Question

If the 25th, 50th and 75th percentiles of a dataset are x, y and z, which of the following is always TRUE? *

A: y-x=z-y.

B: y-x>z-y.

C: y-x
D: (y-x)(y-z)<=0.

Answer 1

Answer:

y-x=z-y

where x=25

y=50

z=75

Answer 2

Concept Introduction:-

It might resemble a word or a number representation of the quantity's arithmetic value. It could resemble a word or a number that represents the numerical value of the quantity.

Given Information:-

We have been given that the $25$th, $50$th and $75$th percentiles of a dataset are $x,y$, y and $z$.

To Find:-

We have to find that which of the following is always TRUE.

Solution:-

According to the problem

The correct answer is “$D$”. That’s why, since $25$th is the first quartile $(Q1)$ and the $50$th is the second quartile $(Q2)$ and the $75$th is the third quartile $(Q3)$, let’s give them numbers in a positive skewed example, $Q1=70=X$, $Q2=100=Y$ $Q3=140=Z$ (apparently, it’s not normally distributed, because if it was, the answer would be an in that case only, and that is not the case as the question requested “ALWAYS” true)

so try calculating

$Y-X=100-70=30, Y-Z=100-140=-40$

$(Y-X)(Y-Z)=30\times(-40)=-1200$

Then it will always give a negative number in case of both negative and positive skewness, and it would give $0$ if it’s normally distributed. (Try it yourself by a slight adjustment to the $Q3$, make it $130$ and recalculate, the answer would be zero)

Final Answer:-

The correct answer is “$D$” which is $(y-x)(y-z) < =0$ is always TRUE.

#SPJ2