Statistics and Probability: Dispersion and Events
Dispersion
Range=maximum−minimum. For data x₁,…,xₙ with mean x̄, variance σ²=Σ(xᵢ−x̄)²/n and standard deviation σ=√variance. Standard deviation is in the same units as the data; variance is in squared units. A small SD means the values cluster near the mean.
Probability
For equally likely outcomes, P(E)=number of favourable outcomes/total outcomes. Always 0≤P(E)≤1; P(not E)=1−P(E). For compound events, P(A∪B)=P(A)+P(B)−P(A∩B). If A and B are mutually exclusive, P(A∩B)=0. For independent events, P(A∩B)=P(A)P(B).
Worked example
For the data 2,4,6,8,10: mean=30/5=6. Deviations are −4,−2,0,2,4; squares are 16,4,0,4,16, total 40. Variance=40/5=8; SD=√8=2√2≈2.83. Range=10−2=8.
Traps
Dividing by n−1 when the school formula uses n; forgetting to square deviations before adding; taking the square root of the sum without dividing by n; and adding probabilities of overlapping events without subtracting the intersection. In probability, count outcomes carefully and do not count the same outcome twice.