Central Limit Theorem Explained by a Sociologist

To reach a certain information in the social sciences inferential statistics is a commonly used method. In most basic terms, inferential statistics is taking a smaller number of people than the whole population of the research topic. That small number of people would provide information to reaching to some conclusions about the general population. Because we infer from the sample about the population, it is called inferential statistics.

Before getting started it is important to remember that the value of one observation does not depend on the value of another observation. They are generally independent from each other. For example the amount of red lights blink on a main street will not change depending on the yearly average salary of a nurse.

To make experiments with real life data one does not need to know each single subject’s value exactly. To be honest it may be impossible to collect that amount of accurate data. However, a representative of real life can be created with a sample. The sample is a group of randomly selected events. Because the real life values consist of randomly occurring multiple events, the sample has a great potential to resemble the real life population. This means while extreme low and high values can happen, remaining values and all of their mean would be similar to the real life values. Usually statistics classes give the example of n >= 30 as large samples. The reason is that in a group of 30 distinct people/events there would be such values that it would have the power to resemble the variety and the tendency of the population.

As a sociology graduate it makes so much sense to call the curve created in the light of Central Limit Theorem the Normal Curve. Because randomly selected independent observations/events are able to reflect the real population of the constructed sample. Norm is defined as social rules that are produced culturally. They are commonly shared and expected to be practiced by all members of the society¹. However not everyone reproduces the norms the way society expects or some individuals may be performing them very extremely while most of the members of society perform and expect them to be performed. Then the normal is understood as the product of most commonly practiced norms.

Similar to the sociological understanding the normal curve is called normal to reflect the values considered to be the norm in the population. If we are creating a sample, how can they become normal compared to the huge number of amounts of real life events? The theory suggests that too high and too low values in real life are very few compared to the middle values. It is very similar to the fact that among upper, middle and lower classes the middle class is the one which is more populous than others. Then if one would decide to visualize the values obtained in the 30+ sample there would be a tendency in values to mainly gather around the center of the curve. The normal curve would have its hill-like shape due to the middle range values to be seen more frequently than other values; it results as higher frequency in the curve to reach that shape.

For all the visual learners out there let me show this situation in Python with simple codes.

If I were to randomly select the following values would look like this.

According to the theory more the merrier. If the number of values we include in the sample increases, the curve will become a wider looking hill. This would mean that when more observations are gathered they are mostly resulted to be around the middle values.

[1]:Sherif, Muzafer. (1936). The psychology of social norms. NewYork: Harper.

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