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The T-Value is the score obtained when you perform a T-Test. It represents the difference between the mean or average scores of two groups, while taking into account any variation in scores.
For example, if you hypothesize that pet owners are more sociable than non-pet owners, you would find a way to measure sociability so that you would get a sociability score for all participants in each group.
Let's say that the mean score of pet owners is higher than that of non-pet owners. Your question would be: is the sociability score of pet owners significantly higher than the other group? The t-value measures if the difference in scores of the two groups is big enough for you to say that pet owners are indeed more sociable than non-pet owners, or if the result was something that could have just happened by chance.