Statistical Inferences and Creative Thinking

Assessing Sample Values On The Basis Of Sample Statistics

Given a sample of values, how do we assess a specific value with respect to sample distribution. For example, given the sample of student scores in Table 1, and its mean value of 55, how do we rate someone who scores 65? How good it is with respect to rest of the students? More importantly, how do we rate the student’s performance relative to other students in more than one class? Can we simply use the mean value and difference from the mean to compare performance? The problem may be further compounded if the maximum scores were quite different in the two cases, coming up with mean values having a meaning within the context of measuring scale. We can generalize this situation by saying that the two cases under consideration may have a different point of reference and unit of measurement for recorded values. We, therefore, need some kind of a ‘standardizing’ process to compare the two. We already know that any characterization using the mean is not very meaningful without taking into account the variations that the individual scores have with respect to the mean. This has been expressed in terms of variance, standard deviation and standard error for the mean value.

In order to make this discussion less abstract, let us consider a person, named Zia, who is considering making a move to another country. Zia salary in country A, where is residing now, 62,000. The mean value of salary for a sample in country A is 56,000, with a standard deviation of 12,300. Zia has been offered a salary of 99,000 in country B where is considering to move. The corresponding values of mean and standard deviation are 84,000 and 28,600, respectively. We know that the currencies in the two countries may not be same and that there cost of living might be different. Zia associates the quality of life he may have by comparing his own salary with the rest of the population.

Here is a superficial comparison by looking at Zia’s salary and the corresponding means.

The difference from the mean in Zia’s salary in country B is two and a half times that of country A, whereas the mean value of salary for country B is only one and a half times that of country A. Should Zia expect a better quality of life in country B with respect to rest of the people? Inferring this would be both superficial and misleading. In the above comparison, we are neither taking account the points of reference for salary values nor their unit or measurement. In order to standardize the statistics to a common point of reference and unit of measurement, we take into account the variance and standard deviation for sample values.

We convert the value we are examining to common standard as follows:

z = d /s           (23)

Where
z is the standardized value, called z-score,
d is the deviation from the mean for the value being evaluated, and
s is the standard deviation of the distribution where d occurs.
Substituting the values in (23) for country A and B in Zia’a case, we arrive at the following standardized scores:

This is approximately .5 for either case. It would, therefore, be wrong for Zia to assume that he would have a better quality of life looking at his salary versus the rest of the people. We may interpret the results in more precise statistical terms by saying that in either case, whether country A or B, Zia’s salary is only .5 of standard deviation higher than the mean i.e. statistically speaking there is no difference.

Figure 5: Frequency distribution of salaries in Country A

Figure 6: Frequency distribution of salaries in country B  

A clearer view of the above situation may be obtained by looking at the graphs of frequency distributions, as shown in Figures 5, 6 and 7. The salary values are expressed in thousands. The title shown for Figure 7 is ‘A Tale of Two Cities’ whereas Zia was looking at two countries. Since a person living in a country actually resides in a city, we have chosen the title as shown, and it sounds a little more appealing.

We see that the salary values, shown in the form of frequency distribution in Figure 5 are more clustered around the middle or the mean, whereas in Figure 6 they more spread out. This comparative assessment becomes more apparent in Figure 7.

Figure 7: Comparison of frequency distributions of salaries in country A and B

Of course it would be contrary to creative thinking to just take a single measure for something as complex as the quality in making a decision where to live. There are likely to be many other, more important, factors affecting the quality of life. Just because we have a scientific measure of something does not make it a superior measure in making decisions.

1. Introduction

2. Creative Thinking and Statistics

3. Raw Data And Data Aggregations By Categories

4. Measures Of Central Tendency

5. Assessing Sample Values On The Basis Of Sample Statistics

6. Conclusions

7. Cited References