There are quite a few statistics on this subject.
indexcreditcards.com - This might be a good starting point.
The Federal Reserve reports that credit cards are used more than 20 billion times a year in the U.S., with the total value of these transactions at about $1.9 trillion.
Based on the number of transactions and the number of credit card holders, the average card holder uses a credit card 119 times a year, for transactions averaging $88 apiece. This comes to an average annual total of about $10,500 in credit card purchases.
CardWeb reported that in the last year, the average size of credit card transaction tops $100 for the first time. 2003's average of $101.64 is 1.9% higher than last year's $99.76 and represents 16% increase in 10 years.
At the end of 2009 the average purchase made with Visa or MasterCard Credit Cards was $82.56, a number lower than previous years, where the average was slightly over $100. You can order more recent reports on this figure from CardData.
The indexcreditcards.com article might have some useful strategies for calculating these statistics, like combining the Federal Reserve data (for example, from FRED) with Census and industry estimates. That's probably the best place to start for recent data that shouldn't cost anything (or if it does, the cost shouldn't be prohibitive). Cardweb looks like it compiles the information you need too, but according to their site, subscriptions start at $1,495.00.
If you need this data for research purposes, citing a website isn't sufficient, but assuming you describe how to came up with your statistics, combining data sets in the way mentioned above should work.
For similar information relating to debit cards, the Federal Reserve also has information about that, since they work with payment processing networks.
There were approximately 46.7 billion debit card transactions in 2011,
with a value of more than $1.8 trillion. This was a 24 percent
increase from the number of transactions in 2009 (37.6 billion) and a
27 percent increase from the value of transactions in 2009 ($1.4
trillion). Signature debit transactions represented about 63 percent
of transaction volume and 61 percent of transaction value in 2011; the
remainder were PIN debit transactions.
mainstreet.com also cites data provided by mint.com which finds that "the average debit card transaction amount in 2010 was $71." This isn't a random sample because it only represents data on mint.com users. Intuitively, I can think of reasons for why users of mint.com would spend either more or less than the population as a whole, but I'm not sure which is true.
Keep in mind, though, that debit card transactions might qualify as Electronic Fund Transfers, which could complicate the picture. I haven't researched payment processing networks extensively, so I can't make a statement on that one way or another.