Interpreting And Describing Frequency Tables And Column Charts

Theory

To interpret a frequency table or column chart, read each bar's height (or table entry), identify the mode (most common), and use percentages $\frac{frequency}{total} \times 100 %$ to compare. For 'at most' or 'at least' questions, add the appropriate frequencies. For comparing two groups of different sizes, convert to percentages within each group.

Most exam questions don't ask you to draw a column chart — they give you one and ask you to interpret it.

To read a column chart, find the top of each bar and read across to the vertical axis. Add up all bars for the total; the tallest bar is the mode; the shortest is the least common.

For ordered categorical or discrete numerical data, cumulative counts are common:

"At most $N$ " — add all frequencies up to and including $N$ .
"At least $N$ " — add all frequencies from $N$ onwards.
"Fewer than $N$ " — add all frequencies strictly below $N$ .

When comparing two groups with different totals, don't compare raw frequencies. Convert each to a percentage within its own group (its relative frequency) before comparing.

The first diagram is the cafe drinks chart with the mode (Coffee) highlighted and the total annotated. The second compares two groups by converting raw counts to percentages within each group.

Coffee is the mode (45). Total = 100. Read each bar's height off the vertical axis.

Year 7's

30 / 60

is

50 %

; Year 11's

10 / 60

is

17 %

. The percentages make the comparison fair.

The only formulas you need are the percentage and the cumulative count rules.

Percentage and relative frequency

$percentage = \frac{frequency}{total} \times 100 %$

% = \frac{frequency}{total} \times 100 %

Cumulative counts

Phrase	Add the frequencies of
"at most $N$ "	$0, 1, 2, \dots, N$ (includes $N$ )
"fewer than $N$ "	$0, 1, \dots, N - 1$ (excludes $N$ )
"at least $N$ "	$N, N + 1, N + 2, \dots$ (includes $N$ )
"more than $N$ "	$N + 1, N + 2, \dots$ (excludes $N$ )

Comparing groups. If the two groups have different totals, convert each frequency to a percentage within its own group first. Only then is a comparison meaningful.

Reading a column chart

Identify the axes — categories on the horizontal, frequency (or percentage) on the vertical.
Read each bar by tracing from its top across to the vertical axis.
Find the mode — the tallest bar. State the category, not the frequency.
Total = sum of all bar heights.

Answering "at most" / "at least" questions

Decide which categories or values qualify (be careful with "fewer than $N$ " vs "at most $N$ ").
Add the relevant frequencies from the table or chart.
If asked for a percentage, divide by the total and multiply by $100 %$ .

Comparing two groups

Compute each group's relative frequency: frequency / group total × $100 %$ .
Compare the percentages, not the raw counts.
Write a short comparison sentence (e.g. "Year 7 is more soccer-oriented at $50 %$ compared with $17 %$ for Year 11").

EXAMPLE 1 — READ OFF A CHART

From the cafe column chart above, how many coffees were sold, and what percentage of the total is that?

SOLUTION

The Coffee bar has height $45$ . The total is $100$ .

$%$	$=$	$\frac{45}{100} \times 100 %$
	$=$	$45 %$

Answer: $45$ coffees were sold, which is $45 %$ of the total.

coffee = 45 = 45 %

EXAMPLE 2 — COMPARE TWO GROUPS

Year 7:

30

of

60

students prefer Soccer. Year 11:

10

of

60

prefer Soccer. Compare the two groups.

SOLUTION

Both groups happen to have the same total here, but it's good practice to compute percentages anyway.

Year 7	$=$	$\frac{30}{60} \times 100 % = 50 %$
Year 11	$=$	$\frac{10}{60} \times 100 % \approx 16.7 %$

Answer: Year 7 is much more soccer-oriented ( $50 %$ ) than Year 11 ( $\approx 16.7 %$ ).

Year 7 = 50 %, Year 11 \approx 16.7 %

EXAMPLE 3 — "AT MOST" COUNT

Days off taken by

60

employees:

0 \to 10

,

1 \to 15

,

2 \to 20

,

3 \to 8

,

4 \to 5

,

5 \to 2

. How many took at most

2

days off?

SOLUTION

"At most $2$ " includes $0$ , $1$ , and $2$ days. Add those three frequencies.

At most 2	$=$	$10 + 15 + 20$
	$=$	$45 employees$

Answer: $45$ employees took at most $2$ days off.

at most 2 = 45

EXAMPLE 4 — COMBINED PERCENTAGE

A survey of

80

students: TikTok

32

, Instagram

24

. What percentage chose either TikTok or Instagram?

SOLUTION

Add the two frequencies, then divide by the total and multiply by $100 %$ .

$%$	$=$	$\frac{32 + 24}{80} \times 100 %$
	$=$	$\frac{56}{80} \times 100 %$
	$=$	$70 %$

Answer: $70 %$ of students chose TikTok or Instagram.

combined % = 70 %

Common pitfalls

The mode is the category, not the frequency. If Coffee has frequency

45

and is the most common, the mode is "Coffee" — not

45

.

"At most $N$ " includes $N$ . "At most

2

days" means

0

,

1

, and

2

— three values added, not two. "Fewer than

2

" would be just

0

and

1

. Read these phrases carefully.

Don't compare raw frequencies across groups of different sizes. Always convert to percentages within each group first.

30

out of

100

is very different from

30

out of

50

.

A frequency of $0$ is still data. "Nobody chose this category" is information. Don't drop it from the table.

Truncated axes mislead. If the y-axis doesn't start at

0

, small differences look much bigger than they are. Always note this when interpreting someone else's chart.

Frequently asked questions

How do I read a value off a column chart?

Find the top of the bar you're interested in and read across to the vertical axis. That's the frequency for that category. If the bar's top is between two tick marks, estimate to the nearest sensible value.

What does 'at most' mean for a frequency count?

At most N means N or fewer — include everything from 0 up to and including N. So 'at most 2 days' includes 0, 1, AND 2 days. Be careful: 'fewer than 2' is just 0 and 1.

What does 'at least' mean for a frequency count?

At least N means N or more — add all the frequencies from N upwards. 'At least 3 days' means 3 days plus 4 days plus 5 days plus anything higher.

How do I compare two groups with different totals?

Convert each group's frequencies to PERCENTAGES within its own group (called relative frequencies), then compare. Comparing raw frequencies is misleading because different totals make different numbers mean different things.

Is the mode a number or a category?

The mode is the CATEGORY (like 'Coffee'), not the frequency number (like 45). For a column chart, the mode is whatever the tallest bar is labelled with.

What if two categories have the same highest frequency?

The distribution is bimodal — there are two modes. State both. For example, if Coffee and Tea both have 30 sales and that's the highest count, the modes are Coffee and Tea.

Video Lessons

Practice Questions

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Practice Questions

Interpreting And Describing Frequency Tables And Column Charts

📖 Theory

Percentage and relative frequency

Cumulative counts

Reading a column chart

Answering "at most" / "at least" questions

Comparing two groups

Common pitfalls

Frequently asked questions

🎬 Video Lessons

✏️ Practice Questions

Theory

Video Lessons

Practice Questions