Statistics: Standard Error and Central Limit Theorem

What is standard error of a population? 

Step 1: Definition of standard error

The standard error of the sample mean is:

Standard Error = σ / root(n) 

where  σ  is the population standard deviation and n n is the sample size.

Step 2: Can it be negative? No because standard deviation is always positive and n also be, so, standard error cannot be negative 

A survey about mental health has been conducted on the freshmen class at ABC High School. A sample of  200 students was randomly selected from the freshmen class at ABC High School for the survey. Identify the population in this study

The 200 selected students

All freshmen at ABC High School

All students at ABC High School

Step 1: Understand the terms

Population: The entire group of individuals the study is interested in learning about.

Sample: A subset of the population that is actually surveyed or studied.

Step 2: Identify the population in the question

The survey is about mental health of the freshmen class at ABC High School.

They took a sample of 200 students from the freshmen class.

So:

Population = All freshmen at ABC High School

Sample = The 200 selected students

Central Limit Theorem 

Let's break this down carefully.

---

### **1. What the Central Limit Theorem (CLT) says**

The CLT states that if you take random samples of size \( n \) from **any population** with mean \( \mu \) and finite variance \( \sigma^2 \), then as \( n \) becomes large, the sampling distribution of the sample mean \( \bar{X} \) approaches a **normal distribution** \( N(\mu, \sigma^2/n) \), regardless of the population's original distribution.

---

### **2. Does it matter if the population distribution is continuous or discrete?**

- **No** — the CLT applies to **any population distribution** with finite variance, whether it is **continuous** (e.g., height, weight) or **discrete** (e.g., number of children, test scores, dice rolls).

- The only requirement is:  

  1. Independent and identically distributed (i.i.d.) samples.  

  2. Finite variance \( \sigma^2 \).  

  3. Sample size \( n \) sufficiently large (rule of thumb: \( n \geq 30 \) for strong non-normality, but smaller \( n \) may suffice if population is not too far from normal).

---

### **3. Examples of CLT with discrete distributions**

- Rolling a fair die: population distribution is discrete uniform. The sample mean of many rolls will be approximately normal for large \( n \).

- Bernoulli trials: proportion of successes → approximately normal for large \( n \) (this is actually the De Moivre–Laplace theorem, a special case of CLT for binary data).

---

### **4. Conclusion**

The CLT holds for **both continuous and discrete distributions**.

---

\[

\boxed{\text{Continuous and Discrete distributions both}}

\]