way. This human touch allows for choosing scaling values that work sensibly, and where a best fit area minimizing value is not ideal for comparison, the author can adjust accordingly. The the graphical manually scaled approach allows us to know where biases are present in our data, and after achieving an understanding of their causes we can estimate their effects on our analysis.

One critical limitation of this process is it requires having redshift data and is therefore insensitive to biases in measuring the redshifts of our targets. To attempt to correct for this we also include a similar scaling of a theoretical LGRB distribution using the results of [@form_rate_letter]. Since we cannot exclude a redshift bias in the [@form_rate_letter] results (i.e. inducing an error in the metallicity evolution history of the universe) this is not a perfect solution (see Footnote ).

We then also use normalized cumulative distribution plots with a uniform step height to track the metallicity distribution of LGRB hosts within various redshift bins. This ability to break the sample into different redshift bins is enabled by the enlarged LGRB host metallicity galaxy population, however this analytical method has been used previously.

Analysis of Sample Redshift Distributions 