No One or two More than two

piercings piercings piercings

Fat-Free Milk 47 (40.8) 33 (30.6) 22 (30.6) 102

2% Milk 40 (65.6) 80 (49.2) 44 (49.2) 164

Whole Milk 113 (52.9) 37 (70.2) 84 (70.2) 234

200 150 150 500

No One or two More than two

piercings piercings piercings

Fat-Free Milk 47 33 22 102

2% Milk 40 80 44 164

Whole Milk 113 37 84 234

200 150 150 500

74 chi-squared test

This text was written 300 years before French mathemati-

cian B

LAISE

P

ASCAL

was born. (Some historians believe

that this work in fact dates back 200 years earlier to the

writings of mathematician Jia Xian.) Scholars of this time

routinely used the triangle to approximate nth roots of

numbers using the equivalent of the

BINOMIAL THEOREM

of today. They preferred their procedural methods of

extracting square roots to solve

QUADRATIC

equations,

rather than make use of the general quadratic formula.

Soon after J

OHN

N

APIER

(1550–1617) of the West

published an account of his new calculating aid, the

N

APIER

’

S BONES

, the Chinese developed an analogous

system of graded bamboo rods that could be used to

quickly compute long multiplications and divisions. It

is not known if the Chinese invented this system inde-

pendently, or whether the idea was perhaps brought to

them by 17th-century Jesuit missionaries. Along with

the

ABACUS

developed in China 500 years earlier, the

calculating rods allowed for improved arithmetic com-

putations, especially useful for the precise computa-

tions needed in astronomy.

Early scholar Z

U

C

HONGZHI

(ca. 500

C

.

E

.) com-

puted the volume of a sphere by a principle identical to

that of B

ONAVENTURA

C

AVALIERI

(1598–1647).

See also

MAGIC SQUARE

.

chi-squared test The chi-squared test is a statistical

test (see

STATISTICS

:

INFERENTIAL

) used to determine

whether or not two characteristics of a population are

independent or associated in some way. For example,

imagine a social study looking for a possible correla-

tion between the type of milk people prefer on their

cereal and the number of body piercings they possess.

Five hundred people were surveyed and the results

obtained are displayed in a

CONTINGENCY TABLE

.

Observe, in this study, that 102/500 = 0.204 of the

participants are fat-free milk users. If milk choice

bears no relationship to body piercings, we would

expect then about 0.204 of the 200 folk with no pierc-

ings to use fat-free milk. We observed a value of 47 (the

observed frequency) but expect a value of 0.204 ×200 =

40.8 (the expected frequency). Similarly, the expected

value for fat-free milk users with more than two pierc-

ings is 0.204 ×150 = 30.6 and for whole milk users

with one or two piercings: (234/500) ×150 = 70.2. In

this way we compute all expected frequencies, here

shown in parentheses:

Denoting the observed frequencies by the letter oand

the expected frequencies by e, we compute the chi-

squared statistic, χ2, as:

where the sum is over all entries in the table. (In the

1800s it was customary to convert all differences to a

positive value by use of the squaring function rather

than the

ABSOLUTE VALUE

function. This way, techniques

of calculus could be readily applied—it is straightfor-

ward to differentiate the square function, for example.)

A large value for χ2indicates that there is considerable

discrepancy between observed and expected values, sug-

gesting that the two features of the population are not

independent, i.e., that there is a

CORRELATION

. A small

χ2value suggests that there is no correlation.

Our particular example yields the value: