# Point Estimate Calculator

Created by Bogna Szyk
Reviewed by Steven Wooding
Last updated: Jun 05, 2023

If you have gathered a lot of population data and want to find the "best guess" parameter, this point estimate calculator will be right up your alley. It uses four different point estimate formulas to give you the most exact value possible. You can start using the calculator right ahead or read on to learn more about the principles behind it.

Make sure to check out the example at the end to better understand how to find the point estimate in simple statistical problems.

## What is the point estimate?

Imagine you're tossing a coin. Every time you throw, you note down the result. For an unbiased coin and a sufficiently large number of trials, you should get roughly 50% heads and 50% tails.

But what if the coin is biased – for example, slightly bent? Then, after a large number of tosses, you will discover that one of the sides appears more often. It means that the probability of getting heads differs from 50% for that particular coin.

The point estimate is the probability of getting a 'heads' result in this example. Once you have tossed the coin enough times and have gathered some data on the coin's "behavior", you will be able to find it with our point estimate calculator.

## Point estimate formulas

You can use four different point estimate formulas: the Maximum Likelihood Estimation (MLE), Wilson Estimation, Laplace Estimation, and Jeffrey Estimation. Each gives a slightly different result and should be used in different circumstances. Our point estimate calculator automatically chooses the most relevant result, but you can open the advanced mode to see all of them.

To calculate the point estimate, you will need the following values:

• Number of successes S: for example, the number of heads you got while tossing the coin.
• Number of trials T: in the coin example, it's the total number of tosses.
• Confidence interval: the probability that your best point estimate is correct (within the margin of error). If you're not yet familiar with this notion, make sure to visit Omni's confidence interval calculator.
• Z-score z: it will be calculated automatically from the confidence interval.

Once you know these values, you can start calculating the point estimate according to the following equations:

• Maximum Likelihood Estimation:
MLE = S / T
• Laplace Estimation:
Laplace = (S + 1) / (T + 2)
• Jeffrey Estimation:
Jeffrey = (S + 0.5) / (T + 1)
• Wilson Estimation:
Wilson = (S + z²/2) / (T + z²)

Once you have calculated all four values, you need to choose the most accurate one. You should do this step according to the following rules:

• If MLE ≤ 0.5, the Wilson Estimation is the most accurate.
• If 0.5 < MLE < 0.9, the Maximum Likelihood Estimation is the most accurate.
• If 0.9 < MLE, then the smaller of Jeffrey and Laplace Estimations is the most accurate.

## How to find the point estimate?

If you are still unsure how the procedure of finding the point estimate works, take a look at the example below. We will examine the biased coin problem in more detail.

1. Determine the total number of coin tosses – this will be the number of trials T. Let's assume T = 100.

2. Count the number of times that you got heads. It will be the number of successes S. Let's say S = 92. (You can be sure that the coin is biased even when simply looking at this number.)

3. Decide on your confidence interval. Let's say you need to be only 90% sure that your result is accurate, so you settle for the confidence interval of 90%.

4. The point estimate calculator will find the z-score for you (in advanced mode). If you want more details on how it's calculated, take a look at the p-value calculator. In this case, z = -1.6447.

5. Use the point estimate formulas:

• MLE = S / T = 92 / 100 = 0.92
• Laplace = (S + 1) / (T + 2) = 93 / 102 = 0.9118
• Jeffrey = (S + 0.5) / (T + 1) = 92.5 / 101 = 0.9158
• Wilson = (S + z²/2) / (T + z²) = (92 + (-1.6447)²/2) / (100 + (-1.6447)²) = 0.9089
6. As the Maximum Likelihood Estimation is greater than 0.9, you should choose the smaller of Jeffrey and Laplace Estimations as the best point estimate. In this case, it is equal to 0.9118. It means that the probability of getting heads with this coin is equal to 91.18%.

## FAQ

### How do I calculate the maximum likelihood point estimate?

To determine the point estimate via the maximum likelihood method:

1. Write down the number of trials, T.
2. Write down the number of successes, S.
3. Apply the formula MLE = S / T. The result is your point estimate.

### How do I calculate Laplace point estimate?

To find the Laplace point estimate for S successes in T trials, you need to apply the formula (S + 1) / (T + 2).

### How do I calculate the Jeffrey point estimate?

The Jeffrey point estimate for S successes in T trials is given by the formula (S + 0.5) / (T + 1).

### How do I calculate Wilson point estimate?

To determine the Wilson point estimate:

1. Write down the number of trials, T.
2. Write down the number of successes, S.
3. Decide on the confidence level.
4. Compute the Z-score, Z, corresponding to this confidence level.
5. Apply the formula (S + Z²/2) / (T + Z²).

### What is the most accurate point estimate formula?

The best point estimate formula is chosen based on the value of the maximal likelihood estimate:

• If 0.5 < MLE < 0.9, stick to MLE;
• If MLE ≤ 0.5, discard MLE and choose Wilson's estimate; or
• If MLE > 0.9, take the smaller of the Jeffrey and Laplace estimates.

### What is the difference between point estimation and interval estimation?

Point estimation of an unknown parameter returns a single value, while interval estimation returns an interval (range) of values.

Bogna Szyk
No. of successes
No. of trials
Confidence interval
%
Point estimate
Best point estimate
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