For a negatively skewed distribution, the median would be equal to which of…
2023
For a negatively skewed distribution, the median would be equal to which of the following measures?
A. Mean
B. Mode
C. Second quartile
D. Fifth decile
E. 50\(^{th}\) percentile
Choose the most appropriate answer from the options given below:
Answer: C. C, D and E only — ConceptA positional measure locates a value by cumulative rank rather than by averaging magnitudes or counting the most frequent value. The median is the…
- A.
A, B and C only
- B.
B, C and D only
- C.
C, D and E only
- D.
A, B, C, D and E
Show answer & explanation
Correct answer: C
Concept
A positional measure locates a value by cumulative rank rather than by averaging magnitudes or counting the most frequent value. The median is the value at cumulative proportion one-half.
Quantile names are equivalent when their fractional positions are equal. This identity does not depend on whether the distribution is symmetric or skewed.
Application
The second quartile is at 2/4 = 1/2 of the ordered data.
The fifth decile is at 5/10 = 1/2 of the ordered data.
The 50th percentile is at 50/100 = 1/2 of the ordered data.
Cross-check and contrast
Mean: the arithmetic average uses all observed magnitudes and is pulled toward the long tail in a skewed distribution.
Mode: the most frequent value is determined by the highest frequency, not by the halfway cumulative rank.
Second quartile, fifth decile, and 50th percentile: the fractions 2/4, 5/10, and 50/100 each reduce to 1/2.
Therefore, the median equals the second quartile, the fifth decile, and the 50th percentile only.