P vs P Hat: Difference and Comparison

Key Takeaways

1. P is the exact probability distribution, P hat is the estimated probability distribution.
2. P is the true underlying probability, P hat is calculated empirically from samples.
3. P hat converges to P as sample size approaches infinity by the law of large numbers.

What is P?

In statistics, “p” represents a population proportion or probability. It refers to the true, unknown proportion or probability of an event or characteristic within a population. “p” represents a population proportion or probability. It refers to the true, unknown proportion or probability of an event or characteristic within a population. For example, if you want to determine the proportion of people in a city who support a particular policy, “p” would represent the true proportion of supporters in the entire population of the city.

The symbol “p” is commonly used when describing categorical data or binary outcomes, where there are two possible outcomes, referred to as success and failure. For example, “p” can represent the proportion of individuals in a population who have a specific characteristic or exhibit a particular behavior.

What is P Hat?

“p-hat” is used to represent a sample proportion or probability. It is an estimate of the true population proportion, based on data from a sample. The symbol “p-hat” is derived by placing a caret symbol (ˆ) above the letter “p” to distinguish it as an estimate rather than the true population proportion.

When conducting surveys or experiments, it is impractical or impossible to gather data from an entire population. Instead, a representative sample is selected to gather information. The sample proportion, denoted as “p-hat,” is calculated by counting the number of occurrences of a specific characteristic or outcome of interest in the sample and dividing it by the sample size.

Difference Between P and P Hat

1. “P” represents the true, unknown population proportion or probability, whereas “P-hat” represents the sample proportion or probability estimated from the observed data.
2. “P” is a fixed and constant value that describes the entire population, while “P-hat” is a variable that varies from sample to sample, representing the estimate of “P” specific to the sampled data.
3. “P” is unknown and is the target parameter being estimated, whereas “P-hat” is the estimated value obtained from the sample data. “P-hat” is a point estimate used to make inferences about “P.”
4. “P” is considered to be a population parameter with a fixed, precise value. On the other hand, “P-hat” is a sample estimate and is subject to sampling variability. The precision of “P-hat” depends on the sample size, with larger samples providing more precise estimates.
5. “P” is used to make population-level inferences or draw conclusions about the entire population. “P-hat” is used as an estimate to make inferences about “P” based on the sample data. Statistical methods, such as confidence intervals or hypothesis testing, are employed using “P-hat” to draw conclusions about “P” with a certain level of confidence.

Comparison Between P and P Hat

Parameters of Comparison	P	P Hat
Definition	The true, unknown population proportion	Sample proportion estimated from observed data
Representation	Fixed, constant value	Variable value that varies from sample to sample
Known vs. estimated	Typically unknown, the parameter being estimated	Obtained from sample data as an estimate
Precision	Fixed, precise value	Subject to sampling variability
Inference	Used for population-level conclusions	Used to make inferences about “P” based on sample data

References

1. https://onlinelibrary.wiley.com/doi/abs/10.1111/j.1537-2995.2010.02818.x
2. https://sciendo.com/article/10.2478/pjct-2021-0033

Last Updated : 25 November, 2023