Equal interval and quantile method are different approaches to data classification.  Taking as a reference the Frequency Histogram of Disease Rate, the equal interval method uses the lowest and the highest rate values on the x-axis (1 to 6) Then, subtracted those values and divided them by the number of classes selected.  Class # 1 will get the number of observations per geographic unit that fit between the ratio 1 and 2. This means an approximate estimation of 800 counts, taking the majority of the counts of the total frequencies available. When you display this data in the third graph, much of the geographic units of the study contain these value rates (between 1 and 2) so that is why the first bar shows a larger white class comparing with the rest of the other methods of classification. A map using equal intervals will display most of the geographic units in white, so the predominant color shade will be white with a small variation of grays.

On the other hand, the quantile method uses the total of observation per geographic unit in the data set and divided them by the number of selected classes. A total of 900 observations are dived into 5 classes. This is equal to 180 observations per class. The top graph located on the right side of the slide displays classes from 1 to 4 being shorter than class # 5. The number of observations are highly concentrated on the left side of the histogram, so classes between 1 and 4 will be filled before getting to the value of 2 (disease rate x 100).  Class # 5 needs rate values between 2 and 6 to complete the last 180 sets of observations.  Using the quantile method in a map, the # of classes will be equally distributed between the 900 geographic units. This means, there are no big variations between classes because each class contains the same number of geographic units per class. For this particular case, the map will have an equal distribution of gray shades across the map.

Both methods of classification of data can be useful depending on the type of analysis that I want to conduct. The equal interval method will help me to locate those geographic units with highest rates of disease. To observe the extremes cases in my study area or those areas with abnormal rates. The quantile method will help me to see which 180 geographic units have the lowest rates vs 180 units with the highest. This will be useful in ranking my geographic units.