Quartiles, Deciles, and Percentiles

June 9, 2013
By

(This article was first published on Analysis with R, and kindly contributed to R-bloggers)

The measures of position such as quartiles, deciles, and percentiles are available in quantile function. This function has a usage,

where:
• x - the data points
• prob - the location to measure
• na.rm - if FALSE, NA (Not Available) data points are not ignored
• names - for attributes, FALSE means no attributes, hence speeds-up the computation
• type - type of the quantile algorithms
• ... - further arguments
Example 1. The junior BS Stat students of MSU-IIT have the following SASE scores: 88, 84, 83, 80, 94, 90, 81, 79, 79, 81, 85, 87, 86, 89, and 92. Determine and interpret the quartiles of these scores.

Interpretation: Therefore, $Q_1$=25% implies that, 25% of the SASE scores fall below or equal to 81.0, while the other 75% of it is above 81.0. $Q_2$=50% is the median, and thus half of the scores are below or equal to 85.0, while the other half, scores more than 85.0. $Q_3$=75%, implies that three-fourth of the data are below or equal to 88.5, while the remaining one-fourth is above 88.5. And the minimum and maximum values are 79.0 and 94.0, respectively.

Example 2. The surveyed weights (in kilograms) of the students in Stat 131 were the following: 69, 70, 75, 66, 83, 88, 66, 63, 61, 68, 73, 57, 52, 58, and 77. Compute and interpret the deciles of these weights.

Notice the difference between the codes of quartiles and deciles computations. This time the function quantile has an argument type which is set to 5. With this, the quantile algorithm between the quartiles and deciles differ. Hence, the appropriate algorithm for decile is type 5, while the quartile is type 7, which is the default one. For further reading about the quantile algorithm run ?quantile. In addition, the prob argument above is the position to be measured, and since deciles divide the data points into ten parts, then a sequence function, seq, is used for prob's value that is from 0 to 1 of length 11 (length = 11, 11 because zero is included, which is the minimum of the data points).

Interpretation: The first decile is $D_1$=10%, implies that one-tenth of the weights fall below or equal to 57.0, and the remaining nine-tenth fall above 57.0. The $D_5$=50% is the median, thus half of the students' weights weigh below or equal to 68.0, while the other half fall above this. And so on.

Example 3. Compute the $15^{th}$, $25^{th}$, and $35^{th}$ percentiles of weights in Example 2.

Interpretation: The fifteenth percentile $P_{15}$=15% is interpreted as 15% of the samples fall below or equal to 58.3 while 85% fall above 58.3. The thirty-fifth percentile $P_{35}$=35%, implies that 35% of the weights fall below or equal to 65.7, and that 65% of it fall above 65.7.

Reference:

Yau, Chi. R Tutorial: Percentile.