Chapter 9.1 — Introduction to Statistics

Where the Numbers Come From

Data collected first-hand by the investigator, for their own specific purpose, is primary data — a teacher personally recording each student's marks, for instance. Data pulled from an existing record someone else already compiled — registers, past reports, published tables — is secondary data. The distinction matters because primary data can be trusted to match exactly what the investigator intended to measure, while secondary data carries whatever assumptions and limitations the original collector built in, often without those being visible to whoever reuses it later. A school nurse weighing every student personally for a health survey is collecting primary data; a researcher later pulling those same weights from the school's stored records for an entirely different study is working with secondary data, even though the underlying numbers are identical — the label describes the relationship between the collector and the purpose, not any property of the numbers themselves. This distinction matters most at the moment a conclusion gets drawn from the data: a claim built on primary data can be traced back to exactly how and why each figure was recorded, while a claim built on secondary data inherits every gap in that original recording process, gaps that may never be visible to whoever reuses the numbers much later.

Raw Data, Before Any Organizing Happens

Data exactly as collected, before any sorting or grouping, is called raw data. The marks scored by 15 students out of 100 illustrate this directly:

Even before any formal formula gets applied, sorting alone already answers several real questions. The range — highest value minus lowest — is 95 − 46 = 49, a quick sense of how spread out the marks are. The middle value once sorted (the 8th of 15 scores) is 84, giving a rough center. And counting directly from the sorted list, 9 of the 15 students scored above 80. None of this needed a fresh formula; it only needed the raw list rearranged into order first. This is worth noticing precisely because it's easy to skip: sorting feels like a purely mechanical first step before the "real" statistics begins, but the range, the rough middle value, and the above-80 count are all genuine statistical summaries in their own right, obtained here with nothing more than reordering fifteen numbers by hand. Scale this same fifteen-value example up to fifty, five hundred, or five thousand entries, though, and manual sorting stops being practical almost immediately — which is exactly the gap the frequency distribution table introduced next is built to close.

From a List of Numbers to a Frequency Table

Sorting works for 15 values but becomes unwieldy for 50. A frequency distribution table instead counts how often each value occurs, using tally marks during counting and a final tally column. For 50 students' marks out of 10:

This is an ungrouped frequency distribution — every individual mark value gets its own row. The same 50 scores can instead be grouped into ranges:

This second version is a grouped frequency distribution — individual marks are collapsed into class intervals. Both tables describe the exact same 50 scores; grouping trades away some individual precision (a "4–6" row no longer distinguishes a 4 from a 6) in exchange for a table that's easier to scan and summarize at a glance. Which version is more useful depends entirely on the question being asked: if the exact number of students scoring exactly 7 matters, the ungrouped table is the only one that still has that information; if the question is simply "did most students score in the top third," the grouped table answers it in one glance instead of forcing a reader to add up several individual rows first. Neither version is more "correct" than the other — both are built from the exact same underlying counts — so choosing between them is really a decision about which question the table is meant to answer, made before the table is built rather than after.

Where Exactly Does a Boundary Value Belong?

Grouping raises one genuinely tricky question: which class does a value sitting right on a boundary belong to? Classes written as 30–39, 40–49, 50–59 are inclusive classes — non-overlapping, with no ambiguity about where an integer like 45 belongs. But classes are often written instead as continuous boundaries — 39.5–49.5, 49.5–59.5 — and now the boundary value 49.5 itself is genuinely ambiguous: does it belong at the top of the first class or the bottom of the second? By convention, a boundary value is always placed in the class where it appears as the lower limit, so 49.5 goes into 49.5–59.5, never into 39.5–49.5. Classes following this overlapping-boundary style, like 30–40, 40–50, 50–60, are called exclusive classes, and this single convention is what keeps every value in exactly one class with no double-counting anywhere in the table. The two styles are really answering slightly different practical needs: inclusive classes read naturally for data that's already whole numbers (ages in complete years, a count of siblings), while exclusive classes read naturally for continuous measurements that could genuinely fall anywhere along a scale (weights, heights, time durations) — and the choice of length for each class interval is itself guided by the data's range, computed the same way as the range shown for the raw marks above: divide the full spread of the data by however many classes feels manageable, then round to a convenient class length — this is exactly the calculation carried out in several of the worked problems in the next exercise, dividing an actual data range by a chosen class size before deciding how many rows the final table needs.

Where This Chapter Goes Next

Exercise 9.1 builds ungrouped and grouped frequency tables from a wide range of real data — blood groups, coin tosses, opinion polls, electricity bills — practicing exactly the class-interval and boundary decisions introduced here — a list of fifty test scores in the order they happened to be recorded tells a reader almost nothing at a glance, while the exact same fifty numbers, sorted and grouped correctly, reveal a class's overall performance in seconds; the numbers themselves never change, only how legibly they're arranged. Exercise 9.2 then moves from simply organizing data to summarizing it with a single representative number: the mean, median, and mode — the natural next step once the raw list has already been sorted and grouped into a table like the ones built above.