Methods and apparatus to estimate audience sizes of media using deduplication based on binomial sketch data

A method comprising:

• receiving, by a computing system, a first network communication from a server of a database proprietor, wherein the first network communication includes first sketch data representative of first individuals exposed to media;
• determining, by the computing system, coefficient values of a polynomial based on (i) variances in the first sketch data and second sketch data (ii) a first cardinality of the first sketch data, and (iii) a second cardinality of the second sketch data, wherein the second sketch data is representative of second individuals exposed to the media, wherein the first sketch data and the second sketch data are generated to maintain a privacy of the first individuals and the second individuals;
• determining, by the computing system, a root of the polynomial, wherein the root is an estimate of an overlap between the first individuals and the second individuals;
• estimating, by the computing system, a deduplicated audience size for the media based on the overlap, the first cardinality, and the second cardinality;
• and
• transmitting, by the computing system, a second network communication to a third-party entity, wherein the second network communication includes a report based on the deduplicated audience size.

A non-transitory computer-readable medium having stored therein instructions that, when executed by a computing system, cause the computing system to perform a set of acts comprising:

• receiving a first network communication from a server of a database proprietor, wherein the first network communication includes first sketch data representative of first individuals exposed to media;
• determining coefficient values of a polynomial based on (i) variances in the first sketch data and second sketch data (ii) a first cardinality of the first sketch data, and (iii) a second cardinality of the second sketch data, wherein the second sketch data is representative of second individuals exposed to the media, wherein the first sketch data and the second sketch data are generated to maintain a privacy of the first individuals and the second individuals;
• determining a root of the polynomial, wherein the root is an estimate of an overlap between the first individuals and the second individuals;
• estimating a deduplicated audience size for the media based on the overlap, the first cardinality, and the second cardinality;
• and
• transmitting a second network communication to a third-party entity, wherein the second network communication includes a report based on the deduplicated audience size.