Answer to Question #261385 in Statistics and Probability for Lan

Question

5. If exactly one number in a set of data is changed, will this necessarily change the 

median of the set? Explain.

6. If a set of data has a mode, then must the mode be one of the numbers in the set? 

Explain.

Answer 1

5) Let for example we hve the set of numbers X={3,6,8,1,2}. For finding a median me(X) of the st X we write X in incresing order as X={1,2,3,6,8} and take number in the middle so we have me(X)=3 because $card(\{x \in X:x<3\}=card(\{x \in X:x>3\}=2$

If now any number except me(X)=3 slightly changed , for example 1->1.5 we will have the value 3 in the middle again: X'={1.5, 2, 3, 6, 8} and me(x')=me(X)=3. But if the value 3 changed itself for example 3->3.1 then for new set X''={1,2,3.1,5,6} me(X')=3.1$\ne me(X)=3$ .Thus in typical situation small changing any number except median itself doesn't lead to changing the median of the set.

6 )

The mode of a sample is the element that occurs most often in the collection, for example mode({2,6,8,6,7,7,6})=6. Thus if the mode exists it is always ia an element of the set.