The median of a sample will always equal the _.
A Mode
B Mean
C 50th percentile
D All of the answers are correct
The correct answer and explanation is:
The correct answer is:
C) 50th percentile
Explanation:
The median of a data set is defined as the middle value when the data is ordered from smallest to largest. More specifically, it divides the data into two equal halves: 50% of the data values fall below the median and 50% fall above it. This corresponds exactly to the 50th percentile.
- Percentiles are measures that indicate the value below which a given percentage of observations fall. For example, the 25th percentile (also called the first quartile) is the value below which 25% of the data lies, and the 75th percentile (third quartile) is the value below which 75% of the data lies.
- The 50th percentile is the value that splits the data into two equal halves, making it equivalent to the median.
Why not the other options?
- A) Mode: The mode is the value that occurs most frequently in the data set. There can be one mode, multiple modes, or no mode at all if all values are unique. The mode is not necessarily related to the median. In fact, the mode can be very different from the median, especially in skewed distributions.
- B) Mean: The mean (or average) is the sum of all data values divided by the number of observations. It is sensitive to extreme values or outliers. The median, on the other hand, is robust to outliers. In symmetric distributions, the mean and median can be the same, but in skewed distributions, they often differ significantly.
- D) All of the answers are correct: Since the mode and mean do not always equal the median, this option is incorrect.
Summary:
- The median is exactly the 50th percentile of the data.
- It provides a robust measure of central tendency that is not skewed by extreme values.
- It differs from the mode and mean, which can vary independently depending on the distribution.
Thus, the median always equals the 50th percentile.