Question

Which one of the options can be inferred about the mean, median, and mode for the given probability distribution (i.e. probability mass function), $P(x)$, of a variable $x$? 

• A. mean = median $\neq$ mode
• B. mean = median = mode
• C. mean $\neq$ median = mode
• D. mean $\neq$ mode = median

Hint:

For symmetric bimodal distributions, mean and median coincide at the centre, while the mode lies at the peaks away from the centre.

Answer 1

The Correct Option is A

Step 1: Observe the symmetry of the distribution.
The given histogram is symmetric about $x=0$. For a symmetric distribution, the mean and median lie at the centre, i.e., both are equal to 0.
Step 2: Locate the mode.
The mode is the value(s) of $x$ corresponding to the highest frequency bar. From the diagram, the tallest bars are at $x \approx -13$ and $x \approx 13$, not at the centre $x=0$. Thus, the mode is different from the mean and median.
Step 3: Conclusion.
- Mean = Median = 0 (centre of symmetry).
- Mode $\neq$ Mean, Median (since maximum frequencies are at the tails).
Hence, \[ \boxed{\text{Mean = Median $\neq$ Mode}} \]