Question

The article “The Ball-on-Three-Ball Test for Tensile Strength: Refined Methodology and Results for Three Hohokam Ceramic Types” (M. Beck, American Antiquity, 2002:558-569) discusses the strength of ancient ceramics. Several specimens of each of three types of ceramic were tested. The loads (in kg) required to crack the specimens are as follows:

Ceramic Type

Loads (kg)

Sacaton

15.30,51,20. 17, 19, 20,

32. 17,15,23, 19. 15. 18,

16, 22,29, 15, 13, 15

Gila Plain

27. 18, 28,25, 55,21, 18,

34,23, 30,20, 30,31,25,

28, 26,17,19, 16, 24,19,

9,31, 19, 27, 20, 43.15

Casa Grande

20,16,20, 36, 27, 35,

66,15,18,24,21,30,

20,24, 23,21, 13,21

a. Construct comparative boxplots for the three samples.

b. How many outliers does each sample contain?

c. Comment on the features of the three samples.

Answer 1

Sacaton.

The line inside the box shows the median. We see that $Me\approx 18.75$. The first quartile $Q_1=15$ , the third quartile $Q_3\approx 22.5$. The width of the box equals the interquartile range $IQR=Q_3-Q_1\approx 22.5-15=6.5$. 2 lines outside the box restrict the values lying in the interval $[Q_1-1.5*IQR; Q_3+1.5*IQR]=[5.25;32.25].$ The dot on the graph is the outlier. It is larger than 50.

Gila Plain.

The line inside the box shows the median. We see that $Me\approx 26$. The first quartile $Q_1\approx 19$, the third quartile $Q_3\approx 29$. The width of the box equals the interquartile range $IQR=Q_3-Q_1\approx 29-19=10$. 2 lines outside the box restrict the values lying in the interval $[Q_1-1.5*IQR; Q_3+1.5*IQR]=[4;44]$. The dot on the graph is the outlier. It is about 55.

Casa Grande.

The line inside the box shows the median. We see that $Me\approx 21.$ The first quartile $Q_1=20$, the third quartile $Q_3\approx 26.25$. The width of the box equals the interquartile range $IQR=Q_3-Q_1\approx 26.25-20=6.25$. 2 lines outside the box restrict the values lying in the interval $[Q_1-1.5*IQR; Q_3+1.5*IQR]=[10.625;35.625]$. 2 dots on the graph are the outliers. One is $\approx 36$ and another one is $\approx 66$.

We see that the second type (Gila Plain) has the largest median. The first type (Sacaton) has the smallest median. The third type (Casa Grande) has the largest number of outliers (two). The first type and the second type have one outlier each. The second type has the widest $IQR$, the first type has the most narrow $IQR$.