how can you use the concept of mean median and mode to measure the performance of the students in half yearly exam of different subjects?
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Mean and median are often presented both as descriptive statistics, but this is actally not the case. The median is a central value of the data. It is that value for which one expects half of the (possible or observed) values being smaller and the other half being larger.
It is more a coincidence that the mean also is (often, but nor always!) quite in the center of the data, but its derivation is completely different. The mean is the result of a probability model over the "errors". One expects the values scattering around a common center, and this centre is determined as the value for which the observed data has the hhighest likelihood. This likelihood has to be calculated from a probability distribution for the deviations of the observed values from this hypothetical center. Maxwell, Herschell and others derived an approprite probability distribution from the simple assumptions that the expectation about the errors is symmetric (positive and negative errors of the same absolute size are expected with the same probability), and that all errors are congeneric (they share the same probability distribution). The result of this derivation is the normal distribution. Calculating the max. Likelihood using the normal distribution leads to the mean as the "best guess" for this assumed center. This is quite a different interpretation than just being the "center of the data".
Therefore I'd say that you should use the median to describe the center of the data, and you should use the mean if your aim is to model such a common center for which your expectations about the errors are in accordance to the two above given assumptions.
It is more a coincidence that the mean also is (often, but nor always!) quite in the center of the data, but its derivation is completely different. The mean is the result of a probability model over the "errors". One expects the values scattering around a common center, and this centre is determined as the value for which the observed data has the hhighest likelihood. This likelihood has to be calculated from a probability distribution for the deviations of the observed values from this hypothetical center. Maxwell, Herschell and others derived an approprite probability distribution from the simple assumptions that the expectation about the errors is symmetric (positive and negative errors of the same absolute size are expected with the same probability), and that all errors are congeneric (they share the same probability distribution). The result of this derivation is the normal distribution. Calculating the max. Likelihood using the normal distribution leads to the mean as the "best guess" for this assumed center. This is quite a different interpretation than just being the "center of the data".
Therefore I'd say that you should use the median to describe the center of the data, and you should use the mean if your aim is to model such a common center for which your expectations about the errors are in accordance to the two above given assumptions.
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