How To Calculate P-Hat

In statistics, the letter "p" denotes the probability of a certain event occurring or a certain parameter being true for a certain population, but when a population is large, it may be impractical or impossible to measure it directly. As an alternative, statisticians take a sample that they can measure, and they denote the result as "p-hat," which is written as a p with a triangular hat over it ( ^). This sampling strategy is common in political polls that seek to determine how many people in the country agree with a certain policy or approve of the job a government official, such as the president, is doing.

The actual calculation of p-hat is not challenging. To do it, you need two numbers. One is the sample size (n) and the other is the number of occurrences of the event or parameter in question (X). The equation for p-hat is p-hat = X/n. In words: You find p-hat by dividing the number of occurrences of the desired event by the sample size.

An example helps clarify this:

A poll wishes to determine how any Americans agree with the policies of the current president. Pollsters contact 1,000 voters and ask the question: "Do you approve of the president's policies?" The poll produces 175 yes answers and 825 no answers, so p-hat for the poll is 175/1,000 = 0.175. The results are usually reported as a percentage, which in this case would be 0.175 x 100 = 17.5 percent.

While it's possible to determine p-hat, the value of p remains unknown, and the degree to which it's possible to trust p-hat as being an accurate representation of p is known as the confidence level. P-hat is a reliable representation of p only if the sample is large enough and is truly random. While political pollsters make efforts to ensure random samples, it's often difficult to do in practice, and the results are often skewed. Skewing can be countered by taking larger samples or by repeating the poll in different parts of the country.

Another factor that influences the confidence level of p-hat is the number of respondents in a poll who actually answer the question. Many will decline to answer and opt to remain undecided, and the more that do so, the less pollsters can meaningfully relate p-hat to p. One way to counter this is to ask simple questions that require yes or no answers.