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Statistics is another branch of math which was developed to locate & interpret patterns in large volumes of data.
The Subjective Idea of an Average
Let us take a familiar scenario of comparing the academic performance of 2 sections of the same grade in a particular test. Typically, the number of students in each of the sections would different and the scores of the students would be different.
For a long time, the subjective idea of an "average" was used to indicate a typical value which can represent a group. Hence if two teachers were asked for the average height of students in the same class, they could come out with different figures.
Over time the range of data and the purposes for which it was used increased. Simultaneously there was a necessity for a mathematical definition which was clear and values which many people can agree on.
If we think about it, there is no single figure which would help us represent the idea of an average. Hence statistics provides us with several parameters which can give some ideas of the average. It is like having several photographs of a hill from various directions to get a "wholistic" picture of the hill in our mind.
Out of these several measures, the idea of Mean, Mode & Median are sufficient for the primary grades.
Arithmetic Mean or Average
The term "average" has come to mean the "arithmetic mean."
The arithmetic mean score of a class of students is simply the sum of all the scores divided by the number of students. This is the commonly used parameter in comparisons of data. The class whose average is higher is taken to have performed better.
While using the Arithmetic Mean as the "average" we need to keep in mind its limitations.
It works best when the data is uniformly spread among in the range and the increases/ decreases in the data or additive in nature. There should be no extremes in the data.
For example if in a class of 30 students, one student is 1.70m tall whereas most others are around 1.4m, the average may be misleading.
Median
We use the measure of median when the marks are not uniformly spread. For example, in a class of 30, 28 may have scored 70 or less and 2 students may have scored in the 90s. Because of the unusual performance of these 2 students, the average of the class would be higher and hence convey a wrong idea of the class performance.
To get the median score, we arrange all the scores in an ascending order and locate the score which falls exactly in the middle.
If there are two scores which are in the middle, then the average of these scores is taken as the Median score.
Mode
Mode is the score which the maximum number of students score. This can be located if the data is sorted. In some cases, there could be 2 modes for the same data.
Spread
A quick measure is the difference between the maximum & minimum marks in a class. If the spread is smaller then there is less variation in the scores of the students. We could say the class which has a smaller spread has a better performance or that the teaching has been more effective.
Frequency Distribution
If we present the data of a class in terms of a frequency table, an expert can say which of the various measures described above would give the best measure of the performance.
A frequency table, as we have seen earlier, presents the various scores or score intervals on the horizontal scale and shows on the vertical axis, the number of students who scored in that range.
Geometric Mean
Geometric Mean is used when the data are related multiplicatively or exponentially rather than additively. A data set which is approximately {1, 3, 9, 27... } is an example.
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