Assumptions and Estimations

Revealing Space


The essence of the concept is that we should probe the space around us in exactly the same fashion as to study a surface, by following direct paths, making measurements, and recoding what we find, free from any preconceptions. What Einstein called a ‘thought experiment'; an experiment devised and carried out in our imagination. We can say a few word about the nature of thought experiments and their important role in understanding the physical world. A good example of the role of thought experiments is the 'law of inertia', first formulated by Galileo and later adopted by Newton as the first of the three famous ‘Newton’s laws’ of physics. Galileo had carefully observed and measured the motion of objects under various conditions, and finally concluded that the correct description was the exact opposite of the standard 'dogma' that had been accepted for almost two thousand years. That 'dogma', promulgated by Aristotle, stated that a force was needed to maintain motion, and when the force was removed. The motion stopped.

Galileo asserted that the motion would go on forever unless a 'force' was exerted to stop it. The Aristotelian belief was able to hold sway for so long for the simple reason that no actual experiment could be devised to verify Galileo’s assertion. There are always forces acting on an object: gravity, friction, and the force exerted by the earth on a falling object at the moment it hits, to name a few. Galileo had to imagine a situation in which all those forces were removed, and he concluded that under those circumstances an object would continue moving in the same direction at the same speed indefinitely. The importance of this thought experiment cannot be overstated, it allowed 'Newton' to put forces back into the picture in his second and third laws and to state the exact effect they would have on the motion of an object. As a result, the 'qualitative' description of the 'physical world' given by Aristotle was replaced by 'Newton’s precise quantitative' statements in the form of simple mathematical equations that became the basis of all of modern physics.


Sampling

From time to time there is great interest in the UK in the result of a coming general election. In the preceding weeks the results of various opinion polls are published, each of which aims to give an accurate prediction of the way in which the electorate as a whole will vote. These polls are based on interviews with only one or two thousand voters out of a total of over 40 million. The choice of sample voters must obviously be made with care to ensure that the sample is truly representative of electoral opinion in the country at large. It would be unrealistic to expect a reasonable estimate of the voting intentions of the British electorate from a sample of interviews held only in the Highlands of Scotland, a Yorkshire mining area or an outer suburb of London. In this case the reliability of the sample survey is revealed soon after polling day. In other cases, including many geographical ones, the total situation maybe so vast that it can never be studied as a whole. A geomorphologist interested in coastal processes may want to investigate differences in the size and shape of sand grains along a beach. The number of sand grains on a beach, though not infinite, is uncountable large. The geomorphologist will have to be content with studying samples of sand. A number of questions need to be answered; how should these samples be chosen, how large should they be, how much reliance can be placed on the sample measurements as estimates of the characteristics of the millions of other sand grains on the beach?

These are questions that the statistician can help to answer. However, before discussing sampling in detail it may be helpful to introduce the topic by means of a practical exercise.

Exercise:
Looking at a map of part of a town, one observes the area including for example of 200 households distributed in several different districts each of fairly homogenous socio economic character. You are a representative of Azadeh Surveys and you only have time to visit 50 houses in the area to find the answers to four questions:

1- Is the head of the household a New Common wealth immigrant?
2- Does the household have television?
3- Will the head of the household vote for the National Democratic party at the next election?

Various methods of choosing the sample are possible:
1- with a pin (eye closed)
2- by eye,
3- take every fourth house on the list
4- go down each street picking every fourth house you come to
5- place a grid over the map and take an appropriate number of houses from each grid square,
6- put all 200 addresses in a hat and pick out 50
7- any other method you can think of

You can choose the 50 households you are to visit and find the answers to the four questions for each of the 50.

If there are major discrepancies between your sample results and the real results, can you account for them? If the exercise is being carried out by a whole class, which method of sampling produces the most accurate answers?

Sampling Methods

Not all methods of sampling are equally suitable for choosing samples, which are representative of the population. If the grid square method of selection is used in the previous exercise it is likely to produce a sample which contains too few houses from areas of high housing density and too many houses from areas where housing density is low. This might means choosing say 10 houses from one specific road area and 10 houses from another, since they both occupy approximately the same area of land. As there are 60 houses in first choice the representation rate in this area is 10/60, or approximately 17%. For the second choice however, the representation rate is 10/23, or approx 43%. Compared with the intended sampling rate of 25% (59/200), the first choice areas is underrepresented, whereas the second choice is over represented. This would have serious repercussions on the results of the survey, since the second area has for example, very high rate of car and television ownership.

The choice of sampling method should be made so as to avoid as far as possible this sort of bias. The advantages and disadvantages of different sampling methods are also important to be considered. Two elementary distinctions must be made, first between systematic and random methods and secondly between spatial and non spatial methods. The choice of sampling method depends both on the nature of the situation and on the purpose for which the sample is required. it is suggested that random samples are a prerequisite of many inferential statistical techniques. However for some purely descriptive purposes sample, which are not random, may be more useful, as in random sampling method the choice of individuals for inclusion in the sample is left entirely to chance. The choice of any particular individuals is in no way influenced by the researcher at any stage in the sampling process. A systematic sampling is one which is selected in some regular way, such as taking every forth address in above exercise. Intuitively, this would seem to produce an even, and therefore fair, coverage of the population in the sense that sample members are selected evenly from throughout the list, avoiding the bunching that can often occur with random sampling. In spatial sampling the concern is with the spatial distribution of a variable, such as altitude or rainfall, which varies continuously from place to place. For stratified sampling the example is: when the interest is in the way in which the local people’s images of the town centre are affected by how long they have lived in the town. The stratified sampling is used here to carry out a sample survey to measure this influence for the even representatives of the population as a whole. Making meaningful initial assumptions and correct estimates are invariably dependent on the researcher’s intelligence and general knowledge and lies close to the heart of their subject.

References:
Osserman, R., Poetry of The Universe, The Orion Publishig group, London, 1995
Ebdon, D., Statistics in Geography, 1977