$'s to Scrum Master

Definitely give the Scrum Guide a read. The Scrum Master is essential to the success of the Scrum Team, but that success isn't about velocity output at all.

We have no burn downs, velocity tracking, or any of the other core functions of the scrum master. 

The ownership of these tools (and deciding if they are needed or not) comes from the needs of the whole Scrum Team. The Scrum Guide will give you a better picture of what everyone's role is and how they integrate together. 

Scrum Masters support the Product Owner, Dev Team and Organization and teaching/coaching throughout so that they can perform their roles at a high level, as well as help them all obtain the benefits of Scrum. Nobody reports to the Scrum Master and the Scrum Master isn't accountable for Sprint success. That success falls on the PO and Dev Team, both in their ability to complete Sprint Goals and identify the correct ones by priority of obtaining value for the Stakeholders. 

As a process framework, Scrum is intended to help solve complex problems through iteration and feedback, while also making it easy to identify areas of improving product/team/environment. It won't actually make those improvements though; that responsibility falls on all the people within it.

Hope that helps and gets you going. There's a lot to tap into which is why the Scrum Master is the Master of Scrum and not a Scrum Manager, for example.

We have no burn downs, velocity tracking, or any of the other core functions of the scrum master

Why do you think of burn downs and velocity tracking as being core functions of a Scrum Master?

If your organization wants to adopt Scrum, a Scrum Master is essential. Your leadership should not weight the benefit of a single role and its impact on the Product, but rather the impact of Scum's roles, events, artifacts and rules as a whole.

Screen well for a Scrum Master. They are the glue that hold the team together and help with the adoption of Scrum in the organization. I always argue for a team of Scrum Masters over an "Agile Coach" because I find that good Scrum Masters can maximize the effectiveness of Scrum in the organization.

I look at the role as an investment and commitment to the continuous improvement of the organization and teams. Many sports teams hire coaching staff...I'm not sure if they know dollar for dollar how much money an offensive coordinator may make them but the organization understands the value that role brings to the team. 

They could also work this empirically if they have the data...suppose I know that my my current cost of ownership for a product is 1 million per year...maybe the product team makes up 500k of that operating cost. Adding a Scrum Master to the mix and empowering them to do their job as it's intended, while raising the cost of ownership, could show signs of improvement in other KPI's within your key value areas. That would be my hypothesis anyway. 

Important to mention I haven't accounted for the value that can be brought to the Scrum Master's service to the organization and leadership in this example, only the team / product. 

I'm going to play devil's advocate for a bit.  A Scrum Master, no matter how good they are, really won't help a team unless the team wants to be helped and the organization wants to allow it.  Many companies will hire Scrum Masters because they want to make things better but then will continue the practices that have gotten them to where they are. In order for a Scrum Master to be effective, the organization will have to be willing to change the way that they work, the way that they measure value, the way that they make decisions.  If the organization is not willing to do that, then paying for a Scrum Master is basically a waste of money. 

For example, are there studies that state a scrum master will add 10% velocity to a team or anything like that?

I have never seen anything like this because it would be nearly impossible to validate.  The Scrum Master does not add velocity or for that matter any kind of improvement.  A Scrum Master facilitates the teams ability to improve themselves by helping the organization to understand how empowering employees will ultimate lead to better results. A good Scrum Master becomes invisible to the organization but their impact is front and center.  @Tony Divel's coach analogy is a good one that most can understand. But another way to look at it is paying a Scrum Master is like taking out an extended warranty on an appliance.  Doing it for the first time is a leap of faith that you will need one.  But after you see the benefit for derived from one, doing it again is much easier to do a second time. 

I have been in situations where the organization was willing to change and say big benefits. I have also been where the Scrum Master was considered a waste because of the inability or lack of desire for the organization to change. So before your company decides to hire Scrum Masters, someone needs to lead discussions on what the expectations are for the Scrum Masters and whether the organization is truly willing to do what it takes to make that come true.  Then pay what the company feels the desired changes are worth to find someone to help drive those changes. 

My favorite Scrum Master analogy is that of a personal trainer.   

They do not perform, and they cannot directly influence how athletes "score" or win/lose games.   They are there to help the individual and the team improve and perform better, and their expertise can also influence improvement and change within the organization.

Still, it is up to the athletes (individuals, teams) to embrace what the personal trainer is teaching/advising.   The PT is never in control of this, but an experienced PT has a variety of ways to influence.

Scrum master will not add 10% velocity to the team or even any calculated data that management can say it is vital. Aside from the comments above, i think the "influencing" part and impediment removal are things that the scrum master can contribute.

The Scrum Master will create transparency to the team on the product backlog, assisting on maximizing a shared understanding of the items in priority vs the non priority items. He/she will help the team focus on the Sprint Goal and remove any bottlenecks that is impeding the team to be successful. This person will ensure that the scrum ceremonies are being followed as they should; upholding transparency, Inspection and Adaptation. He/She will coach other leaders as to why it is important to have Cross Functional Team's and can answer the why Agile. You cannot necessarily add a dollar for dollar on you ROI but will sure help in your Agile Journey. One last thing, having a Developer upholding scrum and development will not work; he/she will neither performing his/her best in either role. 

I hope this helps. 

Ps,

Miguel