Statistical Analysis: Probability, Distributions and Data
Core concepts
Ext 1 statistics blends probability with data analysis. Know discrete random variables, probability distributions summing to 1, expected value $E(X)=\sum xP(X=x)$ and variance $\operatorname{Var}(X)=E(X^2)-[E(X)]^2$. Understand the binomial distribution conditions and formula $P(X=k)=\binom{n}{k}p^k(1-p)^{n-k}$, the normal distribution with standardisation $z=\frac{x-\mu}{\sigma}$, and descriptive statistics: mean, median, mode, range, interquartile range and standard deviation. Correlation describes association, not causation, and the correlation coefficient $r$ lies between $-1$ and $1$.
Traps
- Forgetting that probabilities in a distribution must total 1 to find an unknown constant.
- Confusing $E(X^2)$ with $[E(X)]^2$ in the variance formula.
- Using the binomial formula when trials are not independent or $p$ changes.
- "At least" binomial questions need $1-P(X=0)$ or a sum of terms.
- Reading $r$ near zero as proof of no relationship - it only measures linear association.
Rule of thumb
For normal problems, sketch the curve, standardise with $z=\frac{x-\mu}{\sigma}$, then use symmetry to convert any area into a standard left-tail form before reading tables.
Worked example
A biased coin has $P(H)=0.3$ and is tossed 5 times. Find $P(\text{at least one head})$. Using the complement, $P(\text{no heads})=0.7^5=0.16807$. So the answer is $1-0.16807=0.83193$. For exactly two heads: $\binom{5}{2}(0.3)^2(0.7)^3=10\times0.09\times0.343=0.3087$, consistent with the binomial model.