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Topic outline

 

  • Time: 14 hours
    Level: Introductory

 
 

Introduction

  • Introduction Resource
  • The unit that follows presents two sections from different parts of the Maths for Science teaching text – a course designed to help OU students acquire the knowledge and skills to tackle the mathematical...
 

1 Measurement in science

  • 1.1 Large quantities and small quantities Resource
  • Scientists frequently deal with enormous quantities – and with tiny ones. For example it is estimated that the Earth came into being about four and a half thousand million years ago. It took another six...
  • 1.2 Units of measurement Resource
  • In the UK, two systems of units are in common use. We still use old imperial measures for some things: milk is sold in pints and signposts indicate distances in miles. But for many other everyday measurements...
  • 1.3 Scales of measurement Resource
  • In thinking about the sizes of things, it is sometimes useful to do so in quite rough terms, just to the nearest power of ten. For example, 200 is nearer to 100 than it is to 1000, but 850 is nearer to...
  • 1.4 How precise are the measurements? Resource
  • Scientists are always trying to get better and more reliable data. One way of getting a more precise measurement might be to switch to an instrument with a more finely divided scale. Figure 4 shows parts...
 

2 Probability and descriptive statistics

  • Preamble Resource
  • Statistical information is a familiar aspect of modern life, which features routinely in, for example, news reports, sports commentaries and advertising. Scientists who have collected large amounts of...
  • 2.1 Chance and probability Resource
  • ‘Probability is expectation founded upon partial knowledge. A perfect acquaintance with all the circumstances affecting the occurrence of an event would change expectation into certainty, and leave neither...
  • 2.1.1 Calculating probability Resource
  • If a process is repeated in identical fashion a very large number of times, the probability of a given outcome is defined as the fraction of the results corresponding to that particular outcome.
  • 2.1.2 Probability and common sense Resource
  • The concept of probability is a purely theoretical one. Strictly speaking, no experiment measures a probability: all that we can measure is the fraction of times a particular outcome occurs in a finite...
  • 2.1.3 Expressing probability Resource
  • According to Equation 1, probability is defined as a fraction. However a fraction such as may also be expressed as a decimal number or as a percentage:
  • 2.1.4 Combining probabilities Resource
  • The probabilities described in Sections 2.1.1 and 2.1.2 related to the outcomes of a single process, such as repeatedly tossing one coin. Now suppose you were to toss three separate coins simultaneously....
  • 2.1.5 Probability ratios Resource
  • Probability calculations are important in many branches of science, but nowhere more so than in genetics. Box 4 describes early work in the field and provides some illustrative data, based on plant-breeding...
  • 2.2 Descriptive statistics Resource
  • Scientists collect many different types of information, but sets of data may be very loosely classified into two different types. In the first type, so-called ‘repeated measurement’, an individual quantity...
  • 2.2.1 Repeated measurements Resource
  • Scientists are always concerned with the reliability and precision of their data, and this is the prime reason for them to repeat measurements many times. Consider the photograph shown in Figure 6, which...
  • 2.2.2 The distribution of repeated measurements Resource
  • As noted in the previous section, if the same quantity is measured repeatedly, the results will generally be scattered across a range of values. This is perhaps best illustrated using a real example. Table...
  • 2.2.3 Mean and standard deviation for repeated measurements Resource
  • In everyday terms, everybody is familiar with the word ‘average’, but in science and statistics there are actually several different kinds of average used for different purposes. In the kind of situation...
  • 2.2.4 Using a calculator for statistical calculations Resource
  • Table 3 shows all the values for each step in the process of calculating a standard deviation, so that you can see what the operations encapsulated by Equation 7 actually entail, but you will probably...
  • 2.2.5 How likely are particular results? Resource
  • In real experiments, as opposed to hypothetical ones, it is very rare that scientists make a sufficiently large number of measurements to obtain a smooth continuous distribution like that shown in Figure...
  • 2.2.6 Different types of ‘average’ Resource
  • Figure 8 showed that if the data have a normal distribution the mean value corresponds to the peak of the distribution. Normal distributions of data are very common in science, but by no means universal....
  • 2.2.7 Samples and populations Resource
  • It is no accident that the examples used in Sections 2.2.3 and 2.2.4 to illustrate the statistics for repeated measurements of individual quantities were drawn from chemistry and physics. Experiments involving...
 

References and Acknowledgements

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