How to Create a Box and Whisker Plot

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See the skew of your data in minutes with this easy graph
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A box and whisker plot, or a box plot, is a diagram that shows the distribution of a set of data by plotting its averages. This makes it easy to analyze how the data you collected is spread out along a number line. To draw one yourself, all you need to do is order the numbers from least to greatest, find the average of the entire set, calculate the averages of the largest and the smallest halves of the data, and plot it all on a number line. We’ll also tell you how to calculate the interquartile range and plot any outliers. Keep reading for more!

Section 1 of 2:

Creating a Box and Whisker Plot

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  1. Take all the numbers from your set of data and line them up in order, so that the smallest numbers are on the left and the largest numbers are on the right. [1]
    • Example 1: Organize the data set “2, 5, 1, 3, 4” to get “1, 2, 3, 4, 5.”
    • Example 2: Reorder the data set “18, 39, 19, 34, 28, 78, 22, 24” to get “18, 19, 22, 24, 28, 34, 39, 78.”
  2. Cross out one number on each side of the data set to find the middle number, or the median. If you have an even amount of numbers in your data set, add the two middle numbers together and divide the answer by 2. [2]
    • Example 1: The median of “1, 2, 3, 4, 5” is 3.
    • Example 2: The median of “18, 19, 22, 24, 28, 34, 39, 78” is 24 and 28.
      • 24 + 28 = 52
      • 52 / 2 = 26. The median is 26.
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  3. The median marks the middle of the data set, dividing the numbers into a lower region and an upper region. Find the median of the numbers in the lower region to get the first quartile (Q1), then find the median of the numbers in the upper region to get the third quartile (Q3). [3]
    • Example 1: Since the median is 3, the lower region is 1 and 2 and the upper region is 4 and 5. The median of each is “1 and 2” and “4 and 5,” so add both numbers and divide by 2.
      • Q1: 1 + 2 = 3 and 3 / 2 = 1.5
      • Q3: 4 + 5 = 9 and 9 / 2 = 4.5
    • Example 2: The median is between 24 and 28, so the lower region is “18, 19, 22, 24” and the upper region is “28, 34, 39, 78.” The median of each region is “19 and 22” and “34 and 39,” so add both numbers and divide by 2.
      • Q1: 19 + 22 = 41 and 41 / 2 = 20.5
      • Q3: 34 + 39 = 73 and 73 / 2 = 36.5
  4. Draw a long horizontal line and number it evenly from 0 to a number above the highest number in your data set. Plot Q1, Q2, and Q3 above the line with small vertical lines. Connect the top and bottom of the first quartile to the top and bottom of the second and third quartiles to make a box. [4]
    • If you're dealing with decimals like 1.5 or 20.5, label them on the number line as well.
    • Optionally, label each quartile line as Q1, Q2, and Q3.
  5. Find the IQR by subtracting Q1 from Q3. [5] Use the formulas Q1 - 1.5 x IQR and Q3 + 1.5 x IQR to find the range of the box plot. If a number goes outside of this range, then it’s an outlier. Mark outliers as dots on your number line below the box plot. [6]
    • Example 1: Q3 - Q1 = IQR
      Q1 = 1.5 and Q3 = 4.5
      4.5 - 1.5 = 3, so the IQR is 3.
      • Q1 - 1.5 x IQR = the lower part of the range and Q3 +1.5 x IQR = the upper part of the range.
        1.5 - 1.5 x 3 = 0
        4.5 + 1.5 x 3 = 9
      • The range of the box plot is 0 to 9 or [0, 9] . The lowest number (1) and the highest number (5) in the data set both fit in that range, so there are no outliers.
    • Example 2: Q3 - Q1 = IQR
      Q1 = 20.5 and Q3 = 36.5
      36.5 - 20.5 = 16, so the IQR is 16.
      • Q1 - 1.5 x IQR = the lower part of the range and Q3 + 1.5 x IQR = the upper part of the range.
        20.5 - 1.5 x 16 = -3.5
        36.5 + 2.5 x 16 = 76.5
      • The range of the box plot is -3.5 to 76.5 or [-3.5, 76.5] . The lowest number (18) fits in that range, but the highest number (78) doesn’t. Plot 78 as a dot along the number line.
  6. Plot the highest and lowest numbers within the range of the box plot as small vertical lines above the number line. Connect those lines to the sides of the box plot to create “whiskers.” [7] If you have outliers, connect the next number within the range to the box plot.
    • Example 1: Plot 1 and 5 as small vertical lines slightly above the number line, then connect them to the box plot.
    • Example 2: Plot 18 as a small line above the number line. Since 78 is an outlier, plot the next lowest number, 39, as another small line. Connect both to the box plot.
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Section 2 of 2:

Benefits of a Box and Whisker Plot

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  1. The box plot, representing the averages of the data set, shows where the data is concentrated along a number line. [8] Not only can show you the general trend of the numbers you collected, it can also make it easier to compare multiple sets of data to each other.
    • If the second example we used above represents the age of people who attend music festivals, you can tell from a glance that most people are ages 20.5 (Q1) to 36.5 (Q3) and the average age is 26 (Q2).
      • We could then compare this box plot to another one, which says that most people are ages 45.5 (Q1) to 67.5 (Q3) and the average age is 56.5 (Q2), and see a huge difference between the two.
  2. While many charts and graphs choose to leave them out completely, box and whisker plots show outliers as points along the same number line. [9] Outliers can be important for professionals like scientists and psychologists because they reveal other patterns that might otherwise go unnoticed. [10]
    • There might be an illness that targets people between ages 50 (Q1) to 90 (Q3), but an outlier could be 14 years old. Scientists might then investigate why the 14-year-old got sick.
    • Box plots are also useful for any professional who uses statistics, like research analysts, forensic technicians, and researchers. Meteorologists also use box plots.
  3. Math teacher Joseph Meyer says that a box and whisker plot can be a great alternative to a bar graph or histogram because it’s “more about the quartiles” and represents “where everyone's falling” on a number line.
    • While bar graphs and histograms represent values in certain categories, box plots give you a summary of a large amount of data. [11]
    • However, if you have a smaller set of data or variables that need to be represented, use a bar chart or histogram instead. [12]
    • Box and whisker plots also take up much less space than other charts and graphs.
    EXPERT TIP

    Joseph Meyer

    Math Teacher
    Joseph Meyer is a High School Math Teacher based in Pittsburgh, Pennsylvania. He is an educator at City Charter High School, where he has been teaching for over 7 years. Joseph is also the founder of Sandbox Math, an online learning community dedicated to helping students succeed in Algebra. His site is set apart by its focus on fostering genuine comprehension through step-by-step understanding (instead of just getting the correct final answer), enabling learners to identify and overcome misunderstandings and confidently take on any test they face. He received his MA in Physics from Case Western Reserve University and his BA in Physics from Baldwin Wallace University.
    Joseph Meyer
    Math Teacher

    Box plots provide a clear way to display data. These plots, also called whisker plots, display the spread of your data concisely by using quartiles. They work particularly well to compare distributions across multiple datasets. However, histograms offer a more detailed picture of the data's entire distribution.

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  • Question
    Can you explain what a box and whisker plot is?
    Joseph Meyer
    Math Teacher
    Joseph Meyer is a High School Math Teacher based in Pittsburgh, Pennsylvania. He is an educator at City Charter High School, where he has been teaching for over 7 years. Joseph is also the founder of Sandbox Math, an online learning community dedicated to helping students succeed in Algebra. His site is set apart by its focus on fostering genuine comprehension through step-by-step understanding (instead of just getting the correct final answer), enabling learners to identify and overcome misunderstandings and confidently take on any test they face. He received his MA in Physics from Case Western Reserve University and his BA in Physics from Baldwin Wallace University.
    Joseph Meyer
    Math Teacher
    Expert Answer
    A box-and-whisker plot primarily focuses on illustrating the distribution of data through quartiles, providing insights into where values lie within a dataset. In comparison to histograms, which display frequency distributions, a box and whisker plot offers a more comprehensive representation of the data's spread and central tendency. By depicting the minimum, maximum, and quartiles, this graphical tool not only highlights the central tendency but also reveals the spread and skewness of the data. Consequently, it serves as a valuable alternative to histograms, offering a more nuanced understanding of the distribution and variability within a dataset.
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  • Question
    How do you know what the upper quadrant is?
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    The upper quadrant would be the set of numbers to the right of your median, if they are in order from least to greatest.
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  • Question
    What if I have an even amount of numbers, and the two middle numbers are the same?
    Community Answer
    The median is the middle number or the average of the two numbers if you have an even amount of numbers. If you have two that are the same, you do not need to average since they are the same. If they weren't, you would average the two to get the median.
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      Thanks for reading our article! If you’d like to learn more about math, check out our in-depth interview with Joseph Meyer .

      References

      1. ↑ https://www.purplemath.com/modules/boxwhisk.htm
      2. ↑ https://www.purplemath.com/modules/boxwhisk.htm
      3. ↑ https://www.purplemath.com/modules/boxwhisk.htm
      4. ↑ https://www.mathsisfun.com/definitions/box-and-whisker-plot.html
      5. ↑ https://stattrek.com/statistics/charts/boxplot
      6. ↑ https://www.khanacademy.org/math/statistics-probability/summarizing-quantitative-data/box-whisker-plots/a/identifying-outliers-iqr-rule
      7. ↑ https://www.mathsisfun.com/definitions/box-and-whisker-plot.html
      8. ↑ https://stattrek.com/statistics/charts/boxplot
      9. ↑ https://stattrek.com/statistics/charts/boxplot
      1. ↑ https://www.psychologytoday.com/us/blog/singular-perspective/202102/the-importance-outliers
      2. ↑ https://www.nature.com/articles/nmeth.2807
      3. ↑ https://www.jmp.com/en_us/statistics-knowledge-portal/exploratory-data-analysis/box-plot.html

      About This Article

      Article Summary X

      To make a box and whisker plot, start by organizing the numbers in your data set from least to greatest and finding the median. Then, find the first quartile, which is the median of the beginning of the data set, and the third quartile, which is the median of the end of the data set. Once you've done that, draw a plot line and mark the quartiles and the median on it. Finally, connect the quartiles and median with horizontal lines to make a box, and then mark the outliers. To learn how to interpret a box and whisker plot, scroll down!

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