9. The figure below shows a frequency histogram for the variable X. Which of these
statements is correct for this histogram?
a. mean < median < mode
b. mode > mean > median
c. mode > mean > median
d. mean = mode, mode > median
The histogram is left‑skewed (negatively skewed), so the correct statement is mean < median < mode (option a).
Understanding the histogram
The curve in the question has a long tail stretching towards smaller values of X and a peak near the higher values, which is the typical shape of a left‑skewed (negatively skewed) distribution.
In such distributions, extreme low values pull the mean towards the left, while the median stays between mean and mode, and the mode remains near the main peak on the right.
Correct option: a. mean < median < mode
In a left‑skewed distribution, the mean is the smallest, the mode is the largest, and the median lies in between, giving the order mean < median < mode.
This matches option (a), so option (a) is correct for the given histogram.
Why the other options are wrong
Option b: mode > mean > median
This option says mode is greatest, then mean, then median.
For left‑skewed data, the mean is pulled left and becomes the smallest, so it cannot be greater than the median; therefore mode > mean > median is inconsistent with left skewness.
Option c: mode > mean > median
Option (c) states mode > mean > median, which implies the median is the smallest.
In a negatively skewed distribution, the mean is the smallest measure, not the median, so this order contradicts the known relationship mean < median < mode.
Option d: mean = mode, mode > median
If mean = mode and both are greater than the median, the distribution would be close to symmetric or have a very different shape, not clearly left‑skewed.
For symmetric distributions, all three measures are equal (mean = median = mode), not mean = mode with a smaller median, so this option cannot describe the histogram shown.
Brief introduction for SEO
In statistics, exam questions often test the relationship between mean, median and mode in skewed histograms, especially distinguishing left‑skewed and right‑skewed distributions.
For a left‑skewed histogram like the one in this question, remembering the simple rule mean < median < mode helps quickly identify the correct option in multiple‑choice questions for CSIR NET, GATE, and other competitive exams.
