Draw a Box and Whisker Plot to Display the Data
Quartiles, Boxes, and Whiskers
For many computations in statistics, it is causeless that your information points (that is, the numbers in your list) are clustered around some central value; in other words, information technology is assumed that there is an "average" of some sort. The "box" in the box-and-whisker plot contains, and thereby highlights, the eye portion of these information points.
To create a box-and-whisker plot, we start past ordering our data (that is, putting the values) in numerical order, if they aren't ordered already. So nosotros find the median of our data.
The median divides the data into two halves. To divide the data into quarters, we then notice the medians of these two halves.
Note: If we have an even number of values, then the first median was the average of the two centre values, then we include the middle values in our sub-median computations. If we have an odd number of values, so the commencement median was an actual data point, and so we do not include that value in our sub-median computations. That is, to find the sub-medians, nosotros're only looking at the values that take not yet been used.
And so we have 3 points: the first middle betoken (the median), and the middle points of the two halves (what I've been calling the "sub-medians"). These three points divide the unabridged data set into quarters, called "quartiles".
The summit indicate of each quartile has a proper noun, being a "Q" followed by the number of the quarter. And so the top signal of the outset quarter of the data points is "Qi ", and then forth. Note that Qone is also the middle number for the first half of the list, Q2 is also the middle number for the whole listing, Qthree is the middle number for the 2nd one-half of the list, and Q4 is the largest value in the list.
Once we have found these three points, Qi , Q2 , and Q3 , nosotros have all we demand in order to draw a simple box-and-whisker plot. Hither'due south an case of how it works.
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Depict a box-and-whisker plot for the following data set:
four.3, 5.ane, 3.9, 4.5, 4.4, four.ix, 5.0, 4.7, 4.1, iv.six, four.4, 4.three, 4.8, iv.iv, four.2, four.5, iv.4
My first step is to society the set. This gives me:
3.ix, 4.1, 4.2, 4.3, 4.iii, four.4, four.4, 4.iv, iv.iv, 4.5, 4.five, 4.six, iv.7, 4.8, 4.9, 5.0, 5.ane
The first value I need to observe from this ordered list is the median of the entire gear up. Since there are seventeen values in this list, the ninth value is the middle value of the list, and is therefore my median:
3.9, iv.1, 4.ii, 4.3, 4.3, 4.4, four.four, 4.iv, 4.4,iv.5, 4.5, 4.half-dozen, four.7, 4.8, 4.ix, 5.0, 5.ane
The median is Q2 = 4.4
The next two numbers I demand are the medians of the 2 halves. Since I used the "4.four" in the middle of the list, I tin't re-use it, so my two remaining data sets are:
3.9, 4.1, 4.ii, 4.3, 4.3, 4.4, 4.four, 4.iv
...and:
iv.5, 4.v, 4.6, 4.vii, 4.8, 4.9, v.0, 5.one
The beginning half has viii values, then the median is the average of the center ii values:
Q1 = (4.3 + 4.three)/two = 4.3
The median of the 2nd half is:
Q3 = (4.7 + 4.8)/two = 4.75
To describe my box-and-whisker plot, I'll need to decide on a scale for my measurements. Since the values in my list are written with ane decimal place and range from three.9 to 5.1, I won't use a scale of, say, zero to x, marked off by ones. Instead, I'll draw a number line from 3.5tov.5, and mark off past tenths.
(Yous might choose to mensurate from, say, 3 to half-dozen. Your choice would be equally practiced every bit mine. The thought here is to exist "reasonable", which allows you some flexibility.)
Now I'll mark off the minimum and maximum values, and Qi , Q2 , and Q3 :
The "box" office of the plot goes from Qi to Qiii , with a line drawn within the box to indicate the location of the median, Q2 :
And then the "whiskers" are drawn to the endpoints:
Past the way, box-and-whisker plots don't have to exist drawn horizontally as I did to a higher place; they can be vertical, too.
As mentioned at the commencement of this lesson, the "box" contains the middle portion of your data. Every bit you tin encounter in the graph higher up, the "whiskers" prove how large is the "spread" of the data.
If yous've got a wide box and long whiskers, then perhaps the data doesn't cluster as you'd hoped (or at least assumed). If your box is pocket-sized and the whiskers are curt, then probably your information does indeed cluster. If your box is small and the whiskers are long, and then mayhap the data clusters, merely you've got some "outliers" that you might demand to investigate further — or, as we'll see later, you may desire to discard some of your results.
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Draw the box-and-whisker plot for the following data:
98, 77, 85, 88, 82, 83, 87
My beginning stride is to society the data:
77, 82, 83, 85, 87, 88, 98
Side by side, I'll discover the median. This set has seven values, so the fourth value is the median:
Qtwo = 85
The median splits the remaining data into 2 sets. The offset set is 77, 82, 83. The median of this gear up is:
Q1 = 82
The other ready is 87, 88, 98. The median of this ready is:
Q3 = 88
I now have all the values I need for my box-and-whisker plot. At present I need to figure out what sort of scale I'll use for this. Since all the values are 2-digit whole numbers, I won't bother with decimal places. Because the extreme values (that is, the smallest and largest values) are 77 and 98 (xx-2 units autonomously), I'll use 75 to 100 for min and max values, and I'll count past two'southward for my scale. (There'south nothing special about these values; they're just what feel "reasonable" to me. Your choices may differ. Just don't go using something dizzy like l to 150 or 76.5 to 98.1.)
My set-upward looks similar this:
The kleptomaniacal portion at the lesser of the vertical axis indicates that there is a portion of the number-line that's been omitted. In other words, this notation makes clear that the units for the vertical centrality practice not start from zero.
(This zig-zag portion of the axis appears generally to go by the proper noun "zig-zag" or "break". If there'southward a proper term for this note, I haven't constitute it yet. The closest thing to a "standard" term for this sort of plot appears to exist "a cleaved-centrality graph". I call the squiggly function of the axis "the hicky-bob affair".)
My side by side step is to describe the lines for the median (which is Q2 ) and the two sub-medians (beingness the other quartiles, Q1 and Q3 ), as well as the two extremes:
Then I draw vertical lines to form my box and my whiskers:
I used a graphics plan (and its "snap to grid" setting) to brand my graphs above dainty and neat. For your homework, employ a ruler. And it would probably be a proficient idea to have a half dozen-inch (or fifteen-centimeter) ruler on hand for your next exam. Yes, neatness counts.
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Source: https://www.purplemath.com/modules/boxwhisk.htm