Basic data inference

Data inference uses a small, well-chosen sample to draw conclusions about a much larger population. If the sample is random and representative, the fraction found in the sample carries over to the whole population.
For example, if $15$ of $60$ sampled shoppers prefer a brand, that is $\frac{1}{4}$, so out of $800$ shoppers you would estimate $\frac{1}{4} \times 800 = 200$ prefer it.

The catch is representativeness: an inference can only be extended to the exact population the sample stands for. A survey of male cricket fans in India says nothing reliable about all cricket fans in India, or about male fans worldwide, because the sample does not represent those groups.

Because a sample is only an estimate, results are often reported as a confidence interval: a range within which the population's true value is likely to lie. Two intervals overlap when they share values, as shown below, which means the polls do not necessarily disagree.

In a random sample of $200$ students, $40$ play soccer, a proportion of $\frac{40}{200} = \frac{1}{5}$. Applied to the full population of $1000$ students, the estimate is $\frac{1}{5} \times 1000 = 200$ soccer players.

Poll A gives a confidence interval of $56\%$ to $64\%$, and Poll B gives $52\%$ to $58\%$. They share the values from $57\%$ to $58\%$, so the intervals overlap there, meaning the two polls do not necessarily disagree.