Enter the number of successes and sample size to calculate different point estimates for a population proportion.

**Maximum Likelihood Estimate (MLE):**

**Wilson Estimate:**

**Jeffreys Estimate:**

**Laplace Estimate:**

## Understanding Different Point Estimates for Proportion

### 1. Maximum Likelihood Estimate (MLE)

The **MLE** is the ratio of observed successes to the total sample size:

### 2. Wilson Estimate

The **Wilson Estimate** adjusts for small sample sizes and is calculated as:

where \( Z \) is the Z-score for the selected confidence level.

### 3. Jeffreys Estimate

The **Jeffreys Estimate** is a Bayesian estimator that assumes a prior of Beta(0.5, 0.5):

### 4. Laplace Estimate

The **Laplace Estimate** also uses a Bayesian approach, with a prior of Beta(1, 1):

### Real-Life Example: Customer Satisfaction Survey

Suppose a company conducts a survey among 50 customers to determine the proportion of satisfied customers. Out of these 50 customers, 35 report being satisfied.

**Step 1:**The observed number of successes \( x = 35 \) and sample size \( n = 50 \).**Step 2:**Calculate each estimate:**MLE:**\( \hat{p}_{MLE} = \frac{35}{50} = 0.70 \)**Wilson Estimate:**Using \( Z = 1.96 \) for 95% confidence, \[ \hat{p}_{Wilson} = \frac{35 + \frac{1.96^2}{2}}{50 + 1.96^2} \approx 0.701 \]**Jeffreys Estimate:**\[ \hat{p}_{Jeffreys} = \frac{35 + 0.5}{50 + 1} = 0.695 \]**Laplace Estimate:**\[ \hat{p}_{Laplace} = \frac{35 + 1}{50 + 2} = 0.692 \]**Interpretation:**The company can use these different estimates to better understand the true proportion of satisfied customers. Each estimate provides slightly different values based on assumptions and adjustments for sample size.

Suf is a senior advisor in data science with deep expertise in Natural Language Processing, Complex Networks, and Anomaly Detection. Formerly a postdoctoral research fellow, he applied advanced physics techniques to tackle real-world, data-heavy industry challenges. Before that, he was a particle physicist at the ATLAS Experiment of the Large Hadron Collider. Now, he’s focused on bringing more fun and curiosity to the world of science and research online.