# Statistical Analysis of Experimental Data

Physics I · Principles of Laboratory Practice · https://tryals.app/learn/physics-i/statistical-analysis-of-experimental-data

## Mean and Spread

When you repeat a measurement $N$ times, **statistics** summarises the results.

The **mean** ($\bar{x}$) is your best estimate:

$$\bar{x} = \frac{1}{N}\sum_{i=1}^N x_i$$

The **standard deviation** ($s$) measures the spread of individual measurements:

$$s = \sqrt{\frac{1}{N-1}\sum_{i=1}^N (x_i - \bar{x})^2}$$

The **standard error of the mean** ($\sigma_{\bar{x}}$) measures the uncertainty in $\bar{x}$:

$$\sigma_{\bar{x}} = \frac{s}{\sqrt{N}}$$

| Quantity | What it measures |
| --- | --- |
| $s$ | Spread of the data |
| $\sigma_{\bar{x}}$ | Uncertainty in the mean |
| $N$ | More repeats $\to$ smaller $\sigma_{\bar{x}}$ |

## Distributions and Pitfalls

For random errors, a **Gaussian distribution** forms a bell curve where $\approx 68\%$ of values lie within $\bar{x} \pm s$ and $\approx 95\%$ within $\bar{x} \pm 2s$.

An **outlier** more than $3s$ from the mean is suspicious. Use **Chauvenet's criterion** or the **Grubbs test** to decide whether to reject it.

**Diminishing returns:** Increasing $N$ from 4 to 16 halves the standard error. But going from 100 to 400 only halves it again.

**Common pitfall:** The standard deviation describes single-reading scatter, while the standard error describes **mean** reliability. Quoting the wrong one misstates precision by $\sqrt{N}$.

## Practice questions

3 of this lesson's 11 practice questions, with answers. The full set is in the app.

### 1. Sort each concept into descriptive statistics or inferential statistics.

**Answer:**

- Descriptive: Calculating the sample mean, Computing the standard deviation, Making a histogram
- Inferential: Hypothesis testing (t-test), Constructing a confidence interval, Chi-squared goodness-of-fit test

**Why:** Descriptive statistics summarize data; inferential statistics draw conclusions about a population.

Page: https://tryals.app/practice/physics-i/statistical-analysis-of-experimental-data/sort-each-concept-into-descriptive-statistics-or-inferential

### 2. The standard deviation describes the spread of measurements; the standard error describes the uncertainty in their mean. Why is the second smaller?

A. Because the standard error ignores outliers
B. Because the mean is always closer to the true value than any single measurement
C. Because averaging cancels random fluctuations as $\sigma/\sqrt{n}$
D. Because the standard deviation includes systematic errors and the standard error does not

**Answer:** C. Because averaging cancels random fluctuations as $\sigma/\sqrt{n}$

**Why:** They answer different questions. Reporting $\sigma$ when $\sigma/\sqrt{n}$ is meant, or the reverse, is the commonest error in experimental write-ups.

Page: https://tryals.app/practice/physics-i/statistical-analysis-of-experimental-data/the-standard-deviation-describes-the-spread-of-measurements-the

### 3. Match each statistical term to its definition.

**Answer:**

- Mean → Sum of values divided by the number of values
- Median → Middle value when data is sorted
- Standard deviation → Root-mean-square deviation from the mean
- Variance → Average of the squared deviations from the mean

**Why:** These are the most common descriptive statistics in laboratory work.

Page: https://tryals.app/practice/physics-i/statistical-analysis-of-experimental-data/match-each-statistical-term-to-its-definition