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Box and Whiskers Graphs

Gen Nelson

Box and Whiskers graphs are a simple, linear way to analyze sets of continuous numerical data. They are especially helpful in determining whether or not there are significant differences between sets of data. I teach this technique to ALL my students early in the year, and they use it to analyze all sorts of data quickly and efficiently. The following description will "walk you through" the way I teach this method of analysis to my students. Then, there is a brief description of some possible applications and highlights of this method.
  1. Students are given a piece of paper with rows of "o's" on it like this:

    They are instructed to use their dominant hand to write "x's" in as many of the "o's" as they can in ten seconds.

  2. Each student counts how many "x's" they have written, and this data is tabulated on the board. The table is then rewritten so the data appear in RANK ORDER from lowest to highest.


            20     23     23     24     27     27     30     34     40

  3. Five important data points are identified:

    The LOW VALUE (LV) 20
    The HIGH VALUE (HV) 40
    The MEDIAN (middle value) (M) 27
    The LOWER QUARTILE (LQ) (the median value between the low value and the whole-set median)   23
    The UPPER QUARTILE (UQ) (the median value between the high value and the whole-set median)   30

  4. Draw a number line that spans all the values and plot each of the five points identified in step three:

        19-20 -21-22-23 -24-25-26-27 -28-29-30 -31-32-33-34-35-36-27-38-39-40 -41-42

  5. Draw a "BOX" from the lower quartile to the upper quartile through the median, and then draw "WHISKERS" that extend from the ends of the box to the high and low values:

  6. Students repeat the exercise of filling in "o's", this time using their NON-dominant hand. The new data are tabulated in rank order and the LV, HV, M, LQ and UQ are identified:


        16 LV=15
        16 HV=22
        17 M= 17
        18 LQ=16
        19 UQ=19

  7. A new box and whiskers plot is constructed, using the same scale so the two plots can be aligned vertically:

  8. The following conclusions can be drawn from this analysis:

    1. The "BOX" from data set 2 does NOT overlap the "BOX" from data set one. Therefore, the differences between these two sets of data are likely to be significant (NOT due to random chance). If the boxes DID overlap (to any degree), then the two sets of data are not significantly different.
    2. The "BOX" from set 2 is smaller than the "BOX" for set 1. Therefore, there is less variability in set 2 than in set 1.


This technique can be used to analyze virtually any type of continuous numerical data. For example, in an experiment about the effect of gibberellin on plant height, students compared heights of 10 untreated seedlings (data set 1) to heights of 10 seedlings treated with gibberellin (data set 2). Often, students see that the "BOX" is quite wide, indicating highly variable data, and this observation leads them to go back and reconsider their experimental design in an effort to minimize this variability. Once they figure out how to make these graphs, my students use them all the time, and I find that their ability to interpret data and draw logical conclusions improves rapidly. Many thanks to Steve Randak for showing me this technique!

Landwehr, J.M. and Watkins, A.E., Exploring Data , Dale Seymour Publications
Dale Seymour Publications
P. O. Box 10888
Palo Alto, CA 94303

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