= rank i B (3 + 2 + 4)/3 = 3.00 = rank ii C (4 + 4 + 6)/3 = 4.67 = rank iii D (5 + 4 + 8)/3 = 5.67 = rank iv Now take the ranking order and substitute in new values A iv iii i B i i ii C ii iii iii D iii ii iv becomes: A 5.67 4.67 2.00 B 2.00 2.00 3.00 C 3.00 4.67 4.67 D 4.67 3.00 5.67 These are the new normalized values. However, note that when, as in column two, values are tied in rank, they should instead be assigned the mean of the values corresponding to the ranks they would normally represent if they were different. In the case of column 2, they represent ranks iii and iv. So we assign the two tied rank iii entries the mean of 4.67 for rank iii and 5.67 for rank iv, which is 5.17. And so we arrive at the following set of normalized values: A 5.67 5.17 2.00 B 2.00 2.00 3.00 C 3.00 5.17 4.67 D 4.67 3.00 5.67 The new values have the same distribution and can now be easily compared. Here are the summary statistics for each of the three columns: Min. :2.000 Min. :2.000 Min. :2.000 1st Qu.:2.750 1st Qu.:2.750 1st Qu.:2.750 Median :3.833 Median :4.083 Median :3.833 Mean :3.833 Mean :3.833 Mean :3.833 3rd Qu.:4.917 3rd Qu.:5.167 3rd Qu.:4.917 Max. :5.667 Max. :5.167 Max. :5.667 References External links Normalization of Affymetrix Chips Statistical data transformation Equivalence (mathematics)