We'll start with some more terminology.
The top end of your box may also be called the "upper hinge"; the lower end may also be called the "lower hinge". The lower hinge is also called "the 25th percentile"; the median is "the 50th percentile"; the upper hinge is "the 75th percentile". This means that 25%, 50% and 75% of the data, respectively, is at or below that point.
The distance between the hinges may be referred to as the "H-spread" or, as you will see on the following page, the "Interquartile Range", abbreviated "IQR". ("Hinge" actually has a different technical definition, but the term is sometimes used informally. You should use whichever terminology your book or instructor uses.)
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Also, some books and software will include the overall median (Q_{2}) when computing Q_{1} and Q_{3} for data sets with an odd number of elements. The Texas Instruments calculators do not include Q_{2} in this case, so you may encounter a book answer that doesn't match the calculator answer. And different software packages use all different sorts of formulas. Be careful to use the formula from your book when doing your homework!
Additionally, the box-and-whisker plot may include a cross or an "X" marking the mean value of the data, in addition to the line inside the box that marks the median. The difference between the "X" and the median line can then be used as a measure of "skew".
Please don't ask me to explain "skew".
77, 79, 80, 86, 87, 87, 94, 99
My first step is to find the median. Since there are eight data points, the median will be the average of the two middle values:
(86 + 87) ÷ 2 = 86.5 = Q_{2}
This splits the list into two halves:
77, 79, 80, 86
...and:
87, 87, 94, 99
Since the halves of the data set each contain an even number of values, the sub-medians (or first and third quartiles) will be the averages of the middle two values of each half.
Q_{1} = (79 + 80) ÷ 2 = 79.5
Q_{3} = (87 + 94) ÷ 2 = 90.5
The minimum value is 77 and the maximum value is 99, so I have:
min: 77
Q_{1}: 79.5
Q_{2}: 86.5
Q_{3}: 90.5
max: 99
Then my plot looks like this:
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As you can see, you only need the five values listed above (min, Q_{1}, Q_{2}, Q_{3}, and max) in order to draw your box-and-whisker plot. This set of five values has been given the name "the five-number summary".
If you are asked to produce the five-number summary, they're wanting the same values as I found in the example above, but without doing the associated box-and-whisker plot.
79, 53, 82, 91, 87, 98, 80, 93
The five-number summary consists of the numbers I need for the box-and-whisker plot: the minimum value, Q_{1} (being the bottom of the box), Q_{2} (being the median of the entire set), Q_{3} (being the top of the box), and the maximum value (which is also Q_{4}). So I need to order the set, find the median and the sub-medians (or hinges), note the largest and smallest values, and then list the required values.
I'll start with ordering the list:
53, 79, 80, 82, 87, 91, 93, 98
...so the minimum is 53 and the maximum is 98.
To find the median, I'll take the average of the middle two values of the ordered list:
(82 + 87) ÷ 2 = 84.5 = Q_{2}
The median splits the data into two halves. The lower half of the list is:
53, 79, 80, 82
With an even number of values, the first quartile's value will be the average of the middle two values of this half of the list:
Q_{1} = (79 + 80) ÷ 2 = 79.5
I'll do the same thing with the upper half of the list, which is:
87, 91, 93, 98
Q_{3} = (91 + 93) ÷ 2 = 92
This gives me all the values for the five-number summary. Then my answer (with labelling, just to be clear) is:
five-number summary:
min: 53, Q_{1}: 79.5, Q_{2}: 84.5, Q_{3}: 92, max: 98
Part of the point of a box-and-whisker plot is to show how spread out your values are. But what if one or another of your values is way out of line? For this, we need to consider "outliers"....
You can use the Mathway widget below to practice finding the five-number summary (or skip the widget and continue with the lesson). Try the entered exercise, or type in your own exercise. Then click the button and scroll down to "Find the Five Number Summary" to compare your answer to Mathway's.
(Click "Tap to view steps" to be taken directly to the Mathway site for a paid upgrade.)
URL: http://www.purplemath.com/modules/boxwhisk2.htm
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