Henry Berthold Mann was a professor of mathematics and statistics at Ohio State University

Ohio State University

The Ohio State University, commonly referred to as Ohio State, is a public research university located in Columbus, Ohio. It was originally founded in 1870 as a land-grant university and is currently the third largest university campus in the United States...

. Mann proved the Schnirelmann-Landau conjecture in number theory, and as a result earned the 1946 Cole Prize

Cole Prize

The Frank Nelson Cole Prize, or Cole Prize for short, is one of two prizes awarded to mathematicians by the American Mathematical Society, one for an outstanding contribution to algebra, and the other for an outstanding contribution to number theory. The prize is named after Frank Nelson Cole, who...

. He and his student developed the ("Mann-Whitney") U-statistic of nonparametric statistics. Mann wrote the first mathematical book on the design of experiments

Design of experiments

In general usage, design of experiments or experimental design is the design of any information-gathering exercises where variation is present, whether under the full control of the experimenter or not. However, in statistics, these terms are usually used for controlled experiments...

, .

Early life of a number theorist

Born in Vienna

Vienna

Vienna is the capital and largest city of the Republic of Austria and one of the nine states of Austria. Vienna is Austria's primary city, with a population of about 1.723 million , and is by far the largest city in Austria, as well as its cultural, economic, and political centre...

, Austria-Hungary

Austria-Hungary

Austria-Hungary , more formally known as the Kingdoms and Lands Represented in the Imperial Council and the Lands of the Holy Hungarian Crown of Saint Stephen, was a constitutional monarchic union between the crowns of the Austrian Empire and the Kingdom of Hungary in...

, to a Jewish family, Mann earned his Ph.D. degree in mathematics in 1935 from the University of Vienna

University of Vienna

The University of Vienna is a public university located in Vienna, Austria. It was founded by Duke Rudolph IV in 1365 and is the oldest university in the German-speaking world...

 under the supervision of Philipp Furtwängler

Philipp Furtwängler

Philipp Furtwängler was a German number theorist.He wrote an 1896 doctoral dissertation at the University of Göttingen on cubic forms , under Felix Klein...

. Mann immigrated to the United States in 1938, and lived in New York, supporting himself by tutoring students.

In additive number theory

Additive number theory

In number theory, the specialty additive number theory studies subsets of integers and their behavior under addition. More abstractly, the field of "additive number theory" includes the study of Abelian groups and commutative semigroups with an operation of addition. Additive number theory has...

, Mann proved the Schnirelmann-Landau conjecture on the asymptotic density of sumset

Sumset

In additive combinatorics, the sumset of two subsets A and B of an abelian group G is defined to be the set of all sums of an element from A with an element from B...

s in 1942. By doing so he established Mann's theorem and earned the 1946 Cole Prize

Cole Prize

The Frank Nelson Cole Prize, or Cole Prize for short, is one of two prizes awarded to mathematicians by the American Mathematical Society, one for an outstanding contribution to algebra, and the other for an outstanding contribution to number theory. The prize is named after Frank Nelson Cole, who...

.

Statistics

In 1942 the Carnegie Foundation

Carnegie Foundation

The Carnegie Foundation is an organization based in The Hague, Netherlands. It was founded in 1903 by Andrew Carnegie in order to manage his donation of US$1.5 million, which was used for the construction, management and maintenance of the Peace Palace...

 awarded Mann a fellowship to learn statistics while assisting the operations research

Operations research

Operations research is an interdisciplinary mathematical science that focuses on the effective use of technology by organizations...

 group of Harold Hotelling

Harold Hotelling

Harold Hotelling was a mathematical statistician and an influential economic theorist.He was Associate Professor of Mathematics at Stanford University from 1927 until 1931, a member of the faculty of Columbia University from 1931 until 1946, and a Professor of Mathematical Statistics at the...

 at Columbia University

Columbia University

Columbia University in the City of New York is a private, Ivy League university in Manhattan, New York City. Columbia is the oldest institution of higher learning in the state of New York, the fifth oldest in the United States, and one of the country's nine Colonial Colleges founded before the...

. His group also supported Abraham Wald

Abraham Wald

- See also :* Sequential probability ratio test * Wald distribution* Wald–Wolfowitz runs test...

, and Wald and Mann collaborated on several papers. In statistics, Mann is known for the ("Mann-Whitney") U-statistic and its associated hypothesis test for nonparametric statistics. Collaborating with Wald, Mann developed the Mann–Wald theorem of asymptotic statistics

Asymptotic theory (statistics)

In statistics, asymptotic theory, or large sample theory, is a generic framework for assessment of properties of estimators and statistical tests...

 and econometrics

Econometrics

Econometrics has been defined as "the application of mathematics and statistical methods to economic data" and described as the branch of economics "that aims to give empirical content to economic relations." More precisely, it is "the quantitative analysis of actual economic phenomena based on...

.

Mann wrote the first mathematical book on the design of experiments

Design of experiments

In general usage, design of experiments or experimental design is the design of any information-gathering exercises where variation is present, whether under the full control of the experimenter or not. However, in statistics, these terms are usually used for controlled experiments...

 , whose principles allowed later statisticians to design and to analyze customized experiments. Like contemporary "self help" and "how to" books, the earlier books gave easy-to-follow examples but little theory beyond exhortations to follow three principles of Ronald A. Fisher—to "replicate

Replication (statistics)

In engineering, science, and statistics, replication is the repetition of an experimental condition so that the variability associated with the phenomenon can be estimated. ASTM, in standard E1847, defines replication as "the repetition of the set of all the treatment combinations to be compared in...

", to "establish control" (for example with blocking), and to "randomize

Randomization

Randomization is the process of making something random; this means:* Generating a random permutation of a sequence .* Selecting a random sample of a population ....

" (assignment of treatments to units

Random assignment

Random assignment or random placement is an experimental technique for assigning subjects to different treatments . The thinking behind random assignment is that by randomizing treatment assignment, then the group attributes for the different treatments will be roughly equivalent and therefore any...

). Earlier books provided useful examples of designed experiments along with the design's analysis of variance

Analysis of variance

In statistics, analysis of variance is a collection of statistical models, and their associated procedures, in which the observed variance in a particular variable is partitioned into components attributable to different sources of variation...

, but no basis for constructing new designs for new problems, according to .

According to ,

Ronald A. Fisher was "interested in application and in the popularization
of statistical methods and his early book Statistical Methods for Research Workers

Statistical Methods for Research Workers

Statistical Methods for Research Workers is a classic 1925 book on statistics by the statistician R.A. Fisher. It is considered by some to be one of the 20th century's most influential books on statistical methods. According to ,...

, published in 1925, went through many editions and
motivated and influenced the practical use of statistics in many fields of
study. His Design of Experiments (1935) [promoted] statistical technique and application. In that book he
emphasized examples and how to design experiments systematically from
a statistical point of view. The mathematical justification of the methods
described was not stressed and, indeed, proofs were often barely sketched
or omitted altogether ..., a fact which led H. B. Mann to fill the gaps with a rigorous mathematical treatment in his well known treatise, ."

Later life

In 1946 Mann returned to Ohio State University, where he served as a professor mathematics until his retirement in 1964. Mann then became a professor at the U.S. Army's Mathematics Research Center at the University of Wisconsin (Madison) 1964–1971. Mann was professor at the University of Arizona from 1971–1975.