Find the interpolated sample median, quartiles, or specific quantiles for a vector, matrix, or data frame

For data with a limited number of response categories (e.g., attitude items), it is useful treat each response category as range with width, w and linearly interpolate the median, quartiles, or any quantile value within the median response.

interp.median(x, w = 1,na.rm=TRUE)
interp.quantiles(x, q = .5, w = 1,na.rm=TRUE)
interp.quartiles(x,w=1,na.rm=TRUE)
interp.boxplot(x,w=1,na.rm=TRUE)
interp.values(x,w=1,na.rm=TRUE)
interp.qplot.by(y,x,w=1,na.rm=TRUE,xlab="group",ylab="dependent",
               ylim=NULL,arrow.len=.05,typ="b",add=FALSE,...)

Arguments

x

input vector
q

quantile to estimate ( 0 < q < 1
w

category width
y

input vector for interp.qplot.by
na.rm

should missing values be removed
xlab

x label
ylab

Y label
ylim

limits for the y axis
arrow.len

length of arrow in interp.qplot.by
typ

plot type in interp.qplot.by
add

add the plot or not
...

additional parameters to plotting function

Details

If the total number of responses is N, with median, M, and the number of responses at the median value, Nm >1, and Nb= the number of responses less than the median, then with the assumption that the responses are distributed uniformly within the category, the interpolated median is M - .5w + w*(N/2 - Nb)/Nm.

The generalization to 1st, 2nd and 3rd quartiles as well as the general quantiles is straightforward.

A somewhat different generalization allows for graphic presentation of the difference between interpolated and non-interpolated points. This uses the interp.values function.

If the input is a matrix or data frame, quantiles are reported for each variable.

Value

im

interpolated median(quantile)

v

interpolated values for all data points

See also

median

Examples

interp.median(c(1,2,3,3,3))  # compare with median = 3
#> [1] 2.666667
interp.median(c(1,2,2,5))
#> [1] 2
interp.quantiles(c(1,2,2,5),.25)
x <- sample(10,100,TRUE)
interp.quartiles(x)
#>       Q1   Median       Q3 
#> 3.318182 5.772727 7.666667 
#
x <-  c(1,1,2,2,2,3,3,3,3,4,5,1,1,1,2,2,3,3,3,3,4,5,1,1,1,2,2,3,3,3,3,4,2)
y <-  c(1,2,3,3,3,3,4,4,4,4,4,1,2,3,3,3,3,4,4,4,4,5,1,5,3,3,3,3,4,4,4,4,4)
x <-  x[order(x)]   #sort the data by ascending order to make it clearer
y <- y[order(y)]
xv <- interp.values(x)
yv <- interp.values(y)
barplot(x,space=0,xlab="ordinal position",ylab="value")
lines(1:length(x)-.5,xv)
points(c(length(x)/4,length(x)/2,3*length(x)/4),interp.quartiles(x))
barplot(y,space=0,xlab="ordinal position",ylab="value")
lines(1:length(y)-.5,yv)
points(c(length(y)/4,length(y)/2,3*length(y)/4),interp.quartiles(y))
data(galton)
interp.median(galton)
#>        parent    child
#> [1,] 68.32877 68.18333
interp.qplot.by(galton$child,galton$parent,ylab="child height"
,xlab="Mid parent height")