Copyright law varies from country to country, but the best way recognized by everyone is to sign a contract that includes what will be used from the party that will provide you with the statistics.

This is not legal advice, because it depends on a lot of information that is missing, but just to give you a general understanding:
Citing/referring to the source of the information does not mean it is ok for you to use that information - even if the information is "public". If I leave my car out in a public parking lot, it is still mine and leaving it there doesn't mean that you can use it. The same applies to me putting something on the internet. Even if it's public (like on Google), doesn't mean I am allowing you to use it. Each piece of information (statistics, pictures, music...) is owned by someone, and either they offer a license which tells you how you can/can't use it, or you have to contact them and get their explicit permission.

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If you want to legally use published statistics on a t-shirt for a print on demand business, you must ensure that you have the necessary rights to use the statistics. This may involve obtaining permission from the owner of the statistics or ensuring that the statistics are in the public domain.

Assuming that the statistics are in the public domain, you can use them on a t-shirt for a print on demand business without seeking permission. However, it is still important to cite your sources to avoid any potential legal issues. Here are some steps you can take to properly cite your sources:

Determine the source of the statistics, such as a government agency or research organization.

Locate the specific publication or report that contains the statistics.

Include the name of the source and the title of the publication or report on your t-shirt design. For example, you could use the following format: "Source: [Name of Source], [Title of Publication/Report], [Date of Publication]."

If possible, include a link to the source of the statistics on your t-shirt design or on your online store where the t-shirt is sold. This can help customers verify the accuracy of the statistics and ensure that you are citing your sources correctly.

By properly citing your sources, you can help ensure that you are using published statistics legally and avoid any potential legal issues. It's always a good idea to consult with a lawyer if you have any questions about the legality of using published statistics on a t-shirt or other product for commercial purposes.