BGCSE Statistics 4040 cheat sheet

Primary data are collected first-hand by the investigator for the current purpose; secondary data were gathered earlier by someone else for another purpose.

A census measures every member of the population; a sample survey measures a subset and is cheaper but subject to sampling error.

Observation records behaviour without intervention; an experiment deliberately imposes a treatment to establish cause and effect.

A questionnaire item is biased if its wording leads respondents toward a particular answer; leading and double-barrelled questions must be avoided.

Open questions allow free responses and are hard to code; closed questions offer fixed options and are easy to tabulate.

Non-response bias arises when selected units fail to reply and differ systematically from those who do reply.

Pilot a questionnaire on a small group to test clarity, ordering, and coding before the main survey.

Data may be classified as qualitative (categorical) or quantitative (discrete counts, continuous measurements); the classification determines which summaries are valid.

For a histogram, class width is the horizontal axis and frequency density = frequency ÷ class width is the vertical axis; area equals frequency.

Bar charts use gaps between bars and are for categorical or discrete data; histograms have contiguous bars and are for continuous grouped data.

A cumulative frequency curve plots upper class boundary against cumulative frequency; median is at n/2, quartiles at n/4 and 3n/4.

An ogive's interquartile range is read as the horizontal distance between the quartile values.

Stem-and-leaf displays retain individual values while showing shape; the key must state the unit, e.g. 3|5 means 35.

Pie chart sector angle = (category frequency ÷ total frequency) × 360°.

A scatter diagram plots paired values to reveal direction, strength, and possible non-linearity before correlation is computed.

Comparative displays should use equal scales; truncated axes exaggerate differences and mislead the reader.

Mean = Σx/n; for grouped data use class midpoints, mean = Σfx/Σf.

Median is the middle value of ordered data (average of the two middle values when n is even); it is resistant to outliers.

Mode is the most frequent value or class; it is the only average usable for nominal categorical data.

Range = largest − smallest; it uses only two values and is highly sensitive to outliers.

Interquartile range = Q3 − Q1; semi-interquartile range = (Q3 − Q1)/2.

Population variance σ² = Σ(x − μ)²/N; sample variance s² = Σ(x − x̄)²/(n − 1) using the divisor n − 1 for unbiasedness.

Standard deviation is the square root of variance and is expressed in the same units as the data.

Adding a constant c to every value adds c to the mean but leaves the standard deviation unchanged; multiplying by k multiplies mean and standard deviation by k.

Coding: if y = (x − a)/b then mean(x) = a + b·mean(y) and sd(x) = |b|·sd(y).

For equally likely outcomes, P(A) = number of outcomes in A ÷ total number of outcomes.

Addition rule: P(A ∪ B) = P(A) + P(B) − P(A ∩ B); for mutually exclusive events the intersection term is zero.

Multiplication rule: P(A ∩ B) = P(A) × P(B|A); for independent events P(A ∩ B) = P(A) × P(B).

Complement rule: P(A′) = 1 − P(A).

Conditional probability: P(A|B) = P(A ∩ B)/P(B), provided P(B) > 0.

Two events are independent when P(A ∩ B) = P(A) × P(B), equivalently P(A|B) = P(A).

Mutually exclusive events cannot occur together; independent events can, so the two concepts are distinct.

In a tree diagram, multiply along a branch and add across branches to obtain total probability.

For sampling without replacement, the denominator and numerator both decrease after each draw.

Pearson product-moment correlation r = Σ(x − x̄)(y − ȳ) / √[Σ(x − x̄)²·Σ(y − ȳ)²]; it lies between −1 and +1.

r measures only the strength of linear association; r near 0 does not exclude a strong curved relationship.

Correlation does not imply causation; a lurking variable may drive both x and y.

The least-squares regression line of y on x is y = a + bx with b = Sxy/Sxx and a = ȳ − b·x̄.

The line passes through the point (x̄, ȳ); the two regression lines (y on x and x on y) differ unless r = ±1.

Predicted values are reliable only within the observed range of x; extrapolation beyond it is unreliable.

Spearman's rank correlation is used for ranked or non-linear monotonic data and applies the same −1 to +1 interpretation.

The coefficient of determination r² gives the proportion of variation in y explained by the regression on x.

A time series has four components: trend (long-term direction), seasonal (regular within-year pattern), cyclical (longer irregular swings), and random (residual).

Additive model: value = trend + seasonal + cyclical + random; multiplicative model: value = trend × seasonal × cyclical × random.

A moving average of period equal to the seasonal length smooths out seasonality and isolates the trend.

For an even period such as 4, a centred moving average is needed so the average aligns with an actual time point.

Seasonal variation = actual value − trend (additive) or actual value ÷ trend × 100 (multiplicative).

Simple price relative = (price in current year ÷ price in base year) × 100.

Laspeyres price index = (Σpₙq₀ ÷ Σp₀q₀) × 100, using base-year quantities; Paasche index uses current-year quantities.

Chain base index = (current value ÷ previous value) × 100; to rebase an index from base B to new base C, multiply by 100 ÷ (index of C on old base).

A simple random sample gives every unit an equal chance of selection and every combination an equal chance of being chosen.

Systematic sampling selects every kth unit after a random start; hidden periodicity in the list can bias it.

Stratified sampling divides the population into homogeneous strata and samples each; it reduces variance when strata differ.

Cluster sampling selects whole groups; it is cheaper but usually less precise than simple random sampling.

Quota sampling is non-random and cannot support calculation of sampling error.

The sampling distribution of the mean has mean μ and standard error σ/√n.

The central limit theorem states that for large n the distribution of the sample mean is approximately normal regardless of the population shape.

For a finite population of size N sampled without replacement, the standard error is (σ/√n)·√[(N − n)/(N − 1)].

Larger samples reduce the standard error in proportion to 1/√n, so quadrupling n halves it.

A confidence interval for a mean is x̄ ± z·(σ/√n) when σ is known, or x̄ ± t·(s/√n) when σ is estimated from a small sample.

A 95% confidence interval means that 95% of intervals built this way would contain the true parameter, not that the parameter has 95% probability of lying in this one.

A null hypothesis H₀ states no effect or no difference; the alternative H₁ states the researcher's claim.

A Type I error rejects a true H₀ (its probability is the significance level α); a Type II error fails to reject a false H₀.

A p-value is the probability of a result at least as extreme as observed, given H₀ is true; reject H₀ when p < α.

A one-tailed test places all α in one tail; a two-tailed test splits α between both tails.

Statistical significance does not guarantee practical importance; report effect size and confidence interval width alongside p-values.

Absence of evidence is not evidence of absence; failing to reject H₀ does not prove H₀ is true.

A simple random sample gives every unit an equal chance of selection and every combination an equal chance of being chosen.

Systematic sampling selects every kth unit after a random start; hidden periodicity in the list can bias it.

Stratified sampling divides the population into homogeneous strata and samples each; it reduces variance when strata differ.

Cluster sampling selects whole groups; it is cheaper but usually less precise than simple random sampling.

Quota sampling is non-random and cannot support calculation of sampling error.

The sampling distribution of the mean has mean μ and standard error σ/√n.

The central limit theorem states that for large n the distribution of the sample mean is approximately normal regardless of the population shape.

For a finite population of size N sampled without replacement, the standard error is (σ/√n)·√[(N − n)/(N − 1)].

Larger samples reduce the standard error in proportion to 1/√n, so quadrupling n halves it.