What you are reffering to is called Net Promoter Score (NPS).
The Net Promoter Score is calculated based on responses to a single
question: How likely is it that you would recommend our
company/product/service to a friend or colleague? The scoring for this
answer is most often based on a 0 to 10 scale.
Those who respond with a score of 9 to 10 are called Promoters, and
are considered likely to exhibit value-creating behaviors, such as
buying more, remaining customers for longer, and making more positive
referrals to other potential customers.
The NPS actually measures the positive feelings customers have for a product, since our friends are very important to us. You would recommend a product to a friend only if you have a very positive experience with the product, and you want to share this with your friend. This sharing bonds the relationship with your friend.
The NPS though does not capture the why and the actual future behavior. There is a considerable amount of critisism as it is stated in this article :
Although the NPS measure can be used as a loyalty indicator, it does
not offer an explanation of the root cause or causes of a low score.
Further, this measure is typically taken at the end of the customer
journey, thus potentially masking the underlying issues of concern,
which form the basis for identifying improvements. Relying solely on a
simple single customer metric is risky and so companies are encouraged
to adopt a more nuanced multidimensional approach to better predict
customer behavior (Keiningham, et al. 2007; Wiesel, Verhoef, and de
Haan 2012). Keiningham et al. (2007) stated that a “combination of VOC
metrics universally outperforms the use of only the NPS’s
recommendation intentions when predicting actual loyalty behavior.”
Clearly, the widely acclaimed NPS is based on a customer’s attitude
rather than his or her actual behavior.
Some other customer satisfaction mertics are the Loyalty Index, Customer Satisfaction Score (CSAT), Customer Effort Score (CES).