数学专业英语[1]－The Real Number System

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Exercise

Ⅰ.Translate the following passages into Chinese:

1.A differential M(x,y) dx +N(x,y) dy ,where M, N are real functions of two variables x and y, is called exact in a domain D when the line integral ∫_c M(x,y) dx +N(x,y) dy is the same for all paths of integration c in D, which have the same endpoints.

Mdx+Ndy is exact if and only if there exists a continuously differentiable function u(x,y) such that M= u/ x, N=u/ y.

2. For any normal first order DE yˊ=F(x,y) and any initial x₀ , the initial valve problem consists of finding the solution or solutions of the DE ,for x>x₀ which assumes a given initial valve f(x₀)=c.

3. To show that the initial valve problem is well-set requires proving theorems of existence (there is a solution), uniqueness (there is only one solution) and continuity (the solution depends continuously on the initial value).

Ⅱ. Translate the following sentences into English:

1) 因为y=ч(x) 是微分方程dy/ dx=f(x,y)的解,故有

dч(x)/dx=f (x,ч(x))

2) 两边从x₀到x取定积分得

ч(x)-ч(x₀)=∫_x0^x f(x,ч(x)) dx x₀<x<x₀+h

3) 把y₀=ч(x₀)代入上式, 即有

ч(x)=y₀+∫_x0^x f(x,ч(x)) dx x₀<x<x₀+h

4) 因此 y=ч(x) 是积分方程

y=y₀+∫_x0^x f (x,y) dx

定义于x₀<x<x₀+h 的连续解.

Ⅲ. Translate the following sentences into English:

1) 现在讨论型如 y=f (x,yˊ) 的微分方程的解,这里假设函数 f (x, dy/dx) 有连续的偏导数.

2) 引入参数dy/dx=p, 则已给方程变为 y=f (x,p).

3) 在 y=f (x,p) x p=dy/dx p= f/ x+f/ p dp/dx

4) 这是一个关于x和p的一阶微分方程,它的解法我们已经知道.

5) 若(A)的通解的形式为p=ч(x,c) ,则原方程的通解为

y=f (x,ч(x,c)).

6) 若(A) 有型如x=ψ(x,c)的通解,则原方程有参数形式的通解

x=ψ(p,c)

y=f(ψ(p,c)p)

其中p是参数,c是任意常数.

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Series are a natural continuation of our study of functions. In the previous chapter we found how

to approximate our elementary functions by polynomials, with a certain error term. Conversely, one can define arbitrary functions by giving a series for them. We shall see how in the sections below.

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In practice, very few tests are used to determine convergence of series. Essentially, the comparision test is the most frequent. Furthermore, the most important series are those which converge absolutely. Thus we shall put greater emphasis on these.

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Convergent Series

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Suppose that we are given a sequcnce of numbers

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a₁,a₂,a₃…

i.e. we are given a number a_n, for each integer ｎ＞1.We form the sums

S_n=a₁+a₂+…+a_n

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It would be meaningless to form an infinite sum

a₁+a₂+a₃+…

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because we do not know how to add infinitely many numbers. However, if our sums S_n approach a limit as n becomes large, then we say that the sum of our sequence converges, and we now define its sum to be that limit.

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The symbols

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∑_a=1 ^∞ a_n ^{" z% p, H) p* g2 z- |) D, f" s}

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will be called a series. We shall say that the series converges if the sums approach a limit as n becomes large. Otherwise, we say that it does not converge, or diverges. If the seriers converges, we say that the value of the series is

∑_a=1^∞=lim_a→∞S_n=lim_a→∞(a₁+a₂+…+a_n)

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In view of the fact that the limit of a sum is the sum of the limits, and other standard properties of limits, we get:

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THEOREM 1. Let{ a_n }and { b_n }(n=1,2,…)

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be two sequences and assume that the series

∑_a=1 ^∞ a_n∑_a=1^∞ b_n

converge. Then ∑_a=1^∞(a_n + b_n ) also converges, and is equal to the sum of the two series. If c is a number, then

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∑ _a=1^∞c a_n =c∑_a=1 ^∞a_n

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Finally, if s_n=a₁+a₂+…+a_n and t_n=b₁+b₂+…+b_n then

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∑_a=1^∞a_n ∑ _a=1^∞b_n=lim_a→∞ s_n t_n

In particular, series can be added term by term. Of course , they cannot be multiplied term by term.

We also observe that a similar theorem holds for the difference of two series.

If a series ∑a_n converges, then the numbers a_n must approach 0 as n becomes large. However, there are examples of sequences {an} for which the series does not converge, and yet lim_a→∞a_n=0

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Series with Positive Terms

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Throughout this section, we shall assume that our numbers a_n are ＞ 0. Then the partial sums

S_n=a₁+a₂+…+a_n

are increasing, i.e.

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s₁＜s₂ ＜s₃＜…＜s_n＜s_n+1＜…

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If they are approach a limit at all, they cannot become arbitrarily large. Thus in that case there is a number B such that

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S_n< B

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for all n. The collection of numbers {s_n} has therefore a least upper bound ,i.e. there is a smallest number S such that

s_n<S

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for all n. In that case , the partial sums s_n approach S as a limit. In other words, given any positive number ε>0, we have

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S –ε< s_n < S

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for all n .sufficiently large. This simply expresses the fact that S is the least of all upper bounds for our collection of numbers s_n. We express this as a theorem.

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THEOREM 2. Let{a_n}(n=1,2,…)be a sequence of numbers>0 and let

S_n=a₁+a₂+…+a_n

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If the sequence of numbers {s_n} is bounded, then it approaches a limit S , which is its least upper bound.

Theorem 3 gives us a very useful criterion to determine when a series with positive terms converges:

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THEOREM 3. Let∑_a=1^∞a_n and∑_a=1^∞ b_n be two series , with a_n>0 for all n and b_n>0 for all n. Assume that there is a number c such that

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a_n< c b_n

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for all n, and that∑_a=1^∞b_n converges. Then ∑_a=1^∞ a_n converges, and

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∑_a=1^∞a_n ≤ c∑_a=1^∞b_n

PROOF. We have

a₁+…+a_n≤cb₁+…+cb_n

=c(b₁+…+b_n)≤ c∑_a=1^∞b_n

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This means that c∑_a=1^∞b_n is a bound for the partial sums a₁+…+a_n.The least upper bound of these sums is therefore ≤ c∑_a=1^∞b_n, thereby proving our theorem.

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Differentiation and Intergration of Power Series.

If we have a polynomial

a₀+a₁x+…+a_nxⁿ

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with numbers a₀,a₁,…,a_n as coefficients, then we know how to find its derivative. It is a₁+2a₂x+…+na_nx^n–1. We would like to say that the derivative of a series can be taken in the same way, and that the derivative converges whenever the series does.

THEOREM 4. Let r be a number >0 and let ∑a_nxⁿ be a series which converges absolutely for ∣x∣<r. Then the series ∑na_nx^n-1 also converges absolutely for∣x∣<r.

A similar result holds for integration, but trivially. Indeed, if we have a series ∑_a=1^∞a_nxⁿ which converges absolutely for ∣x∣<r, then the series

∑_a=1^∞a_n/n+1 xⁿ⁺¹=x∑_a=1^∞a_nxⁿ ∕n+1

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has terms whose absolute value is smaller than in the original series.

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The preceding result can be expressed by saying that an absolutely convergent series can be integrated and differentiated term by term and and still yields an absolutely convergent power series.

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It is natural to expect that if

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f (x)=∑_a=1^∞a_nxⁿ,

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then f is differentiable and its derivative is given by differentiating the series term by term. The next theorem proves this.

THEOREM 5. Let

f (x)=∑_a=1^∞ a_nxⁿ

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be a power series, which converges absolutely for∣x∣<r. Then f is differentiable for ∣x∣<r, and

f′(x)=∑_a=1^∞na_nx^n-1.

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THEOREM 6. Let f (x)=∑_a=1^∞a_nxⁿ be a power series, which converges absolutely for ∣x∣<r. Then the relation

∫f (x)d x=∑_a=1^∞a_nxⁿ⁺¹∕n+1

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is valid in the interval ∣x∣<r.

We omit the proofs of theorems 4,5 and 6.

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Vocabulary G( a" T/ m! s. p; }8 y" M( m) B$ [

sequence 序列 positive term 正项

series 级数 alternate term 交错项

approximate 逼近,近似 partial sum 部分和

elementary functions 初等函数 criterion 判别准则(单数)

section 章节 criteria 判别准则(多数)

convergence 收敛(名词) power series 幂级数

convergent 收敛(形容词) coefficient 系数

absolute convergence 绝对收敛 Cauchy sequence 哥西序列

diverge 发散 radius of convergence 收敛半径

term by term 逐项 M-test M—判别法

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Notes # x: v# K7 n4 p* k9 p2 o; s' [

1. series一词的单数和复数形式都是同一个字.例如:

One can define arbitrary functions by giving a series for them(单数)

The most important series are those which converge absolutely(复数)

2. In view of the fact that the limit of a sum of the limits, and other standard properties of limits, we get:

Theorem 1…

这是叙述定理的一种方式: 即先将事实说明在前面,再引出定理. 此句用in view of the fact that 说明事实,再用we get 引出定理.

3. We express this as a theorem.

这是当需要证明的事实已再前面作了说明或加以证明后,欲吧已证明的事实总结成定理时,常用倒的一个句子,类似的句子还有(参看附录Ⅲ):

We summarize this as the following theorem; Thus we come to the following theorem等等.

4. The least upper bound of these sums is therefore ≤c∑_a=1^∞b_n, thereby proving our theorem.

最一般的定理证明格式是”给出定理…定理证明…定理证毕”,即thereby proving our theorem;或we have thus proves the theorem或This completes the proof等等作结尾(参看附录Ⅲ).

5. 本课文使用较多插入语.数学上常见的插入语有:conversely; in practice; essentially; in particular; indeed; in other words; in short; generally speaking 等等.插入语通常与句中其它成份没有语法上的关系,一般用逗号与句子隔开,用来表示说话者对句子所表达的意思的态度.插入语可以是一个词,一个短语或者一个句子.

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Exercise

Ⅰ. Translate the following exercises into Chinese:

1. In exercise 1 through 4,a sequence f (n) is defined by the formula given. In each case, (ⅰ)

Determine whether the sequence (the formulae are omitted).

2. Assume f is a non–negative function defined for all x>1. Use the method

suggested by the proof of the integral test to show that

∑_k=1^n-1f(k)≤∫₁ⁿf(x)d x ≤∑_k=2ⁿf(k)

Take f(x)=log x and deduce the inequalities

c•nⁿ•c^-n< n！<c•nⁿ⁺¹•c^-n

Ⅱ. The proof of theorem 4 is given in English as follows(Read the proof through and try to learn how a theorem is proved, then translate this proof into Chinese ):

Proof of theorem 4 Since we are interested in the absolute convergence. We may assume that a_n>0 for all n. Let 0<x<r, and let c be a number such that x<c<r. Recall that lim_a→∞n^1/n=1.

We may write n a_n xⁿ =a_n(n^1/nx)ⁿ. Then for all n sufficiently large, we conclude that n^1/nx<c. This is because n1/n comes arbitrarily close to x and x<c. Hence for all n sufficiently large, we have na_nxⁿ<a_ncⁿ. We can then compare the series ∑na_nxⁿ with∑a_ncⁿ to conclude that∑na_nxⁿ converges. Since∑na_nx^n-1=1/x∑na_nxⁿ, we have proved theorem 4.

Ⅲ. Recall from what you have learned in Calculus about (ⅰ) Cauchy sequence and (ⅱ) the radius of convergence of a power series.

Now give the definitions of these two terms respectively.

Ⅳ. Translate the following sentences into Chinese:

1. 一旦我们能证明,幂级数∑a_nzⁿ 在点z=z₁收敛,则容易证明,对每一z₁∣z∣<∣z₁∣ ,级数绝对收敛;

2. 因为∑a_nzⁿ在z=z₁收敛,于是,由weierstrass的M—判别法可立即得到∑a_nzⁿ在点z,∣z∣<z₁的绝对收敛性;

3. 我们知道有限项和中各项可以重新安排而不影响和的值,但对于无穷级数,上述结论却不总是真的

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For the definition that follows we assume that we are given a particular field K. The scalars to be used are to be elements of K.

DEFINITION. A vector space is a set V of elements called vectors satisfying the following axioms.

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(A) To every pair, x and y ,of vectors in V corresponds a vector x+y,called the sum of x and y, in such a way that.

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(1) addition is commutative, x + y = y + x.

(2) addition is associative, x + ( y + z ) = ( x + y ) + z.

(3) there exists in V a unique vector 0 (called the origin ) such that x + 0 = x for every vector x , and

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(4) to every vector x in V there corresponds a unique vector - x such that x + ( - x ) = 0.

(B) To every pair,αand x , where α is a scalar and x is a vector in V ,there corresponds a vector αx in V , called the product of α and x , in such a way that

(1) multiplication by scalars is associative,α(βx ) = (αβ) x

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(2) 1 x = x for every vector x.

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(C) (1) multiplication by scalars is distributive with respect to vector addition,α( x + y ) = αx+βy , and

(2)multiplication by vectors is distributive with respect to scalar addition,(α+β) x = αx + βx .

The relation between a vector space V and the underlying field K is usually described by saying that V is a vector space over K . The associated field of scalars is usually either the real numbers R or the complex numbers C . If V is linear space and M真包含于V , and if α u -v belong to M for every u and v in M and every α∈ K , then M is linear subspace of V . If U = { u 1,u 2,…} is a collection of points in a linear space V , then the (linear) span of the set U is the set of all points o the form ∑ c _i u _i , where c _i∈ K ,and all but a finite number of the scalars c_i are 0.The span of U is always a linear subspace of V.

A key concept in linear algebra is independence. A finite set { u ₁,u ₂,…, u _k } is said to be linearly independent in V if the only way to write 0 = ∑ c _i u _i is by choosing all the c _i = 0 . An infinite set is linearly independent if every finite set is independent . If a set is not independent, it is linearly dependent, and in this case, some point in the set can be written as a linear combination of other points in the set. A basis for a linear space M is an independent set that spans M . A space M is finite-dimensional if it can be spanned by a finite set; it can then be shown that every spanning set contains a basis, and every basis for M has the same number of points in it. This common number is called the dimension of M .

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Another key concept is that of linear transformation. If V and W are linear spaces with the same scalar field K , a mapping L from V into W is called linear if L (u + v ) = L( u ) + L ( v ) and L ( αu ) = α L ( u ) for every u and v in V and α in K . With any I , are associated two special linear spaces:

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ker ( L ) = null space of L = L^-1 (0)

= { all x ∈ V such that L ( X ) = 0 }

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Im ( L ) = image of L = L( V ) = { all L( x ) for x∈ V }.

Then r = dimension of Im ( L ) is called the rank of L. If W also has dimension n, then the following useful criterion results: L is 1-to-1 if and only if L is onto.In particular, if L is a linear map of V into itself, and the only solution of L( x ) = 0 is 0, then L IS onto and is therefore an isomorphism of V onto V , and has an inverse L ^-1 . Such a transformation V is also said to be nonsingular.

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Suppose now that L is a linear transformation from V into W where dim ( V ) = n and dim ( W ) = m . Choose a basis {υ₁ ,υ_{2 ,}…,υ_n} for V and a basis {w ₁ ,w₂ ,…,w _m} for W . Then these define isomorphisms of V onto Kⁿ and W onto K^m , respectively, and these in turn induce a linear transformation A between these. Any linear transformation ( such as A ) between Kⁿ and K^m is described by means of a matrix ( a_ij ), according to the formula A ( x ) = y , where x = { x₁ , x ₂,…, x_n } y = { y₁ , y ₂,…, y _m} and

Y _j =Σⁿ_j=i a_ij x_i I=1,2,…,m.

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The matrix A is said to represent the transformation L and to be the representation induced by the particular basis chosen for V and W .

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If S and T are linear transformations of V into itself, so is the compositic transformation ST . If we choose a basis in V , and use this to obtain matrix representations for these, with A representing S and B representing T , then ST must have a matrix representation C . This is defined to be the product AB of the matrixes A and B , and leads to the standard formula for matrix multiplication.

The least satisfactory aspect of linear algebra is still the theory of determinants even though this is the most ancient portion of the theory, dating back to Leibniz if not to early China. One standard approach to determinants is to regard an n -by- n matrix as an ordered array of vectors( u ₁ , u ₂ ,…, u _n ) and then its determinant det ( A ) as a function F( u ₁ , u ₂ ,…, u _n ) of these n vectors which obeys certain rules.

The determinant of such an array A turns out to be a convenient criterion for characterizing the nonsingularity of the associated linear transformation, since det ( A ) = F ( u ₁ , u ₂ ,…, u _n ) = 0 if and only if the set of vectors u_i are linearly dependent. There are many other useful and elegant properties of determinants, most of which will be found in any classic book on linear algebra. Thus, det ( AB ) = det ( A ) det ( B ), and det ( A ) = det ( A') ,where A' is the transpose of A , obtained by the formula A' =( a _ji ), thereby rotating the array about the main diagonal. If a square matrix is triangular, meaning that all its entries above the main diagonal are 0,then det ( A ) turns out to be exactly the product of the diagonal entries.

Another useful concept is that of eigenvalue. A scalar is said to be an eigenvalue for a transformation T if there is a nonzero vector υ with T (υ) λυ . It is then clear that the eigenvalues will be those numbers λ∈ K such that T -λ I is a singular transformation. Any vector in the null space of T -λ I is called an eigenvector of T associated with eigenvalue λ, and their span the eigenspace, E _λ. It is invariant under the action of T , meaning that T carries E_λ into itself. The eigenvalues of T are then exactly the set of roots of the polynomial p(λ) =det ( T -λ I ).If A is a matrix representing T ,then one has p (λ) det ( A -λI ), which permits one to find the eigenvalues of T easily if the dimension of V is not too large, or if the matrix A is simple enough. The eigenvalues and eigenspaces of T provide a means by which the nature and structure of the linear transformation T can be examined in detail.

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Vocabulary

linear algebra 线性代数 non-singular 非奇异

field 域 isomorphism 同构

vector 向量 isomorphic 同构

scalar 纯量,无向量 matrix 矩阵(单数)

vector space 向量空间 matrices 矩阵(多数)

span 生成,长成 determinant 行列式

independence 无关(性),独立(性) array 阵列

dependence 有关(性) diagonal 对角线

linear combination 线性组合 triangular 三角形的

basis 基(单数) entry 表值,元素

basis 基(多数) eigenvalue 特征值,本征值

dimension 维 eigenvector 特征向量

linear transformation 线性变换 invariant 不变,不变量

null space 零空间 row 行

rank 秩 column 列

singular 奇异 system of equations 方程组

homogeneous 齐次

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Notes

1. If U = { u ₁ , u ₂ ,…}is a collection of points in a linear

space V , then the (linear) span of the set U is the set of all points of the form ∑ c _i u _{i , w}where c _i∈ K ,and all but a finite number of scalars c _I are 0.

意思是:如果 U = { u ₁ , u ₂ ,…}是线性空间 V 的点集,那么集 U 的(线性)生成是所有形如 ∑ c _i u _i 的点集,这里 c _i ∈ K ,且除了有限个 c_i 外均为0.

2. A finite set { u ₁ , u ₂ ,…, u _k} is said to be linearly independent if the only way to write 0 = ∑ c _i u _I is by choosing all the c _i= 0.

这一句可以用更典型的句子表达如下: A finite set { u ₁, u ₂ ,…, u _k } is said to be linearly independent in V if ∑c _i u _i is by choosing all the c _i = 0.

这里independent 是形容词,故用linearly修饰它. 试比较F(x) is a continuous periodic function.这里periodic 是形容词但它前面的词却用continuous 而不用continuously,这是因为continuous 这个词不是修饰periodic而是修饰作为整体的名词periodic function.

3. Then these define isomorphisms of V onto Kⁿ and W onto K^M respectively, and these in turn induce a linear transformation A between these.

这里第一个these代表前句的两个基(basis);第二个these代表isomorphisms;第三个these代表什么留给读者自己分析.

4. The least satisfactory aspect of linear algebra is still the theory of determinants-

意思是:线性代数最令人不满意的方面仍是有关行列式的理论.least satisfactory 意思是:最令人不满意.

5. If a square matrix is triangular, meaning that all its entries above the main diagonal are 0,then det ( A ) turns out to be exactly the product of the diagonal entries.

意思是:如果方阵是三角形的,即所有在主对角线上方的元素均为零,那末det( A ) 刚好就是对角线元素的乘积.这里meaning that 可用that is to say 代替,turns out to be解为”结果是

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Exercise

I. Answer the following questions:

1. How can we define the linear independence of an infinite set?

2. Let T be a linear transformation (T: V → W ) whose associated matrix is A.Give a criterion for the non-singularity of the transformation T.

3. Where is the entry a₄₅ of a m -by- n matrix( m>4; n>5) located ?

4. Let A , B be two rectangular matrices.Under what condition is the product matrix well-defined ?

II.Translate the following two examples and their proofs into Chinese:

1.Example1. Let u_k= t^k ,k=0,1,2,... and t real. Show that the set ｛u ₀,u₁,u₂,…｝ is independent.

Proof: By the definition of independence of an infinite set, it suffices to show that for each n ,the n+1 polynomials u₀,u₁,...,u_n are independent.A relation of the form ∑ⁿ_k=0 c_ku_k=0 means ∑ⁿ_k=0 c_k t^k=0 for all t.When t=0,this gives c₀ =0.Differentiating both sides of ∑ⁿ_k=0 c_k t^k =0 and setting t=0,we find that c₁=0.Repeating the process,we find that each cocfficient is zero

2. Example 2. Let V be afinite dimensional linear space, Then every finite basis for V has the same number of elements.

Proof: Let S and T be two finite bases for V. Suppose S consists of k elemnts and T consists of m elements.Since S is independent and spans V ,every set of k+1 elements in V is dependent.Therefore every set of more than k elements in V is dependent. Since T is an independent set , we must have m<k. The same argument with S and T interchanged shows that k<m. Hence k=m.

III.Translate the following sentences into English:

1.设 A 是一矩阵。若其行数 n 等于其列数 m ，则称 A 是一方阵；若n ≠m，则称A 是一矩形阵。

2.设 T 是一线性变换，则 T 的特征值刚好是多项式 P（λ）=det（T- λE ）的根。

3.形如Σⁿ_k=1c_ik x_k =c_i i=1，2，...,m 的一组方程称为 m 个线性方程，n 个未知数的方程组。若所有 c_i=0，则称上述方程组为一齐次方程组。

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数学专业英语－Statistics 8 k, f$ {# n/ W0 E1 U9 B) T3 C7 _ 7 J( H" F" N/ f6 e0 |8 v0 S

The term statistics is used in either of two senses.In common parlance it is generally employed synonymously with the word data.Thus someone may say that he has seen”statistics of industrial accidents in the United States.” It would be conducive to greater precision of meaning if we were not to use statistics in this sense,but rather to say “data (or figures ) of industrial accidents in the United States.”

“Statistics” also refers to the statistical principles and methods which have been developed for handling numerical data and which form the subject matter of this text.Statistical methods,or statistics, range form the most elementary descriptive devices, which may be understood by anyone , to those extremely complicated mathematical procedures which are comprehended by only the most expert theoreticians.It is the purpose of this volume not to enter into the highly mathematical and theoretical aspects of the subject but rather to treat of its more elementary and more frequently used phases.

Statistics may be defined as the collection, presentation, analysis, and interpretation of numerical data.The facts which are dealt with must be capable of numerical expression.We can make little use statistically of the information that dwellings are built of brick, stone, wood, and other materials; however, if we are able to determine how many or what proportion of,dwellings are constructed of each type of material, we have numerical data suitable for statistical analysis.

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Statistics should not be thought of as a subject correlative with physics, chemistry, economics, and sociology. Statistics is not a science; it is a scientific method. The methods and procedures which we are about to examine constitute a useful and often indispensable tool for the research worker. Without an adequate understanding of statistics, the investigator in the social sciences may frequently be like the blind man groping in a dark closet for a black cat that isn’t there. The methods of statistics are useful in an ever---widening range of human activities, in any field of thought in which numerical data may be had.

In defining statistics it was pointed out that the numerical data are collected, presented, analyzed, and interpreted. Let us briefly examine each of these four procedures.

COLLECTION Statistical data may be obtained from existing published or unpublished sources, such as government agencies, trade associations, research bureaus, magazines, newspapers, individual research workers, and elsewhere. On the other hand, the investigator may collect his own information, going perhaps from house to house or from firm to firm to obtain his data. The first-hand collection of statistical data is one of the most difficult and important tasks which a statistician must face. The soundness of his procedure determines in an overwhelming degree the usefulness of the data which he obtains.

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It should be emphasized, however, that the investigator who has experience and good common sense is at a distinct advantage if original data must be collected. There is much which may be taught about this phase of statistics, but there is much more which can be learned only through experience. Although a person may never collect statistical data for his own use and may always use published sources, it is essential that he have a working knowledge of the processes of collection and that he be able to evaluate the reliability of the data he proposes to use. Untrustworthy data do not constitute a satisfactory base upon which to rest a conclusion.

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It is to be regretted that many people have a tendency to accept statistical data without question. To them, any statement which is presented in numerical terms is correct and its authenticity is automatically established.

PRESENTATION Either for one’s own use or for the use of others, the data must be presented in some suitable form. Usually the figures are arranged in tables or presented by graphic devices.

ANALYSIS In the process of analysis, data must be classified into useful and logical categories. The possible categories must be considered when plans are made for collecting the data, and the data must be classified as they are tabulated and before they can be shown graphically. Thus the process of analysis is partially concurrent with collection and presentation.

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There are four important bases of classification of statistical data: (1) qualitative, (2) quantitative, (3) chronological, and (4) geographical, each of which will be examined in turn.

Qualitative When, for example, employees are classified as union or non—union, we have a qualitative differentiation. The distinction is one of kind rather than of amount. Individuals may be classified concerning marital status, as single, married, widowed, divorced, and separated. Farm operators may be classified as full owners, part owners, managers, and tenants. Natural rubber may be designated as plantation or wild according to its source.

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Quantitative When items vary in respect to some measurable characteristics, a quantitative classification is appropriate. Families may be classified according to the number of children. Manufacturing concerns may be classified according to the number of workers employed, and also according to the values of goods produced. Individuals may be classified according to the amount of income tax paid.

Chronological Chronological data or time series show figures concerning a particular phenomenon at various specified times. For example, the closing price of a certain stock may be shown for each day over a period of months of years; the birth rate in the United States may be listed for each of a number of years; production of coal may be shown monthly for a span of years. The analysis of time series, involving a consideration of trend, cyclical period (seasonal ), and irregular movements, will be discussed.

In a certain sense, time series are somewhat akin to quantitative distributions in that each succeeding year or month of a series is one year or one month further removed from some earlier point of reference. However, periods of time—or, rather, the events occurring within these periods—differ qualitatively from each other also. The essential arrangement of the figures in a time sequence is inherent in the nature of the data under consideration.

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Geographical The geographical distribution is essentially a type of qualitative distribution, but is generally considered as a distinct classification. When the population is shown for each of the states in the United States, we have data which are classified geographically. Although there is a qualitative difference between any two states, the distinction that is being made is not so much of kind as of location.

The presentation of classified data in tabular and graphic form is but one elementary step in the analysis of statistical data. Many other processes are described in the following passages of this book. Statistical investigation frequently endeavors to ascertain what is typical in a given situation. Hence all type of occurrences must be considered, both the usual and the unusual.

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In forming an opinion, most individuals are apt to be unduly influenced by unusual occurrences and to disregard the ordinary happenings. In any sort or investigation, statistical or otherwise, the unusual cases must not exert undue influence. Many people are of the opinion that to break a mirror brings bad luck. Having broken a mirror, a person is apt to be on the lookout for the unexpected”bad luck “ and to attribute any untoward event to the breaking of the mirror. If nothing happens after the mirror has been broken, there is nothing to remember and this result (perhaps the usual result )is disregarded. If bad luck occurs, it is so unusual that it is remembered, and consequently the belief is reinforced. The scienticfic procedure would include all happenings following the breaking of the mirror, and would compare the “resulting” bad luck to the amount of bad luck occurring when a mirror has not been broken.

Statistics, then, must include in its analysis all sorts of happenings. If we are studying the duration of cases of pncumonia, we may study what is typical by determining the average length and possibly also the divergence below and above the average. When considering a time series showing steel—mill activity, we may give attention to the typical seasonal pattern of the series, to the growth factor( trend) present, and to the cyclical behaviour. Sometimes it is found that two sets of statistical data tend to be associated.

Occasionlly a statistical investigation may be exhaustive and include all possible occurrences. More frequently, however, it is necessary to study a small group or sample. If we desire to study the expenditures of lawyers for life insurance, it would hardly be possible to include all lawyers in the United States. Resort must be had to a sample;and it is essential that the sample be as nearly representative as possible of the entire group, so that we may be able to make a reasonable inference as to the results to be expected for an entire population. The problem of selecting a sample is discussed in the following chapter.

Sometimes the statistician is faced with the task of forecasting. He may be required to prognosticate the sales of automobile tires a year hence, or to forecast the population some years in advance. Several years ago a student appeared in summer session class of one of the writers. In a private talk he announced that he had come to the course for a single purpose: to get a formula which would enable him to forecast the price of cotton. It was important to him and his employers to have some advance information on cotton prices, since the concern purchased enormous quantities of cotton. Regrettably, the young man had to be disillusioned. To our knowledge, there are no magic formulae for forecasting. This does not mean that forecasting is impossible; rather it means that forecasting is a complicated process of which a formula is but a small part. And forecasting is uncertain and dangerous. To attempt to say what will happen in the future requires a thorough grasp of the subject to be forecast, up-to-the-minute knowledge of developments in allied fields, and recognition of the limitations of any mechanica forecasting device.

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INTERPRETATION The final step in an investigation consists of interpreting the data which have been obtained. What are the conclusions growing out of the analysis? What do the figures tell us that is new or that reinforces or casts doubt upon previous hypotheses? The results must be interpreted in the light of the limitations of the original material. Too exact conclusions must not be drawn from data which themselves are but approximations. It is essential, however, that the investigator discover and clarify all the useful and applicable meaning which is present in his data.

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