what happens when a test company gets it wrong

[Han]

(10-13-2014 04:50 PM)Don Shepherd Wrote:  

(10-12-2014 08:40 PM)Tim Wessman Wrote:  but there are at least 3 ways to do it that I am aware of. All of them are mathematically and statistically valid.

Tim, two points.

First, I accept that the change to the CAS calculation of median in the Prime will also be effective in the Home mode, when the next update is released. The larger problem is students who have already purchased a Prime calculator and do not upgrade. If they use their Prime to calculate median they will get the wrong answer ("wrong" as defined by their teacher, undoubtedly).

Second, you cannot say that all 3 methods of calculating median are mathematically and statistically valid: they return 3 different answers for the same data set (I am assuming that the 3 methods you refer to are the "correct" one [mean of the two middle values], the one that returns the smaller of the two middle values, and the one that returns the larger of the two middle values). There cannot be 3 valid values for median of the same data set, that is just basic to any discussion of median.

Here's one def. from wikipedia:

"In statistics and probability theory, the median is the numerical value separating the higher half of a data sample, a population, or a probability distribution, from the lower half."

If the sample is { 1, 5, 7, 9 } then ANY number strictly between 5 and 7 would satisfy this statement.

The (most common) mathematical definition of median is:

Quote:Given order statistics \( Y_1=min_j X_j, Y_2, ..., Y_{N-1}, Y_N=max_j X_j, \) the statistical median of the random sample is defined by
\[ \begin{cases}
Y_{(N+1)/2} & \text{if \(N\) is odd};\\
\frac{1}{2}(Y_{N/2}+Y_{1+N/2}) & \text{if \(N\) is even}
\end{cases} \]

The problem with this definition is that it doesn't give a good intuitive explanation of median. For odd N we pick the "middle" term yet for even N we pick a number that isn't even in the list sample space.

The definition of the term median determines whether the computed value of the median is "right" or "wrong." However, you insist that that mathematics does not allow for multiple interpretations (definitions, if you will) of a particular term. There are reasonable rationales for using the less typical definitions of median. If your domain is the set of integers, then a rational median may seem misplaced. Moreover, in the case of even elements, you are guaranteed that the median is within your set. I am not suggesting that these are the more common interpretations of median, but strictly speaking from a mathematical point of view, there is nothing wrong with "median" being interpreted differently. I do agree, however, that the more common interpretation should have been implemented.

Joke: How does a mathematician catch a lion and place him into a cage? The mathematician goes into the cage, and defines "the inside" as the opposite side of where he is.