Introductory Statistics course

1. 1.Descriptive Statistics

   Descriptive statistics summarise a data set with a few numbers. Measures of centre say where the data cluster: the mean (the balance point) and the median (the middle value). Measures of spread say how far values typically sit from the centre: range, interquartile range and standard deviation.

   • Mean $\bar{x} = \frac{\sum x_i}{n}$.
   • Sample standard deviation: $s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n - 1}}$.
   • IQR $= Q_3 - Q_1$; outliers fall more than $1.5 \times$ IQR beyond the quartiles.
   Quick check: Find the mean of $4, 8, 15, 16, 23, 42$.

   $18$

   Includes a worked example, 3 common mistakes and 3 practice questions with answers. Study descriptive statistics

2. 2.The Normal Distribution

   The normal distribution is the symmetric bell-shaped curve that describes many measurements, from heights to test scores to measurement errors. It is completely determined by its mean $\mu$ and standard deviation $\sigma$. About $68\%$ of values fall within one $\sigma$ of the mean, $95\%$ within two and $99.7\%$ within three.

   • Standardise: $z = \frac{x - \mu}{\sigma}$; unstandardise: $x = \mu + z\sigma$.
   • The empirical rule: $68$-$95$-$99.7$.
   • Probabilities are areas under the curve; total area is 1.
   Quick check: Find $z$ for $x = 130$ when $\mu = 100$ and $\sigma = 15$.

   $2$

   Includes a worked example, 3 common mistakes and 3 practice questions with answers. Study the normal distribution

3. 3.Confidence Intervals

   A confidence interval estimates an unknown population value from a sample and reports how uncertain the estimate is. It has the form estimate $\pm$ margin of error, for example $\bar{x} \pm z^* \frac{\sigma}{\sqrt{n}}$ for a mean. A $95\%$ confidence level means the method captures the true value in $95\%$ of repeated samples.

   • Critical values: $z^* = 1.645$ (90%), $1.96$ (95%), $2.576$ (99%).
   • Use $t^*$ with $n - 1$ degrees of freedom when $\sigma$ is estimated by $s$.
   • Proportion: $\hat{p} \pm z^*\sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}$.
   Quick check: What is the standard error when $\sigma = 12$ and $n = 36$?

   $2$

   Includes a worked example, 3 common mistakes and 3 practice questions with answers. Study confidence intervals

These topics belong to the course outline. Full explainers for them have not been written yet, so they are summarised here rather than given pages of their own.

Correlation and Regression

Least-squares line and the correlation coefficient.

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