The level of mathematics required

The level of mathematics necessary for understanding the techniques described in this book is quite modest. Yet, using these techniques we are able to solve many difficult numerical and statistical problems. We are able to solve non-linear equations in one or more unknowns, integrate and differentiate a given function (including empirical functions), smooth crude data, fit curves and interpolate. Some of the techniques we develop can be used to solve other complex problems. For example, the method of finite differences (chapter 6) is used extensively for solving differential equations.

Readers will be familiar with the mathematical concepts of differentiation (obtaining the slope of a curve) and integration (finding the area under a curve). There are certain other basic concepts and formulae which bear repeating and these are summarised in the remainder of this chapter.

The Taylor series expansion

This formula uses the value of a function f(x) and its derivatives at a particular point x to produce the value of that function at a neighbouring point x + h.

The normal distribution

Random errors or fluctuations in the results of scientific experiments frequently follow a distribution which approximates the normal (or Gaussian) form. There are good reasons to expect this to be so. The central limit theorem states that the distribution function of a random variable which is the sum of n independent identically-distributed random variables with means μ and variances ρ2 approaches the normal distribution function with mean nμ and variance nρ2 as n becomes large. A particular variable which we measure is often the result of combining a large number of variables which we cannot or do not measure; the deviation of an observation from the mean value or expected value is thus a sum of a number of deviations, some positive and some negative, and often of comparable magnitude.

The normal distribution is a continuous distribution completely defined by its mean μ and variance ρ2.

Introduction

The result of numerical calculation may differ from the exact answer to the mathematical problem for one or more of three basic reasons:

1. The calculation formula may be derived by cutting off an infinite series after a finite number of terms; the errors introduced in this manner are called truncation errors.

2. A calculating device is only able to retain a certain number of decimal digits and the less significant digits are dropped; errors introduced in this manner are called round-off errors.

3. Mistakes may be made by man or machine in performing the calculation or recording the result. The word ‘mistake’ is used to distinguish this source of discrepancy due to human or mechanical fallibility from the largely unavoidable ‘errors’ caused by the necessity to truncate an infinite series or the finite capacity of the calculating device.

The research worker must ensure that the final results of a calculation are not rendered useless by errors or mistakes. Intermediate checks are advisable a long calculation.

Introduction

Many different shapes of discrete distribution can be obtained by varying the parameters R, B and n in the hypergeometric distribution (section 10.13). When, for example, R = B and both are very large, and a reasonably large random sample of size n is drawn, the distribution of the number of red objects chosen is effectively binomial with parameters n (large) and p = ½. We then have a symmetric discrete distribution closely resembling the normal. By varying the hypergeometric parameters, we are able to produce distributions with different levels of skewness and kurtosis (section 8.6). Karl Pearson observed this, and he devised the Pearson system of probability-density functions by using a differential equation similar to the difference equation of the discrete hypergeometric distribution.

The process of fitting a Pearson probability-density function usually involves a large amount of arithmetic, but the work is nowhere near as onerous as previously now that electronic desk calculators are available with exponential, logarithmic and trigonometric keys.

Introduction

Non-linear equations often arise in scientific work. Graphical methods may be used to solve these equations, but the order of accuracy is usually such that the answer obtained can only be regarded as a first approximation. Trialand-error methods can also be used. Iterative methods are usually most convenient. These are procedures whereby xn the nth approximation to the root is obtained by evaluating a function of the earlier approximation xn–1.

The iterative procedures described in this chapter are usually quite satisfactory, but the reader is warned that they may fail in certain circumstances; for example, with multiple roots and close roots. The practical research worker will try a reasonable method. If this fails, he will try another method. If this also fails, he may be advised to seek the assistance of a numerical analyst.

The method of false position

Let us imagine that we wish to solve the non-linear equation f(x) = 0 and we have found that f(a) is positive whilst f(b) is negative.

Regrouping grouped data

Data are often grouped. This may be done to produce a concise table or because the data are sparse or some other reason. The user of a particular table may find that the grouping in the table is not suitable and he needs to regroup the data. This can be a very difficult task, particularly when the numbers in certain groups considerably exceed those in neighbouring groups. The best approach is to apply an interpolation method to the cumulative total.

Example 7.1.1. Births to unmarried mothers in Australia in 1968 totalled 18980. The ages of these mothers are shown in table 7.1.1, in broad age groups. Estimate the number of births to unmarried mothers in each of the quinquennial age groups 12–16, 17–21, 22–26, …, 47–51.