GCSE Statistics cheat sheet

Population = the whole set of individuals or items of interest; a census measures every member, while a sample measures only a part of it.

Simple random sampling gives every member of the population an equal chance of selection and needs a sampling frame listing the whole population.

Systematic sampling takes every kth item from an ordered list after a random start, where k = population size ÷ sample size.

Stratified sampling: the number taken from each stratum = (stratum size ÷ population size) × total sample size.

Quota sampling fills fixed target numbers in each subgroup chosen by the interviewer; because selection is not random, bias cannot be measured.

Cluster sampling divides the population into groups, then a random selection of whole clusters is surveyed.

Capture-recapture estimate of population size = (number in first sample × number in second sample) ÷ number of marked individuals recaptured.

Primary data is collected first-hand for the current purpose; secondary data was collected by someone else for a different purpose and may be out of date.

A leading question suggests a particular answer; open questions allow any response, closed questions offer fixed options only.

Frequency density = frequency ÷ class width; in a histogram with unequal class widths the area of each bar is proportional to its frequency.

Class mid-point = (lower class boundary + upper class boundary) ÷ 2, and is used to estimate the mean of grouped data.

Cumulative frequency at a value is the running total of all observations at or below that value; it is plotted against upper class boundaries.

On a cumulative frequency curve, the median is at n/2, the lower quartile at n/4 and the upper quartile at 3n/4.

A stem-and-leaf diagram keeps the original values while ordering them, and must be labelled with a key such as 3 | 4 = 34.

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

Two-way tables record counts for two categorical variables; row and column totals allow comparison of distributions between categories.

A choropleth map shades regions by a rate (for example per 1000 people) rather than a raw count, so regions of different sizes can be compared fairly.

Outliers are values unusually far from the rest of the data; they should be checked for recording error rather than deleted automatically.

Mean = Σx ÷ n; for grouped data, estimated mean = Σ(f × class mid-point) ÷ Σf.

Median = the middle value of the ordered data; when n is even it is the mean of the two middle values.

Mode = the most frequent value; modal class = the class with the highest frequency.

Range = largest value − smallest value; interquartile range = upper quartile − lower quartile.

Standard deviation measures the typical spread of values about the mean; variance = (standard deviation)².

Outlier test: a value is an outlier if it is below LQ − 1.5 × IQR or above UQ + 1.5 × IQR.

Adding a constant c to every value adds c to the mean and median but leaves the range, interquartile range and standard deviation unchanged.

Multiplying every value by k multiplies the mean, range, interquartile range and standard deviation by k, and the variance by k².

Combined mean of two groups = (n₁ × mean₁ + n₂ × mean₂) ÷ (n₁ + n₂).

Plot the explanatory (independent) variable on the x-axis and the response variable on the y-axis.

Positive correlation: as x increases y tends to increase; negative correlation: as x increases y tends to decrease.

Correlation strength runs from −1 (perfect negative) through 0 (no linear correlation) to +1 (perfect positive).

The line of best fit is drawn by eye through the mean point (x̄, ȳ) with roughly equal numbers of points above and below the line.

Interpolation estimates a value inside the range of the data and is reliable; extrapolation goes outside the range and is unreliable.

Spearman's rank correlation coefficient: r_s = 1 − 6Σd² ÷ (n(n² − 1)), where d is the difference between the ranks of each pair.

A correlation between two variables does not imply that one variable causes the other.

An unusual point on a scatter diagram may be a genuine extreme value or a recording error, so it is checked before removal.

A value of r close to 0 indicates little or no linear relationship, but a strong non-linear relationship may still exist.

A time series is a set of values recorded at successive, equally spaced points in time.

Trend is the underlying long-term movement; seasonal variation is a regular pattern that repeats within a fixed period such as a year.

A 4-point moving average is used for quarterly data because averaging over four quarters removes the seasonal effect.

With an even number of points per cycle the moving average is centred by averaging two consecutive moving averages so it lines up with a data point.

Seasonal component = actual value − trend value for the same period.

The mean seasonal component for a quarter is the average of all its seasonal components; the four means should sum to approximately 0.

Forecast value = trend value for the period + mean seasonal component for that period.

Moving averages smooth out short-term fluctuations so that the trend is easier to identify.

A 12-point moving average is used for monthly data because it removes an annual cycle.

Probability of an event = number of favourable outcomes ÷ total number of equally likely outcomes.

The probabilities of all mutually exclusive outcomes sum to 1, so P(not A) = 1 − P(A).

For mutually exclusive events, P(A or B) = P(A) + P(B).

For independent events, P(A and B) = P(A) × P(B).

For events that are not independent, P(A and B) = P(A) × P(B | A), where P(B | A) is a conditional probability.

On a tree diagram, multiply along the branches and add the results of different branches.

Experimental probability (relative frequency) = number of times the event occurred ÷ number of trials, and it approaches the theoretical value as trials increase.

Expected number of occurrences = number of trials × probability of the event.

Two-way tables and Venn diagrams give probabilities, including conditional probabilities taken from a restricted set of outcomes.