Which one of the options can be inferred about the mean, median, and mode for the given probability distribution (i.e. probability mass function), P(z), of a variable z?
[A right-skewed (positively skewed) discrete probability mass function is shown, with the distribution having a longer tail on the right.]
A. mean >= median >= mode
B. mean = median = mode
C. mean <= median <= mode
D. mean >= mode >= median
GATE 2023 · General Aptitude · Statistics · medium
Answer: mean >= median >= mode (Option A)
Identify the shape of the distribution: The given PMF has its peak (mode) toward the lower values of z and a longer tail extending to the right (higher z values). This is a right-skewed (positively skewed) distribution.
Apply the skewness rule for central tendency ordering: Since the distribution is right-skewed, the mode is at the peak (leftmost central tendency), median lies between mode and mean, and the mean is pulled rightward by the long tail. Therefore: mean >= median >= mode.