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

Definition Two symbols we lean on all page

Tally marks. A tally is a single vertical stroke | meaning "one observation". You draw them as you count, one per event. When you reach the fifth stroke you draw it across the previous four to make a bundle of five. In a strikethrough font that bundle looks like ~~||||~~ — the "strike" is the diagonal 5th stroke (on paper you'd draw it as a real slash 卌). So ~~||||~~ ~~||||~~ | reads as $5 + 5 + 1 = 11$ . We bundle in fives because the human eye can recognise "a group of five" instantly but cannot reliably count nine loose strokes.

The summation sign $\sum$ . The symbol $\sum f$ (say "sum of the $f$ 's") is shorthand for "add up all the frequencies". If the frequencies are $f_{1}, f_{2}, f_{3}$ then $\sum f = f_{1} + f_{2} + f_{3}$ . The stretched version $\sum_{k = 1}^{i} f_{k}$ means "add the $f$ 's starting from the first ( $k = 1$ ) up to the $i$ -th" — a running total that stops at row $i$ . Whenever you see $\sum$ , mentally replace it with "add them all".

The scenario matrix

Think of the table below as a matrix in the everyday sense — a grid that lays out every distinct case. The rows are the eight kinds of situation this topic can throw at you; the columns tell you why each is tricky and which worked example clears it. I label the rows Cell A through Cell H so that each example below can point back to exactly the cell it covers — that way you can see at a glance that no case is left uncovered.

Cell	What makes it tricky	Covered by
A. Basic build	Raw list → tally → frequency, everything positive and clean	Ex 1
B. Zero category	A category that got zero observations — how do you show "nothing"?	Ex 2
C. Total mismatch	Your frequencies don't add to the sample size — spot & fix the error	Ex 3
D. Relative freq / %	Turning counts into proportions; do they add to 100%?	Ex 4
E. Cumulative freq	Running totals; "how many up to X?" and the last row must equal $\sum f$	Ex 5
F. Primary vs secondary decision	A word problem: which collection method, and why	Ex 6
G. Degenerate / one-category	All observations fall in one bin — limiting case of the whole method	Ex 7
H. Exam twist (back-solving)	You're given the frequency table with a hole and must reconstruct a missing value	Ex 8

Recall Why enumerate cases at all?

Question ::: So that when a new problem appears you can ask "which cell is this?" instead of freezing. Every real question is one of these eight shapes.

Ex 1 — Basic build (Cell A)

Forecast: Which face do you bet appears most? Glance at the list — count the 4's in your head before reading on.

Step 1 — List the categories. The faces are $1, 2, 3, 4, 5, 6$ . These are our rows. Why this step? A tally chart needs its categories fixed before you count, so every observation has exactly one home. Miss a row and observations get lost.

Step 2 — Sweep the list once, one tally per value. Going left to right, drop one mark in the matching row. Every 5th mark in a row crosses the previous four — the ~~||||~~ bundle-of-five defined in the box above.

Face	Tally	Frequency
1	\|\|	2
2	\|\|\|\|	4
3	\|\|\|	3
4	~~\|\|\|\|~~ \|	6
5	\|\|	2
6	\|\|\|	3

Why this step? One pass, one direction — you never revisit a number, so you never double-count.

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

What the figure shows. The horizontal axis lists the six die faces; the vertical axis is the frequency (how many times each face came up). Each bar's height is the count from the table — face 1 reaches 2, face 4 reaches 6, and so on. The bars turn the tally column into shape: the tallest bar (highlighted yellow) instantly flags face 4 as the winner, which is exactly what "mode" means below.

Step 3 — Read frequencies off the tallies. $Frequency = (groups of 5) \times 5 + leftover marks$ . Face 4 has one group crossed plus one mark: $1 \times 5 + 1 = 6$ .

Verify: $2 + 4 + 3 + 6 + 2 + 3 = 20$ ✓ — equals the 20 rolls. The mode (tallest bar in the figure) is face 4 with frequency 6, matching the forecast.

Ex 2 — A category with zero (Cell B)

Forecast: How many XL did the shop sell? What do you write in that row — a blank, a dash, or a number?

Step 1 — Fix all four categories, even ones you might not use. Rows: S, M, L, XL. Why this step? The size XL exists as a possible answer, so it must have a row. Leaving it out hides the fact that XL sold nothing — which is real, useful information.

Step 2 — Tally the 8 sales.

Size	Tally	Frequency
S	\|	1
M	~~\|\|\|\|~~	5
L	\|\|	2
XL	(none)	0

Why this step? A zero is a legitimate frequency, not a mistake. You write the number 0, never a blank — a blank looks like "we forgot to check", a 0 says "we checked, it's genuinely none".

Verify: $1 + 5 + 2 + 0 = 8$ ✓ — the zero counts as $0$ in the sum, so the total still matches the 8 sales.

Ex 3 — Total mismatch: spot and fix (Cell C)

Worked example Statement

A student surveyed 30 people on their favourite season and wrote:

Season	Frequency
Spring	6
Summer	11
Autumn	4
Winter	8
The teacher says "one of these is wrong." Find and fix it, given that Autumn was recounted correctly as 4 and the true total is 30.

Forecast: Add the four numbers in your head. Do they hit 30?

Step 1 — Sum what's written. $6 + 11 + 4 + 8 = 29.$ Why this step? The first check on any frequency table is $\sum f = n$ . Here $29 \neq = 30$ , so there is exactly one missing observation somewhere.

Step 2 — Locate the trustworthy rows. Autumn is confirmed at 4. That leaves Spring, Summer, Winter as suspects, short by $30 - 29 = 1$ . Why this step? You can only fix an error you can isolate. Knowing Autumn is fixed narrows the hunt.

Step 3 — Recount reveals Winter was undercounted by 1. Suppose the recount of the raw tallies gives Winter $= 9$ , not 8.

Season	Frequency
Spring	6
Summer	11
Autumn	4
Winter	9

Why this step? We add the missing 1 to the row where the raw tallies actually showed it — never just "adjust the biggest number to make it fit". Fixing must come from the data.

Verify: $6 + 11 + 4 + 9 = 30$ ✓ — total now equals the sample size.

Ex 4 — Relative frequency and percentages (Cell D)

Forecast: The six percentages must add up to a special number. Which?

Step 1 — Recall the formula. $Relative freq = \frac{f _{i}}{\sum f}, Percentage = \frac{f _{i}}{\sum f} \times 100%.$ Why this step? Counts like "6 out of 20" are hard to compare across surveys of different sizes. Dividing by the total $\sum f = 20$ rescales every category onto the same $0 - 1$ ruler.

Step 2 — Divide each frequency by 20.

Face	$f$	$f /20$	Percentage
1	2	$0.10$	$10%$
2	4	$0.20$	$20%$
3	3	$0.15$	$15%$
4	6	$0.30$	$30%$
5	2	$0.10$	$10%$
6	3	$0.15$	$15%$

Why this step? Each row now answers "what share of all rolls landed here?" — comparable to any other experiment, even one with 200 rolls.

Verify: Proportions sum to $0.10 + 0.20 + 0.15 + 0.30 + 0.10 + 0.15 = 1.00$ and percentages to $100%$ ✓. This is the built-in sanity check: relative frequencies of a full dataset always total 1 (or $100%$ ).

Ex 5 — Cumulative frequency (Cell E)

Worked example Statement

Exam marks of 25 students, grouped:

Marks	Frequency
0–20	3
21–40	5
41–60	9
61–80	6
81–100	2
Build the cumulative frequency column, then answer: how many scored 60 or less?

Forecast: Guess the cumulative value at "41–60" before computing.

Step 1 — Cumulative = running total. $Cumulative_{i} = \sum_{k = 1}^{i} f_{k} = (previous cumulative) + f_{i} .$ Why this step? Plain frequency answers "how many in this band?" Cumulative frequency answers "how many up to and including this band?" — needed for questions like "how many scored 60 or less?".

Step 2 — Add downward.

Marks	$f$	Cumulative
0–20	3	$3$
21–40	5	$3 + 5 = 8$
41–60	9	$8 + 9 = 17$
61–80	6	$17 + 6 = 23$
81–100	2	$23 + 2 = 25$

Why this step? Each new cumulative is the last one plus the current row — no need to re-add from the top.

What the figure shows. The horizontal axis is the five marks-bands; the left vertical axis is a count. The faded blue bars are the plain frequency of each band (3, 5, 9, 6, 2). The green line with yellow dots is the cumulative frequency — notice it only ever climbs, because you keep adding. The red arrow marks the answer to our question: at the band ending in 60 the green line sits at 17, so 17 students scored 60 or less. The dashed white line shows the line finishing exactly at $n = 25$ , the total number of students.

Step 3 — Read the answer. "60 or less" is the band ending at 60, i.e. row 41–60, cumulative $= 17$ .

Verify: The final cumulative $= 25$ ✓ equals the total number of students — cumulative frequency must end at $n$ . Students scoring $\leq 60$ : 17.

Ex 6 — Primary vs secondary decision (Cell F)

Forecast: Which one could you realistically collect yourself?

Step 1 — Ask "can I collect this firsthand for my exact question?"

(a) Your own classmates, this week: yes — a short questionnaire reaches everyone.
(b) 50 years of nationwide literacy: no — impossible for one person and it's already measured by the census.

Why this step? The choice pivots on feasibility + fit. Primary data fits your exact question but costs effort; secondary data is convenient but pre-shaped by someone else's purpose.

Step 2 — Assign the method.

(a) → Primary (survey your class). You control the wording and get exactly this week's data.
(b) → Secondary (Census of India reports). Nationwide, decades-long, already gathered.

Why this step? Once feasibility tells you whether firsthand collection is even possible, you commit to the matching label so the reader knows how the data was obtained and how far to trust it. Calling (a) primary signals "I controlled quality myself"; calling (b) secondary signals "reuse an authoritative source and cite it" — the label is a promise about the data's origin, so assigning it correctly is the whole point of the exercise.

Verify: Sanity check each against the definition. (a) is collected by you, firsthand = primary ✓. (b) is collected by someone else, reused by you = secondary ✓. No units to check — this is a categorisation, and both categories are used correctly.

Ex 7 — Degenerate one-category case (Cell G)

Forecast: What is the relative frequency of "False"? Is that a problem?

Step 1 — Keep both categories.

Answer	Frequency	Relative freq
True	15	$15/15 = 1.00$
False	0	$0/15 = 0.00$

Why this step? This is the limiting case: all mass in one bin. The method still works — you don't drop "False" just because it's empty (same lesson as Ex 2). A relative frequency of $1.00$ means "every single observation".

Step 2 — Interpret the extremes. A category with relative frequency $1$ is certain in this sample; one with $0$ never happened. These are the boundary values proportions can take — they always live between $0$ and $1$ .

Verify: $1.00 + 0.00 = 1.00$ ✓. Frequencies $15 + 0 = 15$ ✓ equals class size. The mode is True (trivially, since it holds all the data).

Ex 8 — Exam twist: reconstruct a missing value (Cell H)

Worked example Statement

A frequency table of 40 survey responses has one entry smudged out (shown as $x$ ):

Category	Frequency
A	12
B	$x$
C	9
D	7
You are also told category B has relative frequency $0.30$ . Find $x$ two independent ways.

Forecast: Two methods should give the same $x$ . If they don't, something's wrong.

Step 1 — Method 1: use the total. The frequencies must sum to $n = 40$ : $12 + x + 9 + 7 = 40 \Rightarrow x = 40 - 28 = 12.$ Why this step? $\sum f = n$ is always true, so it's an equation you can solve for the unknown.

Step 2 — Method 2: use the relative frequency. $\frac{x}{40} = 0.30 \Rightarrow x = 0.30 \times 40 = 12.$ Why this step? Relative frequency $= f / \sum f$ ; rearranged, $f = (relative freq) \times n$ . A second, independent route to the same number.

Verify: Both methods give $x = 12$ ✓ — they agree, so the answer is trustworthy. Refill and check: $12 + 12 + 9 + 7 = 40$ ✓ and B's share $= 12/40 = 0.30$ ✓.

Coverage recap

Recall Did we hit every cell of the matrix?

A basic build ::: Ex 1 B zero category ::: Ex 2 C total mismatch / fix ::: Ex 3 D relative frequency and % ::: Ex 4 E cumulative frequency ::: Ex 5 F primary vs secondary decision ::: Ex 6 G degenerate one-category limit ::: Ex 7 H exam-style back-solving ::: Ex 8

Related deep material: Pictorial representation — bar charts and pictograms turns these tables into pictures; Measures of central tendency — mean median mode extracts single-number summaries; Grouped data and class intervals extends Ex 5's banding; Probability from frequency — experimental vs theoretical reads Ex 4's proportions as chances.