Russia 8 класс Probability and Statistics

Chapters: 5

1. Data

Data presentation

Statistics: Asking Questions, Averages and Stacked Bar Graphs – Statistics starts with a question that has many possible answers. We collect data, organise it, show it in a graph, and find one number that sums it up. The mean is the fair share, the median is the middle value, and the mode is the most common value. A weighted average gives more importance to some values. Stacked and 100% stacked bar graphs show how a whole is split into parts.

Sets and operations

Sets: Representation, Types, Subsets, Venn Diagrams and Operations – A set is a well-defined collection of different objects. We write it in roster form {2, 4, 6} or set-builder form {x : x is even}. Sets can be empty, finite, infinite or equal. If every element of B is in A, B is a subset of A (B ⊂ A); a set with n elements has 2ⁿ subsets. Intervals like (a, b) and [a, b] are subsets of real numbers. The universal set U holds everything under study. With Venn diagrams we see union A ∪ B, intersection A ∩ B, difference A − B and complement A′ = U − A.

Variance and standard deviation

Measures of Dispersion: Range, Mean Deviation, Variance and SD – Dispersion means spread: how far the values sit from the centre. Range = largest − smallest. Mean deviation = average distance from the mean (or median). Variance = average of squared distances from the mean. Standard deviation = √variance. The same ideas work for grouped data when every term is multiplied by its frequency.

Elementary events · Operations on events · Conditional probability

Introduction to Probability: Scale, Experiments, Sample Spaces and Trees – Probability is a number from 0 to 1 that tells how likely something is. 0 means it can never happen, 1 means it will surely happen. We can find it by doing an experiment many times (empirical probability), or by listing every possible result (the sample space) and counting the ones we want. Tree diagrams and tables help us list results when two things happen together.
Combined Events, Tree Diagrams and Conditional Probability – When two things happen, list every outcome in a sample space, a table, a Venn diagram or a tree diagram. Multiply along tree branches (the multiplication rule) and add the paths you want. If the first event changes the second, the events are dependent: picking without replacement is the classic case. Conditional probability P(A | B) is the chance of A when we already know B happened: P(A | B) = P(A and B) ÷ P(B).

Trees

Graph Theory: Dots, Lines and Networks – A graph is a set of vertices (dots) joined by edges (lines). The degree of a vertex is how many edges touch it, and the sum of all degrees is twice the number of edges. An Euler trail uses every edge once and exists only when 0 or 2 vertices have odd degree. A tree is a connected graph with no cycles and n − 1 edges. Weighted graphs model roads and networks; Kruskal’s and Prim’s algorithms find a minimum spanning tree.