Zu Chongzhi 537

Category Mean Standard Deviation John’s Score

Performance 730 40 670

Theory 380 35 450

Composition 640 25 660

σ

σ

z-score (z-value) If a set of

DATA

has mean µand

standard deviation σ, then the z-score of a particular

data value xis given by:

This transformation converts the data values of one

set into another set of values with mean 0 and stan-

dard deviation 1. It allows one to effectively compare

two or more independent sets of data. As an example:

a group of freshman college music majors are given

exams in performance, theory, and composition. One

student, John, wishes to judge how well he fared in

each of the categories. The following table summa-

rizes the results:

John’s z-scores are:

Although John obtained his highest score in perfor-

mance, this score is 11/2standard deviations below the

mean grade in this category. Despite the low number,

John scored best in theory, gaining a score 2 standard

deviations above his classmates.

If a set of data is believed to be normally distributed

with mean µand standard deviation σ, converting to z-

values allows one to compare data with the standard

normal distribution. For example, to compute the prob-

ability that a measurement taken at random falls within

a range of values [a,b], one computes the area under the

standard normal curve above the interval .

This can be found by looking at a table of cumulative

distribution values ϕ(z) and calculating the difference

.

See also

NORMAL DISTRIBUTION

;

STATISTICS

:

DESCRIPTIVE

.

Zu Chongzhi (Tsu Chung Chi) (ca. 430–500) Chi-

nese Computation, Astronomy Born in Fan-yang

(now Hopeh), China, scholar Zu Chongzhi is remem-

bered for his invention of the Daming calendar, his

highly accurate calculation of a value for π, and his

derivation of the formula for the volume of a sphere

using a method equivalent to that discovered by Italian

mathematician B

ONEVENTURA

C

AVALIERI

(1598–1647)

a full millennium later.

Zu Chongzhi developed an interest in astronomy at

an early age. As a young scholar he noticed a discrep-

ancy in the position of a sundial’s shadow and the cal-

endar in use in China at the time, motivating him to

work for many years on an improved calendar system.

This work was completed in 462, but it was not until

510, through the efforts of his son, that the Daming

calendar was officially adopted.

Turning his attention to mathematics, Zu

Chongzhi worked to compute a precise value for the

ratio of a circumference of a circle to its diameter, π.

Noting, as a start, that the circumference of a hexagon

is three times its diameter, and that by cutting off its

corners to create a dodecagon yields a ratio slightly

larger than 3, Zu Chongzhi continued to shave corners

to eventually produce a polygon with 192 sides,

which, as he measured, yielded a circumference-to-

diameter ratio of 355/113. This represents an approxi-

mate value of πaccurate to six decimal places. Such

precision was not surpassed for another 1,000 years. It

is worth noting that Zu Chongzhi made his impressive

calculations without the aid of an

ABACUS

(not used in

China until the 1100s) or any other kind of calculating

device.

Through the writings of later scholars, historians

have determined that Zu Chongzhi wrote at least 51

works. Sadly none survive today. Zu Chongzhi is hon-

ored today as one of the few Chinese mathematicians to

have a lunar feature named after him—a 28-kilometer-

wide crater located 20°north of the lunar equator.