IslamOnline - Contemporary Section

How do we use statistics in thinking more creatively about the things we see or record? After all, one may say that statistics is about calculating, describing, manipulating and interpreting mathematical attributes of sets or population. This may be viewed more as formalizing rather than cultivating creative thinking. Before we raise more questions about statistics, let us articulate a little about creative thinking. Let us simply not look into creative thinking but also constructive thinking, critical thinking, and so on. What is creative thinking? This question is asked and answered variously ever so often. We are raising it again to refresh our thoughts and to examine the role of statistics in creative thinking.

Creative thinking is about looking at possibilities in understanding objects or phenomena. It generally involves describing objects or phenomena, making projections on what is likely to happen in observed objects or phenomena with movements in time and space, and taking actions about objects or phenomena in order to move them in a desired direction. Here, the word objects is used to refer to things that interest us. Phenomena are what we experience about things that interest us, not necessarily derived from what those things may be within. The word creative in creative thinking implies emphasis on looking at possibilities. The word constructive in constructive thinking implies emphasis on building solutions from the possibilities, and the word critical in critical thinking implies emphasis on analysis of the possibilities and questioning the interpretations based on the possibilities. Often, we may use creative thinking, constructive thinking, and critical thinking as one and the same, keeping in mind the emphasis if there is one.

Let us do a little experiment. After all statistics is often about experiments and drawing conclusions from data gathered in experiments. In that sense, we may say that statistics is about discovering stories that numbers may tell. Consider, for example, a business is interested in knowing how its employees feel about some corporate values that it espouses. Let us say that this value is, “We take pride in our teamwork.” There are twenty employees. They are asked to rate this value statement in one of the following categories:

        Strongly Disagree,
        Disagree,
        Neither Agree nor Disagree,
        Agree, or
        Strongly Agree.

In order to ‘simplify’ matters, the above choices are assigned a numeric value ranging from 1 to 5 i.e.

Choice	Numeric Value
Strongly Disagree,	1
Disagree,	2
Neither Agree nor Disagree,	3
Agree, or	4
Strongly Agree.	5

The responses from the 20 employees in the business are recorded below as numeric scores for convenience:

5 3 5 5 2 3 3 5 1 5 4 4 2 4 5 4 1 4 2 3 (1)

What can we discern from these responses? Do they support, “We take pride in our teamwork,” the value espoused by the business? We can draw some conclusions in browsing the above scores directly. However, we can try to be a little more creative and arrange the score in a way that makes us easy to see the pattern and draw conclusions. Here are the same scores arranged according to the frequency of their occurrence:

Score 1 2 3 4 5
Frequency 2 3 4 5 6 (2)

We have opened some new possibilities in looking at the scores. We can see easily that more than half the people agree or strongly agree with the stated value, “We take pride in our teamwork.” One fifth neither agree nor disagree with the stated value, and one fourth disagree or strongly disagree with the stated value. In re-arranging the scores as shown, we reduced the number values for study into five groups.

Handling five things in drawing conclusions is a lot easier than handling twenty things. This is simply in the observed nature of human beings. This observation on human beings is based on an empirical study of individuals and organizations by a team consisting of a psychologist and a mathematician. Their study produced a heuristic called “7 plus or minus 2” rule. It means that mental skills work best when we deal with five to nine things that require our attention at the same time.

Attending to less than five things at a time, our mental skills may be under utilized, and attending to more than nine things at a time we may not be able to attend to all of them well. The numbers five to nine presumably are related to the capacity of short-term memory in humans - that part of our memory where we retain things for ready recall when engaged in a task. Creating piles of things and categorizing things into objects is what we do all the time. For example, we may categorize fruits into apples, oranges, and so on rather than talking about them as a single category of all fruits, just as we have tried to do in the above example of scores. Rather than working with twenty individual scores, we arranged them into five categories corresponding to the five categories of responses. It then became easier to talk about what the scores may tell us regarding the espoused corporate value, “We take pride in our teamwork.”

We could keep on analyzing the scores further in their numeric form. However, it is often helpful to visualize what is happening, or the implied patterns of things, when we show them in a pictorial form. We may graph the numeric data into a picture. One such pictorial

representation is shown in Figure 1, corresponding to the numeric data in (1).

Figure 1: A histogram for scores in (1). This is a bar graph in which the area (height) of each bar is proportional to the frequency of values for a particular category of data. The horizontal scale shows the categorization scheme.

The graphical view often allows us to perceive the patterns contained in numeric data more easily. For example, looking at Figure 1, the staff members in this business are not divided evenly between those who strongly disagree with the value statement, "We take pride in our teamwork," and those who strongly agree. Also, one may wonder why as many as four persons out of twenty, a significant number neither agree nor disagree with this statement. Is it because they do not quite understand what the statement says, or that they are simply not interested in the state of affairs? There is some food for further thought. Five out of twenty persons disagree, and two of them strongly, that the statement, "We take pride in our teamwork," reflects the true state of affairs in the business.

Taking the raw data shown in (1) and arranging it as shown in (2) or Figure 1, did open the possibilities to deal with the value statement, "We take pride in our teamwork," for this business. It opened the doors to creative thinking, it allowed critical thinking by facilitating analysis, and in at least some small ways, it allowed the business to constructively think about the possibilities.

The data in (1) includes all employees. Therefore, we can be quite confident that our conclusions reflect the true state of affairs for the entire business as seen from the perspective of employees. What if the number of employees was very large, and it was considered neither necessary nor practical to engage all employees in responding to the value statement, "We take pride in our teamwork." What if we simply took the responses from a reasonable sample of employees and drew conclusions as if they were based on the responses of all employees? This is a new possibility worthy of consideration. However, would we place a lot of confidence in our conclusions if we work with this new possibility? Well, it all depends on how representative of all employees is the sample of employees we chose? To raise questions when something new enters in what we may have been doing unquestionably before is part of creative and critical thinking. We will deal with the issues of taking a sample and making conclusions about the population from which the sample is derived, in later sections. For now, we will say that the sample of values represents a random selection of values from the population i.e. it is free of bias in favor or against any specific segment of population.

Sometimes, we may be interested in very simple categorizations. For example, which way did the majority respond? Did they tend to disagree and strongly disagree or agree and strongly agree? If we assume a center between those who agree and those who disagree then which side weighed more heavily? We can learn this and other interesting features in the scores by simply arranging them in increasing or decreasing order of magnitude. The sorted scores in increasing order are:

More than half the persons responded with agree or strongly agree. The value at this mid point is 4. In statistics, it is called the median. For an even number of observations, the median is the average of two values in the middle. The median of a set of observations is an indication of central tendency. When these observed values are arranged in increasing order of magnitude, half of the observations will be less than this value and the other half will exceed this value. The median value of 4 for the scores means that the majority agreed or strongly agreed with the statement, "We take pride in our teamwork."

There is another statistic, called mode, noting something of interest about observed values. It is a singular categorization picking up the value that occurs most frequently. For the observations listed in (1), the mode is 5, meaning that more persons responded by strongly agreeing with the statement, "We take pride in our teamwork" than those responding in any other single category.

Both median and mode provide insights in what we observe. They are statistics and they require calculations or manipulations in order to gain insights into what we observe. The focus is not on calculations or manipulations. They have to be done. The focus is on gaining insights into what we observe, and this in some small ways is always likely to assist in thinking creatively, constructively, or critically about what we observe.

The sorted arrangement also helps us spot two other simple statistics in the given data i.e. the minimum and maximum values.

Now turning to another possibility. What if the recorded values were not from a discrete set of possible integers such 1, 2, 3, 4, and 5 in the preceding example? Rather the values may take on fractions such as 1.3 or 2.5. Consider the following example of height measurements in feet and fractions of feet for a sample of twenty people:

5.5, 6.5, 6, 5.8, 6.1, 6.2, 5.6, 6.6, 5.4, 5.9, 5, 6.2, 6, 6.7, 4.8, 7.1, 5.3, 5.8, 6, 5.9 (5)

How do we put these values into a small number of categories? As an example, we may define them as follows:

First category gathers all values less than or equal to 5 in height. We may say that this covers the range from 4.5 to 5, with mid point at 4.75. The second category gathers values over 5 and those less than or equal to 5.5, with mid point at 5.25, and so on.

The values listed in (5) and categorized according to (6) produce the following results:

In evaluating the results shown in (6) and Figure 2, we may draw many conclusions as before. But these conclusions may apply only to the sample. Their generalization to the population from which the sample is drawn is well warranted for a good sample. However, these generalizations may have to be qualified with the degree of confidence we may be able to associate with them. For example, the sample indicates clearly that majority of people measured are nearly six feet or taller. How confident are we that this represents the true state of affairs for the population from which the sample measurements have been taken? This leads to questions about the sample itself. For example, does the sample appear to be typical or normal for the kind of thing we are measuring? We know that for a measurement like the heights of people, there some perceived middle or average value that typifies most people. As we move away from this middle or average value, the number of people having those heights would diminish. The pattern of diminishing values away from the middle value represents a tendency in nature, through some mysteries in nature. The resulting pattern is called a bell curve.

The frequency distribution data in (6) and Figure 2 is redrawn in Figure 3, showing a curve that results from connecting the top points of all frequency value bars in Figure 2. It is indicative of a bell shape, although

Figure 3: Frequency distribution of heights shown as a curve indicating the peak and tapering of the values away from the peak.