S-hat is a common model used in statistics when estimating a population.

## Overview: What is an S-hat?

With “hat” meaning an estimation of a value and S representing “Sigma,” S-hat refers to an estimation of the value of Sigma.

## 3 benefits of S-hat Models

There are clear benefits to S-hat Models that are worth recognizing:

### 1. Unbiased

One notable benefit of the S-hat model is that it is unbiased in its estimation. This ensures that it will not underestimate any more than it will overestimate.

### 2. Minimizing performance variation

An S-hat Model is most often used in the early stages of product design, but can actually be utilized in any process where minimizing the performance variation is desired.

### 3. Maximizing mean performance

S-hat Models are also useful for any processes where a goal is maximizing mean performance.

## Why are S-hat models important to understand?

S-hat Models are important to understand for the following reasons:

• Variance – An S-hat Model allows you to model variance.
• Influence of structures – S-hat Models are important to understand because they can help to develop an understanding of the structures that affect the variance of model performance.
• Influence of parameter values – They also help to understand the parameter values that affect variance in model performance.

## An industry example of an S-hat

A large company with several manufacturing plants around the world is wanting to have a clearer understanding of the variance in performance among all their manufacturing plants. They will be working with sampling and estimates and decide to utilize an S-hat Model in order to minimize variance along with maximizing the mean performance.

## 3 best practices when thinking about S-hat Models

Here are some key practices to keep in mind when working with an S-hat Model:

### 1. Large sampling sizes work best

When you have larger sampling sizes, the sampling distribution gets smaller. Therefore, with larger samples, your S-hat Model is likely to be more accurate.

### 2. Keep in mind it is only an estimate

It should be clear to you and those to that you present your work that your findings are only based on estimates.

### 3. Estimating the standard deviation of the sample distribution of the mean

You can use S-hat to estimate this value by dividing it by the square root of N.

## Frequently Asked Questions (FAQ) about S-hat Models

### How do you find the sampling distribution of the mean?

You take many samples from a large population. From each sample, you will compute the sample mean. From those sample means, the distribution is the sample distribution of the mean.

### When working with the S-hat Model, what is the difference between Sigma (σ) and ‘s’ in standard deviation?

Sigma is referencing the ideal population standard deviation as derived from a number of measurements that is infinite, while s is the standard deviation that is pulled from a finite number of measurements.

### What is P-hat?

P-hat is an estimate of the probability of a subset of a population.

## Do not take the S-hat Model for granted

Recent publications have extolled the virtues of S-hat modeling in its usefulness in product design and other processes, yet there is very little discussion of this type of model in data science forums. Some believe that the potential for what can be gleaned from S-hat modeling has yet to be fully tapped. It has already been determined that S-hats can give a deeper understanding of how structures and parameters affect variance in model performance. Most industries may want to have a closer look at the S-hat Model and see what else can be learned from it.