Editing Median (section)

===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 <math> m </math> is 3 since the median is the smallest value of <math> x </math> for which <math> F(x) </math> 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 <math> w </math> 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 <math> f(x) </math> are known, the interpolated median can be calculated from

<math display="block"> m_\text{int} = m + w\left[\frac{1}{2} - \frac{F( m ) - \frac{1}{2} }{f( m )}\right]. </math>

Alternatively, if in an observed sample there are <math> k </math> scores above the median category, <math> j </math> scores in it and <math> i </math> scores below it then the interpolated median is given by

<math display="block"> m_\text{int} = m + \frac{w}{2} \left[\frac{k - i} j\right]. </math>