The pth percentile is a value such that at least p percent of the observations are
a. less than or equal to this value
b. less than this value
c. more than or equal to this value
d. more than this value
The correct answer and explanation is :
The correct answer is a. less than or equal to this value.
Explanation:
In statistics, percentiles are used to understand the relative position of a particular value within a data set. The pth percentile is defined as the value below which p percent of the data fall. This means that if you are at the pth percentile, at least p percent of the observations will be less than or equal to this value, and the remaining (100 – p)% will be greater than or equal to it.
For example, if you are in the 50th percentile (often referred to as the median), this means that 50% of the data points are less than or equal to this value, and 50% are greater than or equal to it. Similarly, in the 90th percentile, at least 90% of the values fall below or are equal to this value, with only 10% of the data being greater.
This percentile concept can be illustrated with a data set. Suppose you have 100 data points sorted in ascending order, and you want to find the 30th percentile (p = 30). The 30th percentile is the value below which 30% of the data points lie. If the data are ranked from lowest to highest, at least 30% of the values will be less than or equal to this percentile value, and the remaining 70% will be greater than or equal to it.
It’s important to note that different methods might exist for calculating percentiles, especially with small data sets or ties, but the general concept remains that the pth percentile is the value where at least p percent of the observations are less than or equal to this value.
Therefore, the correct answer is a, because the definition of a percentile ensures that the value represents a threshold where at least a certain percentage of observations are less than or equal to it.