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R Quantile Calculation Methods

The document discusses quantile calculations in R and how different functions can produce different outputs. It shows examples of calculating quartiles (25th, 50th, and 75th percentiles) using the quantile() and summary() functions on various datasets, with quantile() having 9 different type options that can affect the results. Custom quantile functions are also defined for some of the types. The conclusions are that summary() uses the same algorithm as quantile() with type 7, quantile() sometimes produces the same results for different types, and boxplot() does not use one of the 9 quantile types.

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281 views4 pages

R Quantile Calculation Methods

The document discusses quantile calculations in R and how different functions can produce different outputs. It shows examples of calculating quartiles (25th, 50th, and 75th percentiles) using the quantile() and summary() functions on various datasets, with quantile() having 9 different type options that can affect the results. Custom quantile functions are also defined for some of the types. The conclusions are that summary() uses the same algorithm as quantile() with type 7, quantile() sometimes produces the same results for different types, and boxplot() does not use one of the 9 quantile types.

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QUANTILE CALCULATIONS IN R

Objective:
Showing how quantiles (esp. quartiles) are calculated in R.
R offers different functions to calculate quartiles, which can produce different output.

Examples:
> data <- c(1,12,14,3,96,111)

> summary(data)
Min. 1st Qu. Median Mean 3rd Qu. Max.
1.00 5.25 13.00 39.50 75.50 111.00

> quantile(data, c(0.25, 0.5, 0.75), type = 1)


25% 50% 75%
3 12 96

Sources:
- http://stat.ethz.ch/R-manual/R-patched/library/stats/html/quantile.html
– http://en.wikipedia.org/wiki/Quantile

1.Defining test sets

Q1 MEDIAN Q3
3 3 3 3
2 3 5 7 11 13 17 19 23 29 31 37

> data12 <- c(2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37)

> length(data12) / 4 ## Length of each quartile


[1] 3

Q1 MEDIAN Q3
2.75 2.75 2.75 2.75
2 3 5 7 11 13 17 19 23 29 31

> data11 <- c(2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31)

> length(data11) / 4 ## Length of each quartile


[1] 2.75

Q1 MEDIAN Q3
2.5 2.5 2.5 2.5
2 3 5 7 11 13 17 19 23 29

> data10 <- c(2, 3, 5, 7, 11, 13, 17, 19, 23, 29)

> length(data10) / 4 ## Length of each quartile


[1] 2.5

Q1 MEDIAN Q3
2.25 2.25 2.25 2.25
2 3 5 7 11 13 17 19 23

> data9 <- c(2, 3, 5, 7, 11, 13, 17, 19, 23)

> length(data9) / 4 ## Length of each quartile


[1] 2.25
Comparison of result sets for different functions

DATA FUNCTION Q1 MEDIAN Q3


data12 quantile(data12, c(0.25, 0.5, 0.75), type = 1) 5 13 23
quantile(data12, c(0.25, 0.5, 0.75), type = 2) 6 15 26
quantile(data12, c(0.25, 0.5, 0.75), type = 3) 5 13 23
quantile(data12, c(0.25, 0.5, 0.75), type = 4) 5 13 23
quantile(data12, c(0.25, 0.5, 0.75), type = 5) 6 15 26
quantile(data12, c(0.25, 0.5, 0.75), type = 6) 5.5 15 27.5
quantile(data12, c(0.25, 0.5, 0.75), type = 7) 6.5 15 24.5
quantile(data12, c(0.25, 0.5, 0.75), type = 8) 5.833333 15 26.5
quantile(data12, c(0.25, 0.5, 0.75), type = 9) 5.875 15 26.375
summary(data12) 6.5 15 24.5
boxplot(data12) 6 15 26
data11 quantile(data11, c(0.25, 0.5, 0.75), type = 1) 5 13 23
quantile(data11, c(0.25, 0.5, 0.75), type = 2) 5 13 23
quantile(data11, c(0.25, 0.5, 0.75), type = 3) 5 13 19
quantile(data11, c(0.25, 0.5, 0.75), type = 4) 4.5 12 20
quantile(data11, c(0.25, 0.5, 0.75), type = 5) 5.5 13 22
quantile(data11, c(0.25, 0.5, 0.75), type = 6) 5 13 23
quantile(data11, c(0.25, 0.5, 0.75), type = 7) 6 13 21
quantile(data11, c(0.25, 0.5, 0.75), type = 8) 5.333333 13 22.333333
quantile(data11, c(0.25, 0.5, 0.75), type = 9) 5.375 13 22.25
summary(data11) 6 13 21
boxplot(data11) 6 13 21
data10 quantile(data10, c(0.25, 0.5, 0.75), type = 1) 5 11 19
quantile(data10, c(0.25, 0.5, 0.75), type = 2) 5 12 19
quantile(data10, c(0.25, 0.5, 0.75), type = 3) 3 11 19
quantile(data10, c(0.25, 0.5, 0.75), type = 4) 4 11 18
quantile(data10, c(0.25, 0.5, 0.75), type = 5) 5 12 19
quantile(data10, c(0.25, 0.5, 0.75), type = 6) 4.5 12 20
quantile(data10, c(0.25, 0.5, 0.75), type = 7) 5.5 12 18.5
quantile(data10, c(0.25, 0.5, 0.75), type = 8) 4.833333 12 19.333333
quantile(data10, c(0.25, 0.5, 0.75), type = 9) 4.875 12 19.25
summary(data10) 5.5 12 18.5
boxplot(data10) 5 12 19
data9 quantile(data9, c(0.25, 0.5, 0.75), type = 1) 5 11 17
quantile(data9, c(0.25, 0.5, 0.75), type = 2) 5 11 17
quantile(data9, c(0.25, 0.5, 0.75), type = 3) 3 7 17
quantile(data9, c(0.25, 0.5, 0.75), type = 4) 3.5 9 16
quantile(data9, c(0.25, 0.5, 0.75), type = 5) 4.5 11 17.5
quantile(data9, c(0.25, 0.5, 0.75), type = 6) 4 11 18
quantile(data9, c(0.25, 0.5, 0.75), type = 7) 5 11 17
quantile(data9, c(0.25, 0.5, 0.75), type = 8) 4.333333 11 17.666667
quantile(data9, c(0.25, 0.5, 0.75), type = 9) 4.375 11 17.625
summary(data9) 5 11 17
boxplot(data9) 5 11 17
Custom quantile functions per type
QuantileType1 <- function (v, p) {
v = sort(v)
m = 0
n = length(v)
j = floor((n * p) + m)
g = (n * p) + m - j
y = ifelse (g == 0, 0, 1)
((1 - y) * v[j]) + (y * v[j+1])
}

QuantileType2 <- function (v, p) {


v = sort(v)
m = 0
n = length(v)
j = floor((n * p) + m)
g = (n * p) + m - j
y = ifelse (g == 0, 0.5, 1)
((1 - y) * v[j]) + (y * v[j+1])
}

QuantileType3 <- function (v, p) {


v = sort(v)
m = -0.5
n = length(v)
j = floor((n * p) + m)
g = (n * p) + m - j
y = ifelse(trunc(j/2)*2==j, ifelse(g==0, 0, 1), 1)
((1 - y) * v[j]) + (y * v[j+1])
}

QuantileType7 <- function (v, p) {


v = sort(v)
h = ((length(v)-1)*p)+1
v[floor(h)]+((h-floor(h))*(v[floor(h)+1]- v[floor(h)]))
}

Example:

> data12 <- c(2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37)
> QuantileType1(data12, 0.25)
[1] 5

Conclusions
The function summary() seems to use the same algorithm for calculating Q1, median and Q3 as
does the function quantiles() with type set to 7.

Sometimes, the function quantiles() generates the same results with different types set.

Boxplot does not seem to use one of the 9 types that quantiles() uses to calculate Q1, median
and Q3.
Boxplots
boxplot(data9 , pch=15, main="Boxplot (data9)" , col = "lightblue", pars = list(boxwex = 5))
boxplot(data10, pch=15, main="Boxplot (data10)", col = "lightblue", pars = list(boxwex = 5))
boxplot(data11, pch=15, main="Boxplot (data11)", col = "lightblue", pars = list(boxwex = 5))
boxplot(data12, pch=15, main="Boxplot (data12)", col = "lightblue", pars = list(boxwex = 5))

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