Statistics: free practice, theory and problems

1. Descriptive statistics

What is it about?

Descriptive statistics summarise data with a few numbers: where is the centre, and how much do the data spread?

Concepts and formulas

• Mean: $\bar x = \dfrac{\sum x_i}{n}$.
• Median: the middle value when the data are sorted. Little affected by extreme values.
• Mode: the most frequent value.
• Range: largest minus smallest value.
• Sample standard deviation: $s = \sqrt{\dfrac{\sum (x_i - \bar x)^2}{n - 1}}$.
• For skewed distributions (for example income) the median is often more representative than the mean.

Example

Salaries 400, 450, 500, 520 and 2000 (thousand NOK): mean 774, median 500. One high salary pulls the mean up.

Practise descriptive statistics in the app →

2. Probability distributions

🔗 Shared curriculum with Mathematics S2. The questions are the same, but your progress counts separately for each subject.

What is it about?

A probability distribution tells you how likely each possible outcome is. With the expected value and the standard deviation you describe the "centre" and the "spread". The two most important distributions are the binomial distribution and the normal distribution.

Concepts and formulas

• Expected value: $E(X) = \sum x \cdot P(X = x)$. For a die: $E(X) = 3.5$.
• Variance $\text{Var}(X) = \sum (x - \mu)^2 P(X = x)$, standard deviation $\sigma = \sqrt{\text{Var}(X)}$.
• Binomial distribution ($n$ independent trials, probability $p$ of success): $P(X = k) = \binom{n}{k}p^k(1 - p)^{n - k}$, $E(X) = np$, $\sigma = \sqrt{np(1 - p)}$.
• The normal distribution $N(\mu, \sigma)$: bell-shaped. About 68 % lies within $\mu \pm \sigma$, 95 % within $\mu \pm 2\sigma$ and 99.7 % within $\mu \pm 3\sigma$.
• Standardisation: $Z = \dfrac{X - \mu}{\sigma}$, and $P(X < a) = \Phi\!\left(\dfrac{a - \mu}{\sigma}\right)$. Key values: $\Phi(1.645) = 0.95$, $\Phi(1.96) = 0.975$.
• The central limit theorem: sums and averages of many independent variables become approximately normally distributed.

Example

The height of 18-year-olds is $N(175, 7)$ cm. The share below 182 cm is $\Phi\!\left(\dfrac{182 - 175}{7}\right) = \Phi(1) \approx 0.841$.

Practise probability distributions in the app →

3. Confidence intervals

What is it about?

A sample gives an estimate, but how certain is it? A confidence interval gives a range that, with a given confidence (often 95 %), contains the true value.

Concepts and formulas

• 95 % confidence interval for a mean (known $\sigma$ or large sample): $\bar x \pm 1.96\cdot\dfrac{\sigma}{\sqrt n}$.
• For a proportion $\hat p$: $\hat p \pm 1.96\sqrt{\dfrac{\hat p(1 - \hat p)}{n}}$.
• The margin of error $1.96\,\sigma/\sqrt n$ halves when the sample becomes four times as large.
• 95 % means: if we repeat the survey many times, about 95 % of the intervals contain the true value.

Example

A poll of 1000 people gives 30 % support. Margin of error: $1.96\sqrt{0.3 \cdot 0.7/1000} = 0.028$, i.e. 30 % ± 2.8 percentage points.

Practise confidence intervals in the app →

4. Hypothesis testing

🔗 Shared curriculum with Mathematics S2. The questions are the same, but your progress counts separately for each subject.

What is it about?

A hypothesis test decides whether data give enough reason to reject a claim. Is a coin biased? Does a new medicine work? Has a new teaching method raised the average grade?

Concepts and formulas

• The null hypothesis $H_0$: "no change", for example $p = 0.5$.
• The alternative hypothesis $H_1$: what we want to show, for example $p > 0.5$ (one-sided) or $p \ne 0.5$ (two-sided).
• Significance level $\alpha$: usually 5 %. It is the probability of rejecting $H_0$ even though it is true (type I error).
• P-value: the probability of a result at least as extreme as the one observed, **given that $H_0$ is true**.
• Decision: if the p-value is less than $\alpha$, we reject $H_0$. Otherwise we do not have enough evidence (that does not mean $H_0$ is proved).
• Test statistic for a mean: $Z = \dfrac{\bar x - \mu_0}{\sigma/\sqrt{n}}$. For a proportion: $Z = \dfrac{X - np_0}{\sqrt{np_0(1 - p_0)}}$.
• One-sided test at the 5 % level: reject when $Z > 1.645$. Two-sided: when $|Z| > 1.96$.

Example

A coin gives 60 heads in 100 tosses. $H_0: p = 0.5$, $H_1: p > 0.5$. $Z = \dfrac{60 - 50}{\sqrt{100 \cdot 0.5 \cdot 0.5}} = \dfrac{10}{5} = 2$. The p-value is $1 - \Phi(2) \approx 0.023 < 0.05$, so we reject $H_0$: the coin appears to be biased.

Practise hypothesis testing in the app →