Data collection — primary vs secondary, tally charts, frequency tables: Foundations

This page builds every piece of notation used in the parent topic from absolute zero. If the parent note wrote a symbol and assumed you knew it, we stop and build it here first — the picture, the plain meaning, and why the topic needs it.

0. What is "data"? (before any symbol)

Picture data as a row of little boxes, each holding one answer. Nothing is organized yet — it is a pile.

Figure — Data collection — primary vs secondary, tally charts, frequency tables

1. A "category" (the box you sort into)

The picture: imagine three labelled buckets on the floor. Every answer you hear gets dropped into exactly one bucket.

Why the topic needs it: you cannot count "how many" until you have decided how many of what. Categories are the what. Without them there is nothing to tally.

Two rules a category must obey:

Every fact fits somewhere (the boxes cover all possibilities).
No fact fits two boxes at once (the boxes don't overlap).

2. The tally mark `|` (counting one thing)

The picture: each time an answer drops into a bucket, you scratch one line on that bucket's label. Five answers → five scratches.

Why this symbol and not a number? When data arrives live (people answering one at a time), you don't yet know the final count. You cannot write "8" before the 8th person speaks. A tally lets you add proof of one event instantly, without erasing and rewriting a number every time.

The crossing rule (why the 5th mark is diagonal)

Look at the figure. On the left, ten plain strokes — your eye has to count them one by one, and it will slip. On the right, the same ten drawn in groups of five, where every 5th stroke crosses the previous four ~~||||~~.

3. Frequency and the letter $f$ (counting the whole box)

So " $f = 11$ for Apple" is read: "the frequency of Apple is eleven" — eleven people said apple.

The picture: the frequency is simply the height of the pile in one bucket. Tall pile = big $f$ .

Why a letter and not just the word? Because we will soon have several frequencies at once, and writing "the count of the first category" every time is exhausting. A letter lets us talk about them all with one symbol.

The subscript $f_{i}$ (which box am I talking about?)

$f_{1}$ = frequency of the 1st category
$f_{2}$ = frequency of the 2nd category
$f_{i}$ = frequency of the $i$ -th category, where $i$ is a stand-in for "whichever one you point at"

Why the topic needs it: with three fruits we have three numbers. Calling them $f_{1}, f_{2}, f_{3}$ lets us write one rule that works for all of them instead of three separate sentences.

4. The sum symbol $\sum$ (add up all the boxes)

The picture: empty every bucket onto one big scale and read the total weight. That total must equal the number of people you surveyed.

Why this symbol? Writing $f_{1} + f_{2} + f_{3} + \dots + f_{100}$ is impossible for 100 boxes. Sigma packs "add them all" into one mark.

Reading the full form

$\sum_{k = 1}^{i} f_{k}$

This looks scary; it is just a recipe with three parts:

Part	Plain meaning
$k = 1$ (below)	start the counter $k$ at box 1
$i$ (above)	stop when $k$ reaches box $i$
$f_{k}$ (right)	the thing you add each time — box $k$ 's frequency

So $\sum_{k = 1}^{3} f_{k} = f_{1} + f_{2} + f_{3}$ : start at 1, stop at 3, add each frequency. That is exactly the parent note's cumulative frequency — a running total from the first box up to box $i$ .

5. The fraction bar and $\times 100%$ (turning counts into shares)

The picture: cut a pizza. If Apple is 11 of 23 slices, Apple's slice is $\frac{11}{23}$ of the whole pie.

Why the fraction bar? The bar means "out of". $\frac{11}{23}$ is literally "11 out of 23". Dividing rescales every count onto the same 0-to-1 ruler, so datasets of different sizes become comparable.

6. The $\approx$ sign (honest rounding)

Why not just $=$ ? Using $=$ would be a small lie — the values aren't exactly equal. $\approx$ tells the reader "I trimmed the tail on purpose."

How these foundations feed the topic

Read top to bottom: you cannot reach frequency without first having categories and tallies; you cannot reach relative frequency without both $f$ and $\sum f$ . Each symbol is earned by the one above it.

Equipment checklist

Cover the right side and test yourself. If any answer surprises you, reread that section.

What does a single tally mark | represent?

Exactly one occurrence — one fact landing in a box.

Why is every 5th tally drawn as a diagonal crossing?

Because the eye recognises a group of 5 as one shape, so counting becomes "count fences × 5 + leftovers" instead of one-by-one.

What does the letter

f

stand for?

Frequency — the total count of items in one category.

f_{2}

, what is the small

2

A subscript: an address labelling which category, not a multiplication.

What instruction does

\sum

(sigma) give?

Add up everything that follows.

Read

\sum_{k = 1}^{3} f_{k}

in words.

Start counter

k

at 1, add each frequency

f_{k}

, stop at box 3 — i.e.

f_{1} + f_{2} + f_{3}

Why must

\sum f

equal your sample size?

Every person was counted once into exactly one box, so the totals of the boxes must rebuild the whole group — it is the error check.

What does the fraction

\frac{f _{i}}{\sum f}

tell you?

The share of the whole that one category takes — its relative frequency.

Why multiply relative frequency by

100%

To rescale a 0-to-1 fraction onto the familiar 0-to-100 percent ruler for easy comparison.

What does

\approx

mean and why use it?

"Almost equal after rounding" — used honestly when the exact decimal runs on forever.

Next, see the parent Data collection topic, then build outward to Pictorial representation — bar charts and pictograms, Measures of central tendency — mean median mode, Grouped data and class intervals, Probability from frequency — experimental vs theoretical, and Sampling methods.