Q. 3.189 - Elementary Statistics

Question

The Etruscans. Anthropologists are still trying to unravel the mystery of the origins of the Etruscan empire, a highly advanced Italic civilization formed around the eighth century B.C. in central Italy. Were they native to the Italian peninsula or, as many aspects of their civilization suggest, did they migrate from the East by land or sea? The maximum head breadth, in millimeters, of $70$ modern Italian male skulls and that of $84$ preserved Etruscan male skulls were analyzed to help researchers decide whether the Etruscans were native to Italy. The resulting data can be found on the WeissStats site. [SOURCE: N. Barnicot and D. Brothwell, "The Evaluation of Metrical Data in the Comparison of Ancient and Modern Bones." In Medical Biology and Etruscan Origins, G. Wolstenholme and C. O'Connor, cds., Littlc, Brown \& Co., $1959$]

Step-by-Step Solution

Answer

Italians: mean $132.4$ and median $132$

Etruscans: mean $143.8$ and median $143.5$

The maximum head breadth for Etruscans is higher in general then the maximum head breadth for Italians.

Step 1: Given Information

We are given information of Italians and Etruscans head breadth.

There are $74$ Italians and $84$ Etruscans head breadth.

Step 2: Simplification of Italians data

Taking Italians data to consideration and arranging them from smallest to highest we get,

$$\begin{gathered} 116,120,121,124,125,125,125,125,126,126,127,127,127, \\ 128,128,128,128,129,129,130,130,130,130,130,130,130, \\ 131,131,131,131,132,132,132,132,132,132,133,133,133, \\ 133,133,134,134,134,135,135,135,135,136,136,136,136, \\ 137,137,137,138,138,138,138,138,138,139,139,140,140, \\ 143,144,148 \end{gathered}$$

Mean: Sum of all values divided by number of values

$MEAN=\frac{116+120+121+.........+143+144+148}{70}=\frac{9271}{70}\approx 132.4$
Median: middle value of sorted data

$MEDIAN = 132$

Step 3: Simplification of Etruscans data

Taking Etruscans data to consideration and arranging them from smallest to highest we get,

$$\begin{gathered} 126,131,132,132,134,135,135,136,137,137,137,137, \\ 138,138,139,140,140,140,140,140,140,140,141,141, \\ 141,141,141,141,142,142,142,142,142,142,142,142, \\ 143,143,143,143,143,143,144,144,144,144,144,144, \\ 145,145,145,146,146,146,146,146,146,147,147,147,147, \\ 148,148,148,148,148,149,149,149,149,149,149,150,150, \\ 150,150,152,152,153,154,154,155,158,158, \end{gathered}$$

Mean: Sum of all values divided by number of values

$MEAN=\frac{126+131+132+.........+155+158+158}{84}=\frac{12077}{84}\approx 143.8$

Median: Middle vales of sorted data list

$MEDIAN=\frac{143+144}{2}=\frac{287}{2}=143.5$

Step 4: Analyzing

After using data of mean and median we come to know that The maximum head breadth for Etruscans is higher in general then the maximum head breadth for Italians, because the mean and median of Etruscans is higher than the mean and median of Italians.