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Quantile [Percentile, Decile, Quartile]

Description

quantiles are cut points dividing the range of a probability distribution into continuous intervals with equal probabilities or dividing the observations in a sample in the same way. There is one fewer quantile than the number of groups created. Common quantiles have special names, such as quartiles (four groups), deciles (ten groups), and percentiles (100 groups).

Quartiles (4 Groups)

Divide the number of data points into four parts, or quarters, of more or less equal size. The data must be ordered from smallest to largest to compute quartiles; as such, quartiles are a form of order statistic. The three main quartiles are as follows:

  • The first quartile (\(Q1\)) is defined as the middle number between the smallest number (minimum) and the median of the data set. It is also known as the lower or 25th empirical quartile, as 25% of the data is below this point.
  • The second quartile (\(Q2\)) is the median of a data set; thus 50% of the data lies below this point.
  • The third quartile (\(Q3\)) is the middle value between the median and the highest value (maximum) of the data set. It is known as the upper or 75th empirical quartile, as 75% of the data lies below this point.

Probability density of a normal distribution, with quartiles shown. The area below the red curve is the same in the intervals \((-\infty, Q1)\), \((Q1, Q2)\), \((Q2, Q3)\), and \((Q3, +\infty)\).

Deciles (10 Groups)

It is any of the nine values that divide the sorted data into ten equal parts so that each part represents 1/10 of the sample or population.

Percentiles (100 Groups)

In statistics, a percentile is a term that describes how a score compares to other scores from the same set. While there is no universal definition of percentile, it is commonly expressed as the percentage of values in a set of data scores that fall below a given value.

The 25th percentile is also known as the first quartile (\(Q1\)), the 50th percentile as the median or second quartile (\(Q2\)), and the 75th percentile as the third quartile (\(Q3\)). For example, the 50th percentile (median) is the score below (or at or below, depending on the definition) in which 50% of the scores in the distribution are found.