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The dispersion measure (DM) of fast radio bursts (FRBs) provides a unique way to probe ionised baryons in the intergalactic medium (IGM). Cosmological models with different parameters lead to different DM-redshift ($\mathrm{DM}-z$) relations. Additionally, the over/under-dense regions in the IGM and the circumgalactic medium of intervening galaxies lead to scatter around the mean $\mathrm{DM}-z$ relations. We have used the Evolution and Assembly of GaLaxies and their Environments (EAGLE) simulations to measure the mean $\mathrm{DM}-z$ relation and the scatter around it using over one billion lines-of-sight between redshifts $0<z<3$. We investigated two techniques to estimate line-of-sight DM: `pixel scrambling' and `box transformations'. We find that using box transformations (a technique from the literature) causes strong correlations due to repeated replication of structure. Comparing a linear and non-linear model, we find that the non-linear model with cosmological parameters, provides a better fit to the $\mathrm{DM}-z$ relation. The differences between these models are the most significant at low redshifts ($z<0.5$). The scatter around the $\mathrm{DM}-z$ relation is highly asymmetric, especially at low redshift $\left(z<0.5\right)$, and becomes more Gaussian as redshift approaches $z\sim3$, the limit of this study. The increase in Gaussianity with redshift is indicative of the large scale structures that is better probed with longer lines-of-sight. The minimum simulation size suitable for investigations into the scatter around the $\mathrm{DM}-z$ relation is 100~comoving~Mpc. The $\mathrm{DM}-z$ relation measured in EAGLE is available with an easy-to-use python interface in the open-source FRB redshift estimation package FRUITBAT.