Quartiles, Boxes, and Whiskers
For many computations in statistics, it is assumed that your data points (that is, the numbers in your list) are amassed effectually some primal value; in other words, it is assumed that in that location is an "average" of some sort. The "box" in the box-and-whisker plot contains, and thereby highlights, the heart 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. Then we find the median of our information.
The median divides the data into 2 halves. To split the information into quarters, we then observe the medians of these two halves.
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Notation: If we have an even number of values, and then the first median was the average of the two middle values, then we include the middle values in our sub-median computations. If we have an odd number of values, and so the start median was an actual data point, and then we practise non include that value in our sub-median computations. That is, to notice the sub-medians, nosotros're only looking at the values that have not yet been used.
So nosotros accept three points: the get-go centre point (the median), and the middle points of the ii halves (what I've been calling the "sub-medians"). These iii points divide the unabridged data fix into quarters, called "quartiles".
The pinnacle point of each quartile has a name, being a "Q" followed by the number of the quarter. And so the tiptop point of the first quarter of the information points is "Q1 ", and and then forth. Annotation that Qi is also the middle number for the first half of the listing, Q2 is also the heart number for the whole list, Qthree is the middle number for the second half of the list, and Qiv is the largest value in the list.
Once we take found these iii points, Q1 , Q2 , and Q3 , we have all nosotros need in guild to draw a simple box-and-whisker plot. Here's an case of how information technology works.
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Draw a box-and-whisker plot for the following information prepare:
iv.3, 5.1, iii.ix, iv.five, 4.4, 4.9, five.0, 4.7, 4.one, 4.vi, 4.4, 4.iii, four.eight, 4.4, four.two, iv.5, 4.4
My first step is to order the set. This gives me:
3.9, 4.1, four.2, 4.3, four.3, 4.iv, iv.4, iv.4, four.4, 4.five, 4.5, 4.vi, 4.7, 4.8, iv.9, 5.0, 5.1
The beginning value I need to find from this ordered list is the median of the entire prepare. Since there are seventeen values in this listing, the ninth value is the heart value of the list, and is therefore my median:
3.9, four.i, iv.2, four.3, four.3, 4.4, 4.4, 4.4,4.4,iv.5, 4.5, 4.6, 4.vii, 4.8, iv.ix, 5.0, 5.1
3.9, iv.1, iv.2, four.3, four.three, 4.4, 4.iv, four.4, four.iv,4.v, 4.5, 4.six, four.7, iv.8, iv.9, v.0, 5.1
The median is Qii = 4.iv
The next two numbers I need are the medians of the two halves. Since I used the "4.4" in the centre of the list, I can't re-use it, so my 2 remaining data sets are:
3.nine, iv.one, four.2, four.3, 4.3, four.4, 4.4, 4.4
...and:
4.5, four.v, 4.6, 4.7, 4.8, 4.9, v.0, 5.1
The get-go one-half has eight values, so the median is the average of the center 2 values:
Qone = (4.3 + four.3)/2 = 4.three
The median of the 2nd half is:
Q3 = (4.7 + 4.8)/two = 4.75
To draw my box-and-whisker plot, I'll need to determine on a scale for my measurements. Since the values in my list are written with ane decimal identify and range from 3.9 to five.one, I won't employ a calibration of, say, zero to ten, marked off past ones. Instead, I'll describe a number line from 3.vto5.5, and mark off by tenths.
(You might choose to measure from, say, 3 to 6. Your choice would be equally good as mine. The idea hither is to be "reasonable", which allows you some flexibility.)
At present I'll mark off the minimum and maximum values, and Q1 , Q2 , and Q3 :
The "box" part of the plot goes from Q1 to Q3 , with a line drawn inside the box to point the location of the median, Q2 :
So the "whiskers" are drawn to the endpoints:
By the manner, box-and-whisker plots don't have to be fatigued horizontally every bit I did higher up; they tin can exist vertical, too.
As mentioned at the commencement of this lesson, the "box" contains the middle portion of your data. As you can see in the graph to a higher place, the "whiskers" evidence how large is the "spread" of the data.
If you've got a wide box and long whiskers, so maybe the information doesn't cluster as you'd hoped (or at least assumed). If your box is small and the whiskers are short, then probably your data does indeed cluster. If your box is modest and the whiskers are long, then peradventure the data clusters, merely you've got some "outliers" that you might need to investigate further — or, as nosotros'll see later, you lot may want to discard some of your results.
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Draw the box-and-whisker plot for the post-obit information:
98, 77, 85, 88, 82, 83, 87
My outset footstep is to order the data:
77, 82, 83, 85, 87, 88, 98
Next, I'll discover the median. This set has seven values, and then the fourth value is the median:
Q2 = 85
The median splits the remaining information into two sets. The starting time set is 77, 82, 83. The median of this set is:
Q1 = 82
The other ready is 87, 88, 98. The median of this set is:
Q3 = 88
I now have all the values I need for my box-and-whisker plot. Now I need to figure out what sort of scale I'll utilise for this. Since all the values are two-digit whole numbers, I won't bother with decimal places. Considering the farthermost values (that is, the smallest and largest values) are 77 and 98 (xx-two units apart), I'll use 75 to 100 for min and max values, and I'll count past two's for my calibration. (At that place'southward nil special about these values; they're only what feel "reasonable" to me. Your choices may differ. Just don't go using something silly like 50 to 150 or 76.five to 98.1.)
My prepare-up looks similar this:
The crooked portion at the bottom of the vertical axis indicates that there is a portion of the number-line that's been omitted. In other words, this annotation makes clear that the units for the vertical axis exercise not start from zero.
(This zig-zag portion of the axis appears generally to go by the proper noun "zig-zag" or "intermission". If there's a proper term for this notation, I oasis't found it yet. The closest thing to a "standard" term for this sort of plot appears to exist "a broken-axis graph". I call the squiggly part of the centrality "the hicky-bob thing".)
My next step is to draw the lines for the median (which is Q2 ) and the two sub-medians (existence the other quartiles, Q1 and Q3 ), as well as the two extremes:
And then I draw vertical lines to form my box and my whiskers:
I used a graphics program (and its "snap to grid" setting) to make my graphs to a higher place dainty and bang-up. For your homework, use a ruler. And information technology would probably be a good thought to have a six-inch (or 15-centimeter) ruler on hand for your next exam. Yes, neatness counts.
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