The median of a sample will always equal the
a. mode
b. mean
c. 50th percentile
d. all of the above answers are correct
The correct answer and explanation is :
The correct answer is c. 50th percentile.
Explanation:
The median of a sample is a measure of central tendency that indicates the middle value of a data set when arranged in ascending or descending order. It divides the data set into two equal halves, with half of the values below it and half above it.
The 50th percentile is also defined as the value below which 50% of the data points fall. In other words, the 50th percentile is the data point at the middle of a distribution. By this definition, the median is the same as the 50th percentile. Therefore, the median is always equal to the 50th percentile of a data set, making option c the correct answer.
Why the other options are incorrect:
- a. Mode: The mode is the value that appears most frequently in a data set. The median and mode are not necessarily the same. A data set can have multiple modes, no mode, or a mode that is far from the median. For example, in a dataset like [1, 2, 2, 3, 4], the mode is 2, but the median is 2, showing one case where the mode coincides with the median. However, this is not always the case. Therefore, the median does not always equal the mode.
- b. Mean: The mean, or average, is the sum of all values divided by the number of values. The median and mean are not always the same. In skewed distributions, the mean can be pulled in the direction of the tail, while the median remains unaffected by extreme values. For example, in the dataset [1, 2, 3, 100], the mean is much higher than the median, which is 2. Therefore, the median does not always equal the mean.
In summary, the median is specifically defined as the 50th percentile, while the mode and mean can differ depending on the distribution of the data.