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# How should the median be calculated?!- OCR S1 watch

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1. I have been taught that the median is always taken as the n+1/2 th value of the data set, with the one exception being cumulative frequency graphs, in which case the n/2 th value is estimated. In a past paper I did, one question involved estimating the median from a histogram. I estimated the n+1/2 th value, however the mark scheme only included the use of n/2. Does this mean that for histograms, the median should always be estimated as the n/2 th value? Similarly, for grouped frequency tables, when using linear interpolation to estimate the median, should I use n+1/2 (as I have been taught to do and as my revision guide suggests) or n/2 (as with cumulative freq. and histograms)? What is the rule over which one to use, and if possible for OCR S1 in particular?
2. (Original post by 11owolea)
I have been taught that the median is always taken as the n+1/2 th value of the data set, with the one exception being cumulative frequency graphs, in which case the n/2 th value is estimated. In a past paper I did, one question involved estimating the median from a histogram. I estimated the n+1/2 th value, however the mark scheme only included the use of n/2. Does this mean that for histograms, the median should always be estimated as the n/2 th value? Similarly, for grouped frequency tables, when using linear interpolation to estimate the median, should I use n+1/2 (as I have been taught to do and as my revision guide suggests) or n/2 (as with cumulative freq. and histograms)? What is the rule over which one to use, and if possible for OCR S1 in particular?
The median is always the (n+1)/2,th data point if you actually have all of the individual data points, so that there's an equal number on either side of the median. However, with any method of grouping data, e.g. grouped frequency table, histogram, box plot, cumulative frequency diagram, etc., you can no longer see every individual data point, so we instead treat the dataset as one continuous object rather than a bunch of discrete points, and thus take the (n/2),th value, as this splits the continuous interval 0 to n directly in half. Another rationale for this is that when you have grouped data, you can only find an approximation to the median anyway (since you don't have all of the separate data points), and so it's not worth adding the 1 because it's only approximate.

Therefore to summarise, whenever there's grouped data, or some other circumstance where parts of the data are combined so that you no longer have all of the original points, you should use (n/2) rather than (n+1)/2. Hopefully this helps.

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Updated: January 13, 2017
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