A full comparison always needs two separate statements, written in context, not just as bare numbers:

1. Average — which data set tends to have higher values overall (mean, median or mode).
2. Spread — which data set is more consistent (range or interquartile range).

Example: Class A: median $68$, IQR $14$. Class B: median $55$, IQR $22$. Class A scored higher on average ($68>55$), and Class A's results were more consistent, since its IQR is smaller ($14<22$).

• A small sample may not represent the whole population.
• A sample drawn from only part of a population (one class, not the whole school) may be biased.
• Extrapolating far beyond the range of the collected data is unreliable.
• A survey question that's leading, unclear, or only answerable by people with certain access (e.g. an online-only survey) can skew who responds.

No. Seeing two variables rise together only shows a relationship exists — it never proves that one causes the other. A third, unmentioned factor could be responsible for both.

Example: ice cream sales and drowning incidents both rise in summer months, and a scatter graph of one against the other would show a clear positive correlation. But buying ice cream doesn't cause drownings — a third factor, hot weather, independently increases both the number of people swimming and the number of people buying ice cream. Quoting a real-sounding but unrelated pair like this is a useful way to explain, in an exam answer, exactly why correlation on its own proves nothing about cause.

A single unusually high or low value can pull the mean a long way from where most of the data actually sits, while barely disturbing the median. When a data set is described as being affected by an outlier or being "skewed", that's usually a signal that the median (and interquartile range, rather than the full range) gives the more representative summary of the group.

A researcher surveys $10$ students from one class and concludes "60% of all students at this school prefer online lessons." Explain why this conclusion may not be reliable.

Solution

The sample is very small (only $10$ students), and it was drawn from a single class rather than a cross-section of the whole school, so it may not represent every year group or ability level. The result cannot reliably be generalised to "all students at this school" — a larger sample spanning multiple classes would be needed to support that claim.

A two-way table cross-references two categories at once (e.g. year group against favourite subject). Locate the correct row and column simultaneously to read off a specific joint value, and sum a whole row or column to find a category's overall total. Questions phrased "given that..." restrict you to a single row or column, exactly as with conditional probability from a Venn diagram.