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Interpolated median

When dealing with a discrete variable, it is sometimes useful to regard the observed values as being midpoints of underlying continuous intervals. An example of this is a Likert scale, on which opinions or preferences are expressed on a scale with a set number of possible responses. If the scale consists of the positive integers, an observation of 3 might be regarded as representing the interval from 2.50 to 3.50. It is possible to estimate the median of the underlying variable. If, say, 22% of the observations are of value 2 or below and 55.0% are of 3 or below (so 33% have the value 3), then the median m {\displaystyle m} is 3 since the median is the smallest value of x {\displaystyle x} for which F (x) {\displaystyle F(x)} is greater than a half. But the interpolated median is somewhere between 2.50 and 3.50. First we add half of the interval width w {\displaystyle w} to the median to get the upper bound of the median interval. Then we subtract that proportion of the interval width which equals the proportion of the 33% which lies above the 50% mark. In other words, we split up the interval width pro rata to the numbers of observations. In this case, the 33% is split into 28% below the median and 5% above it so we subtract 5/33 of the interval width from the upper bound of 3.50 to give an interpolated median of 3.35. More formally, if the values f (x) {\displaystyle f(x)} are known, the interpolated median can be calculated from

Alternatively, if in an observed sample there are k {\displaystyle k} scores above the median category, j {\displaystyle j} scores in it and i {\displaystyle i} scores below it then the interpolated median is given by