# Sampling and Representativeness

Psychology I · Statistics · https://tryals.app/learn/psychology-i/sampling-and-representativeness

## The Sampling Distribution

If you draw many random samples from a population and compute each sample's mean, those means form their own distribution: the **sampling distribution of the mean**.

Even if the original population is heavily skewed, the **Central Limit Theorem** guarantees this sampling distribution approaches a **normal distribution** as sample size grows large enough, conventionally at $n \geq 30$.

| Concept | Meaning |
| --- | --- |
| **Sampling distribution** | Distribution of means from many repeated samples |
| **Central Limit Theorem** | Distribution approaches normal as $n$ grows |
| **Standard error** | Spread of the sampling distribution |

This lets researchers use normal-distribution statistics even when the underlying population is non-normal.

## Standard Error and Pitfalls

The spread of the sampling distribution is captured by the **standard error (SE)**, measuring how much sample means vary from the true population mean due to random chance.

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

Where $\sigma$ is the population standard deviation and $n$ is the sample size. **Standard error** shrinks as $n$ grows, which is why larger samples yield more precise estimates.

> **Common pitfall:** Confusing **standard deviation (SD)** with **standard error (SE)**. **SD** describes the spread of individual data points around the mean. **SE** describes the spread of sample means across many hypothetical samples, and **SE** is always smaller than **SD** for the same data.

## Practice questions

4 of this lesson's 9 practice questions, with answers. The full set is in the app.

### 1. A study with 20 participants finds no significant effect. Why is *"there is no effect"* the wrong conclusion?

A. Because non-significant results are never publishable and so never conclusive
B. Because a small sample may be unable to detect a real effect at all
C. Because 20 participants is below the minimum for any statistical test
D. Because the effect must exist; the study simply failed to find it

**Answer:** B. Because a small sample may be unable to detect a real effect at all

**Why:** A null result from an underpowered study is uninformative rather than negative, it is compatible with no effect and with a real effect the design could never see. The fix is to report power or a confidence interval: 'the effect is between −0.1 and +0.5' says something, and 'p > .05' does not.

Page: https://tryals.app/practice/psychology-i/sampling-and-representativeness/a-study-with-20-participants-finds-no-significant-effect-why-is

### 2. Sort each description under standard deviation or standard error.

**Answer:**

- Standard deviation (SD): Spread of individual data points around the mean, Describes the population's own variability, independent of sample size
- Standard error (SE): Spread of sample MEANS around the true population mean, Quadruple the sample and it halves, Computed as sigma divided by the square root of n

**Why:** SD is about individual scores and does not depend on sample size; SE is about sample means, equals sigma/sqrt(n), and shrinks as n grows. Confusing the two is exactly the pitfall this lesson warns against.

Page: https://tryals.app/practice/psychology-i/sampling-and-representativeness/sort-each-description-under-standard-deviation-or-standard-error

### 3. Order the reasoning that builds the sampling distribution and its standard error.

**Answer:**

1. Draw many random samples of the same size from a population.
2. Compute the mean of each sample.
3. Collect these sample means into their own distribution (the sampling distribution of the mean).
4. As sample size grows, this distribution approaches a normal shape (the Central Limit Theorem).
5. Its width narrows as each sample gets larger, and that width is the standard error.

**Why:** Repeated samples yield many means, which form the sampling distribution; that distribution becomes normal as n grows (CLT) and its spread (the standard error) shrinks with n. This chain is why sample means are so predictable.

Page: https://tryals.app/practice/psychology-i/sampling-and-representativeness/order-the-reasoning-that-builds-the-sampling-distribution-and-its

### 4. The Central Limit Theorem says the distribution of sample means approaches normality as n grows, regardless of the population's shape. Why is 'regardless of the population's shape' the remarkable part?

A. Because it removes the need for random sampling, since shape no longer matters.
B. Because it means individual scores also become normally distributed as the sample grows.
C. Because it shows that all populations are ultimately normally distributed once measured accurately.
D. Because normal-based inference then applies to non-normal variables.

**Answer:** D. Because normal-based inference then applies to non-normal variables.

**Why:** Without the theorem, testing a hypothesis would require knowing the population's shape, which is almost never available. It makes that ignorance survivable. Guard against the misreading: individual incomes stay resolutely skewed however large the sample, it is the distribution of sample *means* that becomes normal, and only for samples drawn randomly.

Page: https://tryals.app/practice/psychology-i/sampling-and-representativeness/the-central-limit-theorem-says-the-distribution-of-sample-means