Several statistical techniques are proposed for economically analyzing large masses of data by means of punched-card equipment; most of these techniques require only a counting sorter. The methods proposed are designed especially for situations where data are inexpensive compared to the cost of analysis by means of statistically "efficient" or "most powerful" procedures. The principal technique is the use of functions of order statistics, which we call systematic statistics. It is demonstrated that certain order statistics are asymptotically jointly distributed according to the normal multivariate law. For large samples drawn from normally distributed variables we describe and give the efficiencies of rapid methods: i) for estimating the mean by using $1, 2, \cdots, 10$ suitably chosen order statistics; (cf. p. 386) ii) for estimating the standard deviation by using 2, 4, or 8 suitably chosen order statistics; (cf. p. 389) iii) for estimating the correlation coefficient whether other parameters of the normal bivariate distribution are known or not (three sorting and three counting operations are involved) (cf. p. 394). The efficiencies of procedures ii) and iii) are compared with the efficiencies of other estimates which do not involve sums of squares or products.

The non-central Wishart distribution is the joint distribution of the sums of squares and cross-products of the deviations from the sample means when the observations arise from a set of normal multivariate populations with constant covariance matrix but expected values that vary from observation to observation. The characteristic function for this distribution is obtained from the distribution of the observations (Theorem 1). By using the characteristic functions it is shown that the convolution of several non-central Wishart distributions is another non-central Wishart distribution (Theorem 2). A simple integral representation of the distribution in the general case is given (Theorem 3). The integrand is a function of the roots of a determinantal equation involving the matrix of sums of squares and cross-products of deviations of observations and the matrix of sums of squares and cross-products of deviations of corresponding expected values. The knowledge of the non-central Wishart distribution is applied to two general problems of multivariate normal statistics. The moments of the generalized variance, which is the determinant of sums of squares and cross-products multiplied by a constant, are given for the cases of the expected values of the variates lying on a line (Theorem 4) and lying on a plane (Theorem 5). The likelihood ratio criterion for testing linear hypotheses can be expressed as the ratio of two determinants or as a symmetric function of the roots of a determinantal equation. In either case there is involved a matrix having a Wishart distribution and another matrix independently distributed such that the sum of these two matrices has a non-central Wishart distribution. When the null hypothesis is not true the moments of this criterion are given in the non-central planar case (Theorem 6).

The paper contains some solutions of the weighing problems proposed by Hotelling [1]. The experimental designs are applicable to a broad class of problems of measurement of similar objects. The chemical balance problem (in which objects may be placed in either of the two pans of the balance) is almost completely solved by means of designs constructed from Hadamard matrices. Designs are provided both for a balance which has a bias and for one which has no bias. The spring balance problem (in which objects may be placed in only one pan) is completely solved when the balance is biased. For an unbiased spring balance, designs are given for small numbers of objects and weighing operations. Also the most efficient designs are found for the unbiased spring balance, but it is shown that in some cases these cannot be used unless the number of weighings is as large as the binomial coefficient $\binom {p}{\frac{1}{2}p}$ or $\binom {p}{\frac{1}{2}(p + 1)}$ where $p$ is the number of objects. It is found that when $p$ objects are weighed in $N \geq p$ weighings, the variances of the estimates of the weights are of the order of $\sigma^2/N$ in the chemical balance case $(\sigma^2$ is the variance of a single weighing), and of the order of $4\sigma^2/N$ in the spring balance case.

It is well known that various statistics of a large sample (of size $n$) are approximately distributed according to the normal law. The asymptotic expansion of the distribution of the statistic in a series of powers of $n^{-\frac{1}{2}}$ with a remainder term gives the accuracy of the approximation. H. Cramer [1] first obtained the asymptotic expansion of the mean, and recently P. L. Hsu [2] has obtained that of the variance of a sample. In the present paper we extend the Cramer-Hsu method to Student's statistic. The theorem proved states essentially that if the population distribution is non-singular and if the existence of a sufficient number of moments is assumed, then an asymptotic expansion can be obtained with the appropriate remainder. The first four terms of the expansion are exhibited in formula (35).

Upper and lower limits for the expected number $n$ of observations required by a sequential probability ratio test have been derived in a previous publication [1]. The limits given there, however, are far apart and of little practical value when the expected value of a single term $z$ in the cumulative sum computed at each stage of the sequential test is near zero. In this paper upper and lower limits for the expected value of $n$ are derived which will, in general, be close to each other when the expected value of $z$ is in the neighborhood of zero. These limits are expressed in terms of limits for the expected values of certain functions of the cumulative sum $Z_n$ at the termination of the sequential test. In section 7 a general method is given for determining limits for the expected value of any function of $Z_n$.

In studying the variation of a variable subject to erratic trend effects, it is customary to employ as a measure of variation a statistic that eliminates most of such effects. It is shown in this paper that the statistic $w = \sum^{n - 1}_1 \|x_{i+1} - x_i\| \sqrt{\pi}/2(n - 1)$ is nearly as efficient as the statistic $\delta^2 = \sum^{n - 1}_1 (x_{i+1} - x_i)^2/(n - 1)$ that is customarily employed. The asymptotic variance of w is obtained by integration techniques; the proof of the asymptotic normality of w is based upon a theorem of S. Bernstein on the asymptotic distribution of sums of dependent variables. The method of proof is sufficiently general to prove the asymptotic normality of w, and of $\delta^2$, for $x$ having a distribution for which the third absolute moment exists.

Let $\mu$ and $\sigma^2$ be the unknown mean and variance, respectively, of a normally distributed population on which $N$ independent observations $x_1, \cdots, x_N$ have been made. Let $L_1$ and $L_2, L_1 < L_2$, and $\alpha, 0 < \alpha < 1$, be given constants. We define the following symbols:\begin{equation*}\tag{(a)} \gamma = (\sqrt{2\pi}\sigma)^{-1} \int^{L_2}_{L_1} \exp \big\{-\frac{1}{2} \frac{(y - \mu)^2}{\sigma^2}\big\} dy\end{equation*} \begin{equation*}\tag{(b)} \bar{x} = N^{-1}\sigma\ x_i\end{equation*} \begin{equation*}\tag{(c)} s^2 = (N - 1)^{-1}\sigma (x_i - \bar{x})^2\end{equation*} (d) $\chi^2_{1-\alpha}$ as that number for which $P\{\chi^2 < \chi^2_{1-\alpha}\} = 1 - \alpha$ where $\chi^2$ has $N - 1$ degrees of freedom. \begin{equation*}\tag{(e)} w = \sqrt{N - 1}\frac{s}{\chi 1 - \alpha}\end{equation*} \begin{equation*}\tag{(f)} D = (2\pi)^{-1/2} \int^{(L_2 - {\bar{x})/w}_{(L_1-{\bar{x})/w} \exp \big\{ - \frac{1}{2} y^2\big\} dy\end{equation*} It is proved that, under restrictions stated precisely below, and before the observations are made, the probability that $D \leq \gamma$ differs from $\alpha$ by a number which can be made arbitrarily small by making $N$ sufficiently large. Thus an approximate (large sample) lower confidence limit for $\gamma$ is obtained. Similar methods can be applied to obtain upper and two-sided confidence limits. A problem raised by the present paper (but not attacked here) is to investigate the rapidity of approach to $\alpha$ of $P\{D \leq \gamma\}$. It would perhaps be useful to obtain a series for the latter in powers of $N^{-1\frac{1}{2}$; the first term of such an expansion is obtained here.