewness refers to the asymmetry of the data distribution. Boxplots can help identify skewness:

• Symmetric Distribution: The median is in the center of the box, and the whiskers are approximately equal in length.
• Positive Skew (Right Skew): The median is closer to Q1, the right whisker is longer, and outliers may be present on the right side.
• Negative Skew (Left Skew): The median is closer to Q3, the left whisker is longer, and outliers may be present on the left side.

2.6. Comparing Multiple Boxplots

Boxplots are particularly useful for comparing the distributions of multiple datasets. To compare boxplots:

• Compare Medians: Look at the positions of the medians to compare the central tendencies of the datasets.
• Compare IQRs: Compare the lengths of the boxes to compare the variability of the middle 50% of the data.
• Compare Whiskers: Compare the lengths of the whiskers to compare the overall range of the data.
• Compare Outliers: Compare the number and position of outliers to identify differences in extreme values.

For example, consider the following scenario: You are comparing the test scores of two classes, Class A and Class B. Class A has a median of 80, an IQR of 10, and no outliers. Class B has a median of 75, an IQR of 15, and two outliers above 95. This suggests that Class A performed better overall (higher median), had less variability (smaller IQR), and had more consistent scores (no outliers) compared to Class B.

Alt Text: An illustration comparing two box plots, demonstrating how to analyze differences in medians, IQRs, whiskers, and outliers between two datasets.

3. How to Create a Box and Whisker Plot?

Creating a box and whisker plot involves several steps, from calculating the necessary statistics to drawing the plot itself. Here’s a detailed guide:

3.1. Manual Calculation

1. Order the Data: Arrange the data in ascending order.

2. Find the Median (Q2):

   • If the number of data points is odd, the median is the middle value.
   • If the number of data points is even, the median is the average of the two middle values.

3. Find the First Quartile (Q1):

   • The first quartile is the median of the lower half of the data.
   • If the number of data points in the lower half is odd, Q1 is the middle value.
   • If the number of data points in the lower half is even, Q1 is the average of the two middle values.

4. Find the Third Quartile (Q3):

   • The third quartile is the median of the upper half of the data.
   • If the number of data points in the upper half is odd, Q3 is the middle value.
   • If the number of data points in the upper half is even, Q3 is the average of the two middle values.

5. Calculate the Interquartile Range (IQR):

   • IQR = Q3 – Q1

6. Determine Outliers:

   • Lower Bound: Q1 – 1.5(IQR)
   • Upper Bound: Q3 + 1.5(IQR)
   • Any data points below the Lower Bound or above the Upper Bound are considered outliers.

7. Determine Minimum and Maximum Values (Excluding Outliers):

   • The minimum value is the smallest data point that is not an outlier.
   • The maximum value is the largest data point that is not an outlier.

8. Draw the Boxplot:

   • Draw a number line that covers the range of your data.
   • Draw a box from Q1 to Q3.
   • Draw a line inside the box to represent the median (Q2).
   • Draw whiskers from each end of the box to the minimum and maximum values (excluding outliers).
   • Plot outliers as individual points beyond the whiskers.

3.2. Using Software and Tools

Creating boxplots manually can be time-consuming, especially with large datasets. Fortunately, many software tools and programming languages can automate the process. Here are some popular options:

• Microsoft Excel:

  • Enter your data into a spreadsheet.
  • Select the data and go to Insert > Statistics Chart > Box and Whisker.
  • Customize the chart as needed.

• Google Sheets:

  • Enter your data into a spreadsheet.
  • Select the data and go to Insert > Chart.
  • Choose the Box plot chart type.
  • Customize the chart as needed.

• R:

  • Use the boxplot() function in R.
  • Example: boxplot(data, main="Boxplot", ylab="Values")

• Python (with Matplotlib or Seaborn):

  • Use the boxplot() function in Matplotlib or Seaborn.
  • Example:

import matplotlib.pyplot as plt
import seaborn as sns

sns.boxplot(x=data)
plt.show()

3.3. Step-by-Step Example Using Excel

Let’s create a boxplot using Excel with the following dataset: 2, 3, 5, 6, 7, 8, 9, 10, 12, 14

1. Enter the Data:

   • Open Excel and enter the data into a column (e.g., column A).

2. Calculate the Statistics:

   • In separate cells, calculate the minimum, Q1, median, Q3, and maximum using Excel functions:
     • Minimum: =MIN(A1:A10)
     • Q1: =QUARTILE.INC(A1:A10,1)
     • Median: =MEDIAN(A1:A10)
     • Q3: =QUARTILE.INC(A1:A10,3)
     • Maximum: =MAX(A1:A10)

3. Create the Boxplot:

   • Select the original data (A1:A10).
   • Go to Insert > Statistics Chart > Box and Whisker.
   • Excel will create a boxplot based on the data.

4. Customize the Chart:

   • Click on the chart to customize it. You can change the chart title, axis labels, and other formatting options.
   • To display outliers, ensure that the “Show Outliers” option is enabled in the chart settings.

3.4. Common Mistakes to Avoid

• Incorrectly Calculating Quartiles: Ensure you are using the correct method for calculating quartiles (inclusive or exclusive).
• Misinterpreting Outliers: Remember that outliers are not necessarily errors. They may represent valid but unusual data points.
• Ignoring Skewness: Pay attention to the position of the median and the length of the whiskers to identify skewness.
• Comparing Incompatible Datasets: Ensure that the datasets you are comparing are relevant and comparable.

Alt Text: A screenshot of Microsoft Excel displaying a box and whisker plot, demonstrating how to create the plot using Excel’s built-in charting tools.

4. Advanced Techniques and Considerations

While basic box and whisker plots are useful for summarizing data, advanced techniques can provide even more insights. Here are some advanced considerations and techniques:

4.1. Notched Boxplots

Notched boxplots add notches around the median, providing a visual indication of the confidence interval around the median. The notches extend to +/- 1.58 * IQR / sqrt(n), where n is the sample size.

• Overlapping Notches: If the notches of two boxplots do not overlap, there is strong evidence that the medians of the two groups are significantly different.
• Non-Overlapping Notches: If the notches of two boxplots overlap, there is no strong evidence that the medians are significantly different.

4.2. Variable Width Boxplots

Variable width boxplots make the width of the box proportional to the square root of the sample size. This allows you to visually compare the sample sizes of different groups:

• Wider Box: A wider box indicates a larger sample size.
• Narrower Box: A narrower box indicates a smaller sample size.

4.3. Boxplots with Additional Data Points

Adding individual data points to a boxplot can provide more detailed information about the data distribution. This can be done by overlaying a scatter plot or a strip plot on top of the boxplot:

• Scatter Plot: Plots each data point as an individual point, showing the exact distribution of the data.
• Strip Plot: Similar to a scatter plot but with points slightly jittered to avoid overlap.

4.4. Handling Different Sample Sizes

When comparing boxplots with different sample sizes, it’s important to consider the effect of sample size on the variability of the data. Larger sample sizes tend to produce more stable estimates of the median and quartiles, while smaller sample sizes are more prone to sampling error.

• Use Variable Width Boxplots: As mentioned above, variable width boxplots can help visualize differences in sample size.
• Calculate Confidence Intervals: Calculate confidence intervals for the medians and quartiles to quantify the uncertainty associated with each estimate.

4.5. Dealing with Non-Normal Data

Boxplots are effective for summarizing data regardless of whether it is normally distributed. However, when dealing with non-normal data, it’s important to consider the following:

• Skewness: Pay attention to the position of the median and the length of the whiskers to identify skewness.
• Outliers: Be cautious when interpreting outliers, as they may be more common in non-normal distributions.
• Transformations: Consider transforming the data (e.g., using a logarithmic transformation) to make it more normal before creating boxplots.

4.6. Combining Boxplots with Other Visualizations

Boxplots can be combined with other visualizations to provide a more comprehensive view of the data. For example:

• Histograms: Combine boxplots with histograms to show both the summary statistics and the detailed shape of the distribution.
• Density Plots: Combine boxplots with density plots to show the smooth estimate of the distribution.
• Violin Plots: Violin plots are similar to boxplots but show the estimated probability density of the data at different values.

Alt Text: An example of advanced box plots, including notched boxplots, variable width boxplots, and boxplots combined with scatter plots, showcasing more detailed data analysis.

5. Real-World Examples and Case Studies

To illustrate the practical applications of box and whisker plots, let’s consider a few real-world examples and case studies:

5.1. Case Study: Comparing Student Test Scores

A school district wants to compare the performance of students in two different schools, School A and School B. They collect test scores from a sample of students in each school and create boxplots to compare the distributions:

• School A: Median = 82, Q1 = 75, Q3 = 88, Minimum = 65, Maximum = 95, No outliers
• School B: Median = 78, Q1 = 70, Q3 = 85, Minimum = 60, Maximum = 98, One outlier at 98

Interpretation:

• Students in School A generally performed better than students in School B (higher median).
• The scores in School A were more consistent than in School B (smaller IQR).
• School B had one student who performed exceptionally well (outlier at 98).

5.2. Example: Analyzing Customer Satisfaction Ratings

A company wants to analyze customer satisfaction ratings for two different products, Product X and Product Y. They collect ratings on a scale of 1 to 10 and create boxplots to compare the distributions:

• Product X: Median = 8, Q1 = 7, Q3 = 9, Minimum = 5, Maximum = 10, No outliers
• Product Y: Median = 6, Q1 = 5, Q3 = 7, Minimum = 2, Maximum = 9, Two outliers at 2 and 3

Interpretation:

• Customers are generally more satisfied with Product X than with Product Y (higher median).
• The ratings for Product X are more consistent than for Product Y (smaller IQR).
• Product Y has some customers who are very dissatisfied (outliers at 2 and 3), which may warrant further investigation.

5.3. Application: Monitoring Manufacturing Quality

A manufacturing company monitors the quality of its products by measuring a key performance metric (e.g., diameter of a bolt) and creating boxplots to track the distribution of measurements over time. By comparing boxplots from different production runs, they can identify any changes in the process and take corrective action:

• Production Run 1: Median = 10.0 mm, Q1 = 9.8 mm, Q3 = 10.2 mm, Minimum = 9.6 mm, Maximum = 10.4 mm, No outliers
• Production Run 2: Median = 10.1 mm, Q1 = 9.9 mm, Q3 = 10.3 mm, Minimum = 9.7 mm, Maximum = 10.5 mm, No outliers
• Production Run 3: Median = 9.9 mm, Q1 = 9.7 mm, Q3 = 10.1 mm, Minimum = 9.5 mm, Maximum = 10.3 mm, One outlier at 9.5 mm

Interpretation:

• The process was relatively stable in Production Runs 1 and 2, with consistent medians and IQRs.
• In Production Run 3, the median shifted slightly lower, and there was one outlier below the acceptable range, indicating a potential issue with the manufacturing process that needs to be addressed.

Alt Text: A real-world example of box plots used to compare salaries by job type, illustrating how different distributions can be easily compared using boxplot visualizations.

6. Common Misconceptions About Box and Whisker Plots

Despite their usefulness, box and whisker plots are often misunderstood. Here are some common misconceptions:

6.1. Misconception: Boxplots Show the Distribution of the Data

• Reality: Boxplots summarize the distribution but do not show the detailed shape of the distribution like histograms or density plots. Boxplots highlight key statistics such as the median, quartiles, and outliers, but they do not reveal whether the data is unimodal, bimodal, or has other complex features.

6.2. Misconception: The Whiskers Always Extend to the Minimum and Maximum Values

• Reality: The whiskers extend to the minimum and maximum values excluding outliers. Outliers are plotted as individual points beyond the whiskers. This is a critical distinction, as it prevents extreme values from distorting the visual representation of the main body of the data.

6.3. Misconception: Outliers Are Always Errors

• Reality: Outliers are data points that are significantly different from the rest of the data, but they are not necessarily errors. Outliers may represent valid but unusual data points. They could be due to natural variation, measurement errors, or the presence of a different underlying process. It’s important to investigate outliers to determine their cause before deciding whether to remove them.

6.4. Misconception: Boxplots Can Only Be Used for Normally Distributed Data

• Reality: Boxplots can be used for any type of data, regardless of whether it is normally distributed. In fact, boxplots are particularly useful for identifying skewness and outliers in non-normal data.

6.5. Misconception: A Longer Box Always Means Greater Variability

• Reality: A longer box (larger IQR) indicates greater variability in the middle 50% of the data. However, it does not necessarily mean greater overall variability. The range of the data (as indicated by the whiskers and outliers) also contributes to the overall variability.

6.6. Misconception: Comparing Boxplots Is Always Straightforward

• Reality: Comparing boxplots can be more complex when the sample sizes are different or when the data is highly skewed. It’s important to consider these factors when interpreting boxplots and to use advanced techniques (such as notched boxplots or variable width boxplots) to account for them.

6.7. Misconception: Boxplots Provide All the Information Needed for Data Analysis

• Reality: Boxplots are a useful tool for summarizing and comparing data, but they should not be used in isolation. It’s important to combine boxplots with other visualizations and statistical analyses to gain a more comprehensive understanding of the data.

6.8. Misconception: The Median Is Always the Best Measure of Central Tendency

• Reality: While the median is a robust measure of central tendency (less sensitive to outliers than the mean), it is not always the best choice. The best measure of central tendency depends on the specific characteristics of the data and the goals of the analysis. In some cases, the mean or the mode may be more appropriate.

Understanding these common misconceptions can help you avoid misinterpreting box and whisker plots and make more informed decisions based on the data.

Alt Text: A diagram highlighting common misconceptions about box and whisker plots, emphasizing the importance of accurate interpretation and avoiding misunderstandings.

7. Best Practices for Using Box and Whisker Plots

To ensure that you are using box and whisker plots effectively, follow these best practices:

7.1. Clearly Label Axes and Provide Context

Always label the axes of your boxplots and provide context to help your audience understand the data. Include a clear title, axis labels, and a description of the data being presented.

7.2. Use Consistent Scales

When comparing multiple boxplots, use consistent scales to make it easier to compare the distributions. This is particularly important when comparing boxplots from different sources or with different units of measurement.

7.3. Order Boxplots Logically

When presenting multiple boxplots, order them in a logical way to help your audience follow the story. For example, you might order boxplots by group, time period, or magnitude of the median.

7.4. Highlight Key Findings

Use annotations or callouts to highlight key findings in your boxplots. For example, you might draw attention to outliers, differences in medians, or changes in variability.

7.5. Avoid Clutter

Keep your boxplots clean and uncluttered to make them easier to interpret. Avoid using too many colors or adding unnecessary details.

7.6. Choose the Right Tool

Use the right tool for creating your boxplots. While you can create boxplots manually or with simple tools like Excel, more advanced tools like R or Python offer greater flexibility and customization options.

7.7. Check Your Work

Always check your work to ensure that your boxplots are accurate and that you have not made any mistakes in the calculation or interpretation of the data.

7.8. Consider Your Audience

Consider your audience when creating boxplots. Use language and visuals that are appropriate for their level of expertise. If you are presenting to a non-technical audience, avoid using jargon and focus on the key takeaways.

7.9. Use Boxplots in Combination with Other Visualizations

As mentioned earlier, boxplots are most effective when used in combination with other visualizations. Consider combining boxplots with histograms, density plots, or scatter plots to provide a more comprehensive view of the data.

By following these best practices, you can create box and whisker plots that are clear, accurate, and informative.

Alt Text: An example illustrating best practices for using box and whisker plots, including clear labeling, consistent scales, logical ordering, and highlighting key findings.

8. Frequently Asked Questions (FAQs) About Box and Whisker Plots

Here are some frequently asked questions about box and whisker plots:

Question	Answer
What is the interquartile range (IQR)?	The interquartile range (IQR) is the range between the first quartile (Q1) and the third quartile (Q3). It represents the spread of the middle 50% of the data.
How do you identify outliers in a boxplot?	Outliers are identified as data points that fall below Q1 – 1.5(IQR) or above Q3 + 1.5(IQR). They are plotted as individual points beyond the whiskers.
What does skewness mean in the context of a boxplot?	Skewness refers to the asymmetry of the data distribution. In a boxplot, skewness is indicated by the position of the median within the box and the length of the whiskers. A median closer to Q1 and a longer right whisker indicate positive skew, while a median closer to Q3 and a longer left whisker indicate negative skew.
Can boxplots be used for categorical data?	Boxplots are typically used for numerical data. For categorical data, bar charts or pie charts are more appropriate.
How do you compare two or more boxplots?	To compare boxplots, look at the positions of the medians to compare the central tendencies, compare the lengths of the boxes to compare the variability of the middle 50% of the data, compare the lengths of the whiskers to compare the overall range of the data, and compare the number and position of outliers to identify differences in extreme values.
What are the advantages of using boxplots over histograms?	Boxplots provide a quick summary of the data, highlight key statistics, and easily identify outliers. Histograms show the detailed shape of the distribution but can be more difficult to interpret. Boxplots are particularly useful for comparing distributions between different groups.
How does sample size affect the interpretation of boxplots?	Larger sample sizes tend to produce more stable estimates of the median and quartiles, while smaller sample sizes are more prone to sampling error. When comparing boxplots with different sample sizes, consider using variable width boxplots or calculating confidence intervals for the medians and quartiles.
What is a notched boxplot?	A notched boxplot adds notches around the median, providing a visual indication of the confidence interval around the median. If the notches of two boxplots do not overlap, there is strong evidence that the medians of the two groups are significantly different.
Can boxplots be used to identify bimodal distributions?	Boxplots are not ideal for identifying bimodal distributions, as they only show summary statistics. Histograms or density plots are better suited for visualizing the shape of the distribution and identifying multiple modes.
How do you handle missing data when creating boxplots?	Missing data should be handled appropriately before creating boxplots. You can either remove the missing data points or impute them using appropriate methods. The choice depends on the amount of missing data and the goals of the analysis.
What is the difference between a boxplot and a violin plot?	A boxplot shows summary statistics like median, quartiles, and outliers, while a violin plot shows the probability density of the data at different values, offering a more detailed view of the distribution’s shape.
How can boxplots help in data quality assessment?	Boxplots can reveal data inconsistencies and errors by highlighting outliers, skewness, and unusual distributions. This helps analysts identify potential data quality issues that need further investigation.
Are there any limitations to using boxplots?	Yes, boxplots can oversimplify complex data distributions and may not capture nuances like multimodality. They also require careful interpretation when comparing datasets with unequal sample sizes or non-normal distributions.
In what fields are boxplots most commonly used?	Boxplots are widely used in statistics, data analysis, finance, healthcare, and engineering to compare distributions, identify outliers, and monitor process variations.
How do you create a boxplot in Python?	You can create a boxplot in Python using libraries like Matplotlib or Seaborn. For example, with Seaborn, you would use the sns.boxplot(x=data) function.
What does the length of the “whiskers” indicate in a boxplot?	The length of the whiskers indicates the range of data excluding outliers. Longer whiskers suggest greater variability within the data, while shorter whiskers indicate less variability.
How do you interpret a boxplot when it’s very short or compressed?	A very short or compressed boxplot suggests that the data is tightly clustered around the median with low variability. It may also indicate a uniform distribution or a small dataset.
When should you use a boxplot instead of a bar chart?	Use a boxplot to compare the distribution of continuous numerical data, while a bar chart is more suitable for comparing the frequency or magnitude of categorical data.
Can boxplots be used for time series data?	While not primarily designed for time series, boxplots can be used to compare the distribution of data across different time periods, helping to identify changes in central tendency and variability over time.
How do you explain a boxplot to someone with no statistical knowledge?	Explain that the box shows where most of the typical data points are, the line inside the box marks the middle value, the lines extending from the box show the spread of the rest of the data, and the dots are unusual values that are far away from the rest.

9. Conclusion

Box and whisker plots are a powerful tool for summarizing and comparing data distributions. By understanding the key components of a boxplot and following best practices for their use, you can gain valuable insights into your data and make more informed decisions. Whether you are analyzing student test scores, customer satisfaction ratings, or manufacturing quality, boxplots can help you identify trends, outliers, and differences between groups. Embrace the power of boxplots to enhance your data analysis toolkit.

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