Doing a Box Plot With a Calculator

Stat 201

Suppose that you have the following sample:

308

113

126

195

185

356

195

166

485

155

223

137

183

300

346

700

1000

The box plot for these data is:

We will show below how to calculate these numbers.

1. We need the mean. Using JMP or a calculator we get . This is plotted as an x inside a circle.

2. We need:

Q₁ = first quartile

Q₂ = median

Q₃ = third quartile

To do this we have to order the sample from smallest to largest:

Position	Ordered Sample
1	113
2	126
3	137
4	155
5	166
6	183
7	185
8	195
9	195
10	223
11	300
12	308
13	346
14	356
15	485
16	700
17	1000

Next, we need to calculate the positions and values of

Q₁ = first quartile

Q₂ = median

Q₃ = third quartile

We do this on the following table:

What to Calculate	Position of the quartiles (Note that these are not the quartile.)	Quartile
Q1	(n+1)(.25) = (17+1)(.25) = 4.5	Since the position ends in 0.5 we average the sample value in position 4 and 5 (155+166)/2 = 157.5	This value is the lower side of the box.
Q2	(n+1)(.50) = (17+1)(.5) = 9	We take the value in position 9 which is 195	This value is the line in the middle of the box
Q3	(n+1)(.75) = (17+1)( .75) = 13.5	Since the position ends in 0.5 we average the sample value in position 13 and 14 (346+356)/2 = 351	This value is the upper side of the box.

We can now calculate the interquartile range = IQR=351-157.5=193.5

Now we need to calculate the fences:

Fences (These points are not plotted in the box plot)	Name
	Upper inner fence
	Upper outer fence
	Lower inner fence
	Lower outer fence

For convenience we show the ordered sample again:

Position	Ordered Sample
1	113
2	126
3	137
4	155
5	166
6	183
7	185
8	195
9	195
10	223
11	300
12	308
13	346
14	356
15	485
16	700
17	1000

We now calculate the values of the adjacent points and those of the mild and extreme outliers:

• Upper adjacent point = largest data point still lower than, that is, 485. This is the value at end of the upper whisker.
• Upper mild outliers = data points greater than and smaller than. There is only one of these, namely, 700. This point will be plotted as a small circle with no filling above the upper whisker.
• Upper extreme outliers = data points larger than, There is only one of these, namely, 1000. This point will be plotted above as a dark circle.
• Lower adjacent point = smallest data point still greater than, namely, 113. This is the value at end of the lower whisker.
• Lower outliers. There are not lower outliers be they mild or extreme because there is no value smaller than

We now we have all the values we need to plot the box plot.