Descriptive Statistics and Hypothesis Testing

Basic theory and formulas for measures of location, spread and shape, and for tests on means, proportions, variances and categorical data.

Coming soon

Descriptive statistics summarises a sample with a few numbers and charts. Hypothesis testing decides, based on the same sample, whether the data agree with a claim about the population. This poster brings together the basic theory and formulas of both.

Measures of Location

Mean: $\overline{x} = \frac{1}{n} \sum x_{i}$
Median: the middle value of the ordered observations
Mode: the value with the highest frequency
Quartiles: $Q_{1}$ , $Q_{2}$ (the median) and $Q_{3}$

Measures of Spread

Range: $R = x_{(n)} - x_{(1)}$
Interquartile range: $I Q R = Q_{3} - Q_{1}$
Sample variance: $s^{2} = \frac{1}{n - 1} \sum {(x_{i} - \overline{x})}^{2}$
Standard deviation: $s = \sqrt{s^{2}}$
Coefficient of variation: $C V = s / \overline{x}$

Measures of Shape

Skewness: $g_{1} = m_{3} / m_{2}^{3 / 2}$ , positive when the tail is on the right
Kurtosis: $g_{2} = m_{4} / m_{2}^{2} - 3$ , positive when the tails are heavier than those of the normal

Charts

Histogram: shows the shape of the distribution
Box plot: a box from $Q_{1}$ to $Q_{3}$ , a line at the median, whiskers up to $1.5 \cdot I Q R$ , and outliers beyond them
Scatter plot: shows the relationship between two variables

Basic Concepts

Null and alternative hypothesis: $H_{0}$ and $H_{1}$ , with a two-sided or one-sided alternative
Test statistic: the quantity from the sample on which we base the decision
Rejection region: the values of the test statistic for which we reject $H_{0}$
Significance level: $α$ , usually 0.05 or 0.01
p-value: the probability, under $H_{0}$ , of getting a value as extreme as or more extreme than the observed one; we reject $H_{0}$ when $p < α$

Errors and Power

Type I error: $α = P (reject H_{0} | H_{0} true)$
Type II error: $β = P (do not reject H_{0} | H_{1} true)$
Power: $1 - β$ , increases with the sample size
Neyman–Pearson lemma: the likelihood ratio test is the most powerful for a given $α$

Mean and Proportion

z-test, known variance: $Z = \frac{\overline{x} - μ_{0}}{σ / \sqrt{n}}$
t-test, unknown variance: $T = \frac{\overline{x} - μ_{0}}{s / \sqrt{n}}$ , with $n - 1$ degrees of freedom
Two independent samples: t-test with pooled variance $s_{p}^{2}$
Paired observations: t-test on the differences $d_{i}$
Proportion: $Z = \frac{\hat{p} - p_{0}}{\sqrt{p_{0} (1 - p_{0}) / n}}$

Variance and Categorical Data

One variance: $χ^{2} = \frac{(n - 1) s^{2}}{σ_{0}^{2}}$
Two variances: $F = s_{1}^{2} / s_{2}^{2}$
Goodness of fit: $χ^{2} = \sum \frac{{(O_{i} - E_{i})}^{2}}{E_{i}}$
Independence in a contingency table: same formula, with $(r - 1) (c - 1)$ degrees of freedom