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PART II. SOME ELEMENTARY THEORY FOR
DESIGN
3. MOMENTS AND EXPECTED VALUES
A. MOMENTS OF DISTRIBUTIONS
The moment coefficients of a set of numbers 53
The moment coefficients of a distribution 54
The definition of a moment 56
Moment coefficients of combined distributions 57
Moment coefficients of the area under a continuous curve 59
Relations between moment coefficients about the mean and about some
other axis 		64
Short-cuts in the calculation of the mean and variance 66
B. EXPECTED VALUES
Definitions: random variable, probability or theoretical distribution,
"expected" value 		69
Commutation of E and Σ 71
The coefficient of variation 75
4. SOME VARIANCES IN RANDOM SAMPLING
A. SOME PRINCIPLES OF PROCEDURE
Some definitions: universe, frame, ideal bowl, random sampling 76
The frame and the sampling unit 77
Drawing the sample from the frame 83
Definition of population. Notation for universe and sample 84
Exercises on the use of random numbers and on the biases of certain
sampling procedures 		87
B. SOME GENERAL THEOREMS ON VARIANCE
Calculation of the mean and variance of the probability-distribution of
the estimated sum and mean 		99
Further characteristics of the probability-distribution of the mean 103
Use of the above theory 105
Special results for a universe of two cells 109
The theoretical distribution of r and r/n in samples from a two-celled
universe 		111
Special difficulties encountered in sampling for small proportions 114
Chart for reading coefficients of variation in binomial sampling 117
Some exercises on the binomial 119
C. THE PROPAGATION OF ERROR
An illustration of simple propagation of error 127
The right-triangular relation connecting the mean square error, the vari-
ance, and the bias of an estimate 		129
The propagation of mean square error and variance 130
5. MULTISTAGE SAMPLING, RATIO-ESTIMATES, AND CHOICE OF
SAMPLING UNIT
A. MULTISTAGE SAMPLING
Advantages of two or more stages of sampling 135

-xii-

Questia, a part of Gale, Cengage Learning. www.questia.com

Publication Information: Book Title: Some Theory of Sampling. Contributors: William Edwards Deming - author. Publisher: John Wiley & Sons. Place of Publication: New York. Publication Year: 1950. Page Number: xii.
    
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