how to draw a box and whisker plot

Quartiles, Boxes, and Whiskers

For many computations in statistics, it is assumed that your information points (that is, the numbers in your list) are clustered around some fundamental 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 middle portion of these data points.

To create a box-and-whisker plot, we start by ordering our data (that is, putting the values) in numerical society, if they aren't ordered already. Then we notice the median of our data.

The median divides the data into ii halves. To divide the information into quarters, we then find the medians of these two halves.

Content Continues Below

Notation: If we accept an fifty-fifty number of values, so the beginning median was the average of the two eye values, then we include the middle values in our sub-median computations. If we have an odd number of values, and then the first median was an actual data point, and then nosotros do not include that value in our sub-median computations. That is, to find the sub-medians, we're only looking at the values that have not nevertheless been used.

So we have three points: the first middle point (the median), and the middle points of the two halves (what I've been calling the "sub-medians"). These iii points separate the entire data prepare into quarters, called "quartiles".

The top betoken of each quartile has a name, being a "Q" followed by the number of the quarter. So the top bespeak of the first quarter of the data points is "Q_i ", and and so along. Note that Q_i is also the centre number for the first half of the list, Q_two is also the centre number for the whole listing, Q_three is the middle number for the second half of the list, and Q_four is the largest value in the list.

Once we have institute these three points, Q₁ , Q₂ , and Q_three , we accept all we need in society to draw a simple box-and-whisker plot. Here's an example of how it works.

• Draw a box-and-whisker plot for the following data set:

four.3,  v.ane,  3.9,  iv.v,  4.4,  four.nine,  5.0,  4.7,  four.i,  4.six,  4.4,  4.3,  iv.eight,  iv.4,  4.2,  4.v,  4.iv

My first step is to social club the set. This gives me:

3.ix,  iv.1,  4.2,  4.3,  4.3,  4.4,  4.4,  4.4,  4.4,  four.5,  iv.5,  four.6,  iv.7,  four.8,  iv.ix,  5.0,  v.1

The start value I need to find from this ordered list is the median of the entire set. 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.one,  4.2,  4.iii,  four.3,  4.4,  4.iv,  iv.4,4.four,iv.5,  4.5,  4.6,  4.7,  4.eight,  4.9,  v.0,  v.one

3.ix,  iv.i,  4.2,  iv.3,  4.iii,  4.4,  iv.4,  4.4, four.4,4.5,  four.5,  4.half-dozen,  4.7,  iv.viii,  4.9,  5.0,  5.1

The median is Q₂ = iv.four

The next two numbers I demand are the medians of the two halves. Since I used the "4.four" in the middle of the list, I can't re-use it, and so my ii remaining data sets are:

3.nine,  4.1,  iv.2,  4.iii,  iv.3,  4.iv,  4.4,  4.4

...and:

4.5,  four.5,  4.6,  4.7,  4.8,  iv.9,  5.0,  v.1

The first half has eight values, so the median is the average of the middle ii values:

Q_one = (iv.3 + 4.3)/ii = 4.3

The median of the second half is:

Q_iii = (four.7 + 4.8)/2 = 4.75

To draw 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 3.nine to 5.1, I won't use a calibration of, say, aught to 10, marked off by ones. Instead, I'll draw a number line from 3.5to5.5, and mark off past tenths.

(You might choose to measure from, say, iii to vi. Your pick would be as good as mine. The idea here is to exist "reasonable", which allows you some flexibility.)

Now I'll mark off the minimum and maximum values, and Q₁ , Q₂ , and Q₃ :

The "box" role of the plot goes from Q₁ to Q₃ , with a line drawn inside the box to indicate the location of the median, Q₂ :

And then the "whiskers" are drawn to the endpoints:

By the way, box-and-whisker plots don't take to be drawn horizontally as I did higher up; they can be vertical, likewise.

---

As mentioned at the beginning of this lesson, the "box" contains the middle portion of your data. As y'all can see in the graph above, the "whiskers" prove how large is the "spread" of the data.

If yous've got a broad box and long whiskers, then maybe the data doesn't cluster every bit you'd hoped (or at least assumed). If your box is pocket-sized and the whiskers are short, and so probably your data does indeed cluster. If your box is minor and the whiskers are long, then perchance the data clusters, but y'all've got some "outliers" that you might demand to investigate further — or, as we'll see later, you may want to discard some of your results.

---

• Depict the box-and-whisker plot for the following information:

98,  77,  85,  88,  82,  83,  87

My beginning step is to order the data:

77, 82, 83, 85, 87, 88, 98

Next, I'll find the median. This set has seven values, so the fourth value is the median:

Q₂ = 85

The median splits the remaining data into two sets. The first set is 77, 82, 83. The median of this prepare is:

Q₁ = 82

The other gear up 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 demand 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. Because the farthermost values (that is, the smallest and largest values) are 77 and 98 (twenty-two units apart), I'll use 75 to 100 for min and max values, and I'll count by ii's for my scale. (There's zero special about these values; they're just what feel "reasonable" to me. Your choices may differ. Only don't become using something silly like l to 150 or 76.five to 98.1.)

My fix-up looks like this:

The kleptomaniacal portion at the bottom of the vertical axis indicates that at that place is a portion of the number-line that's been omitted. In other words, this notation makes clear that the units for the vertical axis exercise not starting time from zip.

(This zig-zag portion of the axis appears by and large to become by the name "zig-zag" or "break". If there'due south a proper term for this annotation, I oasis't institute it yet. The closest affair to a "standard" term for this sort of plot appears to exist "a broken-axis graph". I call the squiggly part of the axis "the hicky-bob thing".)

My next footstep is to draw the lines for the median (which is Q_ii ) and the two sub-medians (being the other quartiles, Q₁ and Q₃ ), as well as the two extremes:

And then I describe vertical lines to class my box and my whiskers:

---

I used a graphics program (and its "snap to grid" setting) to make my graphs above nice and neat. For your homework, apply a ruler. And it would probably be a proficient thought to have a vi-inch (or fifteen-centimeter) ruler on hand for your next test. Yeah, neatness counts.

Source: https://www.purplemath.com/modules/boxwhisk.htm

Posted by: burnsyoudly.blogspot.com

Share this post