# Sampling and the Central Limit Theorem

Mathematics I · Data Analysis and Probability · https://tryals.app/learn/mathematics-i/sampling-and-the-central-limit-theorem

## From Sample to Population

Statistics uses a **sample** (a subset of collected data) to infer properties of a **population** (the whole group of interest).

| Term | Symbol | Meaning |
| --- | --- | --- |
| **Parameter** | $\mu$ | True population characteristic |
| **Statistic** | $\bar{x}$ | Sample quantity estimating $\mu$ |

For valid inference, use a **random sample** where every member is equally likely to be chosen, avoiding **bias** (systematic error from non-random selection). 

Because samples vary, $\bar{x}$ is a **random variable**. Its **sampling distribution** has a centre equal to $\mu$ and a spread called the **standard error**:

$SE = \frac{\sigma}{\sqrt{n}}$

Here, $\sigma$ is population standard deviation and $n$ is sample size. Larger samples yield more precision, but only by a factor of $\sqrt{n}$.

## The Central Limit Theorem

The **Central Limit Theorem (CLT)** states that for a sufficiently large sample ($n \ge 30$ typically), the sampling distribution of the mean is **approximately normal**, regardless of the population's underlying shape.

| What Changes with CLT | What Does NOT Change |
| --- | --- |
| Distribution of the **sample mean** ($\bar{x}$) | The **raw data** population shape |
| Averages become bell-shaped | Skewed individual data remain skewed |

> **Common pitfall:** Thinking the CLT makes the **raw data** normal. It only applies to the **sampling distribution of the mean**. The population shape remains unchanged; only the distribution of the _statistic_ turns normal as $n$ grows.

## Practice questions

6 of this lesson's 10 practice questions, with answers. The full set is in the app.

### 1. Polls typically sample around a thousand people whether the population is a city or a country of 60 million. Why does population size barely matter?

A. Because the margin of error is fixed by convention at three points
B. Because polling organisations cannot afford larger samples
C. Because the standard error depends on sample size, not on the fraction
D. Because larger populations are more homogeneous, requiring less sampling

**Answer:** C. Because the standard error depends on sample size, not on the fraction

**Why:** The formula contains n and not N, so what matters is how many you asked rather than what fraction they represent, the soup only needs to be stirred. The intuition that a bigger country needs a bigger poll is among the most persistent errors about sampling, and it is simply not in the mathematics.

Page: https://tryals.app/practice/mathematics-i/sampling-and-the-central-limit-theorem/polls-typically-sample-around-a-thousand-people-whether-the

### 2. Order the stages of a statistical inference from population to conclusion.

**Answer:**

1. Define the population and the parameter you want to know
2. Draw a random sample, avoiding systematic bias in selection
3. Compute the sample statistic that estimates the parameter
4. Use the sampling distribution to describe how much that statistic varies
5. State a conclusion about the parameter together with its uncertainty

**Why:** Representativeness comes first because no analysis repairs a biased sample. The sampling distribution then converts a single number into a number with an honest margin, which is what the following lessons formalise.

Page: https://tryals.app/practice/mathematics-i/sampling-and-the-central-limit-theorem/order-the-stages-of-a-statistical-inference-from-population-to

### 3. Select every true statement about sampling.

A. The Central Limit Theorem requires the population itself to be normal
B. The standard error is proportional to 1 over the square root of n
C. A larger sample repairs a biased sampling method
D. The standard error shrinks as the sample size grows
E. Averaged over many samples, the sample mean lands on the population mean

**Answer:** B. The standard error is proportional to 1 over the square root of n; D. The standard error shrinks as the sample size grows; E. Averaged over many samples, the sample mean lands on the population mean

**Why:** Bias is a systematic error that grows no smaller with more data, a bigger biased sample is just a more confident wrong answer. And the theorem's whole power is that it needs no assumption about the population's shape.

Page: https://tryals.app/practice/mathematics-i/sampling-and-the-central-limit-theorem/select-every-true-statement-about-sampling

### 4. Sort each quantity by whether it describes the population or the sample.

**Answer:**

- Parameter (population): The mean height of every adult in Spain, The true proportion of all voters supporting a party, The standard deviation of all values in the population
- Statistic (sample): The mean height of the 200 adults actually measured, The proportion supporting it among 1000 people polled

**Why:** Parameters describe the whole group and are usually unknowable; statistics are computed from the sample and change every time you resample. Statistical inference exists precisely to cross that gap, and confusing the two collapses the distinction it depends on.

Page: https://tryals.app/practice/mathematics-i/sampling-and-the-central-limit-theorem/sort-each-quantity-by-whether-it-describes-the-population-or-the

### 5. The standard error of the sample mean is σ/√n. Why the square root - why does quadrupling the sample only halve the uncertainty?

A. Because the square root is an approximation used because the exact formula is intractable.
B. Because larger samples contain more errors, which partly offsets the gain from averaging.
C. Because variance falls like 1/n, so its square root falls like 1/√n.
D. Because standard error measures spread in squared units, which must be square-rooted to compare with the mean.

**Answer:** C. Because variance falls like 1/n, so its square root falls like 1/√n.

**Why:** Work in variances, where the arithmetic is clean: independent variances add to nσ², dividing by n to form the mean divides by n², leaving σ²/n, and the square root converts that back into the units of the data. Hence the diminishing return: halving uncertainty costs four times the data, which is why polls settle around a thousand respondents.

Page: https://tryals.app/practice/mathematics-i/sampling-and-the-central-limit-theorem/the-standard-error-of-the-sample-mean-is-n-why-the-square-root

### 6. Complete the statement of the Central Limit Theorem.

**Answer:** The Central Limit Theorem says that for a large enough sample, the distribution of **the sample mean** is approximately normal.

**Why:** It is the distribution of the _statistic_ that turns normal, regardless of the population's own shape. The data stay as skewed as they always were; it is their averages that become bell-shaped.

Page: https://tryals.app/practice/mathematics-i/sampling-and-the-central-limit-theorem/complete-the-statement-of-the-central-limit-theorem