# The Interpolated Median

The Instructor Report includes a column labeled “Your Median.” This value represents an interpolated median, as it provides more information than a standard median.

In computing numerical summaries for responses, the following scale is used for the majority of questions:

• 5 = SA Strongly Agree or Almost Always
• 4 = A Agree or Frequently
• 3 = N Neutral or Sometimes
• 2 = D Disagree or Occasionally
• 1 = SD Strongly Disagree or Rarely

N/A (Not Applicable) responses and responses left blank are not included when computing numerical summaries.

The median is the middle observation in a sorted list of data. Half of the values in the data set are less than or equal to the median and half are greater than or equal to it. The interpolated median (IM) which is used in these reports, adjusts the median slightly upward or downward.

For example, any interpolated median between 3.5 and 4.5 indicates that the actual median rating for the question was 4. An interpolated median between 4.0 and 4.5 also indicates that there were more ratings above 4 than below 4. Similarly, an interpolated median between 3.5 and 4.0 indicates that there were fewer ratings above 4 than below 4.

Why use an interpolated median?

To illustrate the usefulness of the interpolated median, consider two questions with 20 responses for each question. The table below lists the number of respondents for each question that gave each response to a particular question:

Response Question 1 Question 2
5 = Strongly agree 9 1
4 = Agree 10 10
3 = Neither agree nor disagree  0 6
2 = Disagree 1 1
1 = Strongly disagree 0 2

Both question 1 and question 2 have medians of 4 for this question. However, it is quite clear that the overall ratings on question 1 were substantially higher than question 2. The interpolated median provides a way to adjust the median to reflect this. The interpolated median for Question 1 is 4.4 (the median is adjusted upward since 9 respondents gave a rating above the median while only 1 gave a rating below the median. On the other hand, for Question 2, more respondents gave ratings below the median than above it, so the interpolated median adjusts downward to 3.6. The interpolated median clearly represents the differences between the two questions, while the median failed to do so.

How is the interpolated median actually computed?
Define variables as follows:
M = the standard median of the responses
nl = number of responses strictly less than M
ne = number of responses equal to M
ng = number of responses strictly greater than M

The interpolated median IM is then computed as follows:

If ne is nonzero:
IM = M + (ng - nl) / (2ne)

If ne is zero:
IM = M