Data and probability — supplement

Foundations: what and whom are we observing?

The population is the set we want to learn about; a sample is the part we observe. If we want to study travel times of all students at a school, that is the population. The 40 students interviewed form a sample. Sample size matters, but so does selection: choosing only those arriving very early can bias collected times.

A variable is the characteristic recorded for each observed unit. Transport mode is qualitative; number of siblings is discrete quantitative; travel time is continuous quantitative even when rounded to minutes. Numerical category codes do not turn categories into quantities: averaging bus and walking codes does not produce a useful measure.

A random sample does not guarantee perfection, but helps avoid choices systematically favoring certain results. Randomly selecting people differs from randomly assigning a treatment: the first helps represent a population; the second helps compare causes in an experimental study. Survey nonresponse and different measurement methods can also distort conclusions.

Organize and represent a distribution

A value or class's absolute frequency is the number of observations belonging to it. Relative frequency divides that count by total n; percentage multiplies by 100. If 9 of 30 students use bicycles, relative frequency is 9/30 = 0,3 and percentage 30%. Absolute frequencies should sum to n and relative frequencies to 1, apart from small rounding effects.

For categories, a chart with separated bars shows counts or proportions. For a continuous variable grouped into intervals, a histogram uses adjoining bars. If class widths differ, each bar's area, not simply height, must represent frequency; an appropriate density is used. Always declare class boundaries so no observation belongs to no class or to two classes.

A graph needs a title, axes, units and readable scales. Truncating the vertical axis on a bar chart can exaggerate visual differences. Grouping data helps reading but loses information about values within each class. A mean calculated using class midpoints is generally an approximation, not the exact mean of the original data.

Mean, median and mode answer different questions

The arithmetic mean is the sum of values divided by observation count: x̄ = (x₁ + … + xₙ)/n. It can be interpreted as equal redistribution of the total. If each value xᵢ appears fᵢ times, the mean is the sum of products fᵢxᵢ divided by the sum of frequencies. Do not take the simple average of group means when group sizes differ.

The median requires ordered data. For odd n it is the middle value; for even n the average of the two middle values. A mode is a value with greatest frequency and may not be unique. In 2, 3, 3, 4, 18, the mean is 30/5 = 6, median 3 and mode 3. The extreme value 18 pulls the mean even though four of five values lie between 2 and 4.

No single measure is always best. The mean uses all magnitudes and is useful for totals; the median describes a central position less sensitive to the magnitude of extremes; the mode can describe the most frequent category. A conclusion should state which measure it uses and, where relevant, show distribution shape.

The same mean can hide variability

The lists 4, 5, 6 and 0, 5, 10 have mean 5, but the second is more spread out. Range is maximum minus minimum: 2 for the first and 10 for the second. This measure uses only two values. Interquartile range, Q₃ − Q₁, describes spread of the central half; the quartile calculation procedure must be declared because small samples have different conventions.

Standard deviation uses deviations from the mean. For a complete population of N values, variance is σ² = [Σ(xᵢ − μ)²]/N and standard deviation σ = √σ². The symbol Σ means summing the indicated terms and μ is population mean. Squares prevent positive and negative deviations cancelling. The root returns the data's original unit.

When using a sample of n > 1 values to estimate population variance, the usual formula is s² = [Σ(xᵢ − x̄)²]/(n − 1), and s = √s². The different denominator serves estimation; the formula used must be stated. Zero standard deviation means all values are equal, not that their mean is zero.

Outcomes, events and observed frequencies

Sample space Ω contains possible outcomes of a random experiment. Event A is a subset of Ω. Probability P(A) lies between 0 and 1, with P(Ω) = 1. If elementary outcomes are finite and equally likely, P(A) = number of favorable outcomes divided by number of possible outcomes. Equal likelihood is an assumption, not a consequence of simply listing outcomes.

For an ideal six-sided die, Ω = {1, 2, 3, 4, 5, 6}. The event “even” has three outcomes, hence probability 3/6 = 1/2. Complement Aᶜ contains outcomes outside A and has probability 1 − P(A). In union A ∪ B, either event occurring is sufficient; in intersection A ∩ B, both must occur.

The rule P(A ∪ B) = P(A) + P(B) − P(A ∩ B) subtracts the part counted twice. An observed frequency of 47 even results in 100 throws is 0,47, not an automatic change in theoretical probability 0,5. Over many independent repetitions under the same model, relative frequencies tend to approach probability; there is no requirement to immediately compensate for a sequence of results.

Conditional probability: who stays in the denominator

P(A | B) means probability of A given B occurred. The information restricts the universe to B. The denominator therefore becomes the probability of B, which must be positive:

If we randomly choose a student among 40, of whom 25 use public transport and 10 of those 25 arrive before 7, the probability of arrival before 7 given public-transport use is 10/25. Dividing by 40 would answer “uses public transport and arrives before 7”, not the conditional question.

The multiplication rule is P(A ∩ B) = P(B)P(A | B). A and B are independent when knowing one does not change the other's probability; equivalently P(A ∩ B) = P(A)P(B). They are mutually exclusive if they cannot occur together, meaning A ∩ B is empty. Two positive-probability mutually exclusive events are not independent: knowing one occurred makes the other impossible.

In draws without replacement, the set's composition changes and subsequent probabilities must be recalculated. With replacement and a new random selection under the same conditions, independence is possible. The assumption needs justification; “they happen at different times” is insufficient.

Correlation, causation and critical reading

A scatterplot places observation pairs (x, y) and helps reveal positive or negative association, absence of a linear pattern or a curved relationship. Linear correlation coefficient r, when defined, ranges from −1 to 1 and describes direction and strength of linear association. A value near zero does not rule out a nonlinear relationship.

Correlation does not demonstrate causation. In a hypothetical observation, students with more study hours may achieve higher results, but motivation, available support or prior knowledge can influence both variables. Reverse causation or participant selection may also occur. A fitted line describes association; it does not prove that changing x alone will produce the predicted change in y.

Before generalizing, ask who was observed, how data were obtained, which comparisons are fair and which factors were omitted. Do not extrapolate a trend to ages, times or quantities far beyond the data without an additional reason. A responsible statistical answer combines calculation, description of conditions and clear limits on what observations allow us to conclude.

Worked examples, without skipping steps

Follow the reasoning first. Then cover the solution and try to reconstruct it in your own words.

Now try it yourself

There is no timer. Write what you know, choose a relationship and justify each step. Hints and solutions are available whenever you need them.

Mistakes worth avoiding

A large sample eliminates every bias.

Systematically partial selection can retain bias even with many observations.

The mean describes the whole distribution.

Spread, asymmetry, extremes and context must be examined.

In a conditional probability, always divide by the initial total.

Divide by the set or probability of the known condition.

Independence means two events cannot occur together.

That is mutual exclusivity; independence means information about one does not change the other's probability.

Correlation proves one variable causes the other.

Observed association permits confounding, reverse causation and other explanations.