Tally and Frequency Tables

Count raw data with tally marks, organise them in a frequency table, and read the table: totals, the most common value, percentages, ratios, missing frequencies, and test marks and other data grouped into given classes.

Before you start

Raw data are data as they were first collected, before they have been organised (Types of Data). Data can be categorical, such as colours or CCAs, or numerical, such as the number of siblings or a test mark.
In Types of Data you organised raw data by counting each value and setting the counts out in a table. This lesson names that table a frequency table and gives a surer way to count: tally marks.
A percentage of a whole is $\frac{\text{part}}{\text{whole}} \times 100\%$ . For example, 9 out of 36 is $\frac{9}{36} \times 100\% = 25\%$ .
A ratio is written in its simplest form by dividing both terms by their highest common factor (HCF): $12 : 8 = 3 : 2$ .

Tally Marks

When we count data as they are collected, or go through a long list of raw data, it is easy to lose count. Tally marks keep the count as we go: one stroke for each item.

The strokes are grouped in fives. The fifth stroke is drawn across the four before it, making a bundle of five that can be counted at a glance.

Tally marks

Draw one upright stroke for each item.

For the fifth item, draw a stroke across the four upright strokes. Then start a new bundle.

To read tally marks, count the bundles in fives and add the strokes left over.

Tally marks for some numbers
Number	Tally marks
1
2
3
4
5
6
8
10
13

Example 1: Counting tally marks

Mr Tan used tally marks to record the number of Secondary 1 students absent from school on Monday, Tuesday and Wednesday. How many students were absent on each day, and how many absences were there on the three days altogether?

Students absent
Day	Tally
Monday
Tuesday
Wednesday

Solution

Monday: 2 bundles of five and 4 more strokes, so $2 \times 5 + 4 = 14$ .

Tuesday: 1 bundle of five and 2 more strokes, so $5 + 2 = 7$ .

Wednesday: 2 bundles of five and 1 more stroke, so $2 \times 5 + 1 = 11$ .

Altogether: $14 + 7 + 11 = 32$ .

14 students were absent on Monday, 7 on Tuesday and 11 on Wednesday: 32 absences altogether.

Frequency Tables

The frequency of a value or a category is the number of times it occurs in the data. A frequency table lists each value or category with its frequency, so that a long list of raw data can be read at a glance.

The first column lists the values or categories, and is headed with what was recorded (for example, "Number of siblings" or "Colour"). When the table is made by tallying, a tally column comes next. The frequency column is last, with the total at the foot.

Making a tally and frequency table

Step 1: List every value or category in the first column. Numerical values go in order from the smallest to the largest, with none in between left out, even one whose frequency is 0. Categories that have an order (sizes S, M, L, XL) keep that order.

Step 2: Go through the raw data once, in order. For each item, draw a stroke in its row and tick the item off the list, so that no item is missed or counted twice.

Step 3: Count the tally marks in each row and write the frequency.

Step 4: Add the frequencies. The total must equal the number of items in the raw data.

Example 2: A frequency table for numerical data

20 students were asked how many siblings they have. Their answers were:

2, 1, 0, 3, 1, 2, 1, 0, 2, 3, 1, 2, 0, 1, 4, 2, 1, 3, 1, 2

Make a tally and frequency table for these data. Which number of siblings is the most common?

Solution

The answers go from 0 to 4, so the first column lists 0, 1, 2, 3 and 4.

Go through the list in order. The first answer is 2, so draw a stroke in the row for 2. The second answer is 1, so draw a stroke in the row for 1. Carry on to the end of the list, ticking off each answer.

Number of siblings	Tally	Frequency
0		3
1		7
2		6
3		3
4		1
Total		20

Check: $3 + 7 + 6 + 3 + 1 = 20$ , the number of students asked.

The greatest frequency is 7, in the row for 1 sibling.

1 sibling is the most common answer, given by 7 of the 20 students.

What the total can and cannot check

A total that is not 20 shows that an answer was missed or counted twice. A total of 20 does not prove the table right: if one of the answers of 3 were tallied in the row for 1 by mistake, the row for 1 would show 8 and the row for 3 would show 2, and the total would still be 20. So count each row again from the raw data as well.

Example 3: A frequency table for categorical data

15 students were asked their favourite colour. Their answers were:

Blue, Red, Blue, Green, Blue, Red, Blue, Yellow, Green, Red, Blue, Blue, Red, Green, Blue

Make a tally and frequency table for these data.

Solution

The categories are Blue, Red, Green and Yellow. These categories (colours) have no natural order, so they may be listed in the order in which they first appear. Categories that do have an order, such as T-shirt sizes S, M, L and XL, are listed in that order.

Go through the list once, drawing a stroke in the right row for each answer.

Colour	Tally	Frequency
Blue		7
Red		4
Green		3
Yellow		1
Total		15

Check: $7 + 4 + 3 + 1 = 15$ , the number of students asked. Counting each row again from the list: Blue is the 1st, 3rd, 5th, 7th, 11th, 12th and 15th answer (7), and Green the 4th, 9th and 14th (3).

7 students chose blue, 4 chose red, 3 chose green and 1 chose yellow: 15 students in all.

Reading a Frequency Table

Once the data are in a frequency table, many questions can be answered from the table without going back to the raw data. Read each question carefully for the words that say which rows to use.

Reading a frequency table

The total frequency is the sum of the frequencies. It is the number of people or things in the data.

The most common value or category is the one with the greatest frequency.

For whole-number values: at least 2 means 2 or more; at most 2 means 2 or fewer; more than 2 means 3 or more; fewer than 2 means 1 or fewer (2 itself is not included).

Fraction of the whole $= \frac{\text{frequency}}{\text{total frequency}}$ , and percentage $= \frac{\text{frequency}}{\text{total frequency}} \times 100\%$ .

A ratio compares two frequencies, and is written in its simplest form.

Example 4: Reading a frequency table of numbers

The table shows the number of books each student in a class read last month.

Number of books read	Frequency
0	3
1	5
2	8
3	4

(a) How many students are in the class?

(b) What is the most common number of books read?

(d) How many students read at least 2 books?

(e) What fraction of the students read fewer than 2 books?

(f) How many books did the class read altogether?

Solution

(a) Total frequency $= 3 + 5 + 8 + 4 = 20$ .

(b) The greatest frequency is 8, in the row for 2 books.

(d) At least 2 means 2 or 3 books: $8 + 4 = 12$ .

(e) Fewer than 2 means 0 or 1 book: $3 + 5 = 8$ , so the fraction is $\frac{8}{20} = \frac{2}{5}$ .

(f) The 5 students who read 1 book read 5 books between them, the 8 who read 2 books read $2 \times 8 = 16$ , and so on. Multiply each number of books by its frequency and add:

$0 \times 3 + 1 \times 5 + 2 \times 8 + 3 \times 4 = 0 + 5 + 16 + 12 = 33$

(a) 20 students (b) 2 books (c) 20% (d) 12 students (e) $\frac{2}{5}$ (f) 33 books

Example 5: Reading a frequency table of categories

Some students were asked their favourite subject. The table shows their answers.

Subject	Frequency
Mathematics	12
Science	15
English	8
History	5

(a) How many students were asked?

(b) Which subject is the most popular?

(d) Find the ratio of the number who chose Science to the number who chose History.

(e) What percentage of the students did not choose Science?

Solution

(a) $12 + 15 + 8 + 5 = 40$

(b) The greatest frequency is 15, for Science.

(d) Science : History $= 15 : 5 = 3 : 1$ (divide both terms by 5)

(e) The number who did not choose Science is $40 - 15 = 25$ , so the percentage is $\frac{25}{40} \times 100\% = 62.5\%$ .

(a) 40 students (b) Science (c) 30% (d) 3 : 1 (e) 62.5%

Finding a Missing Frequency

Sometimes a frequency is not given but the total is. The frequencies add up to the total, so the missing frequency is the total minus the sum of the others.

When two frequencies are unknown but linked (one is twice the other, say), write both in one unknown and solve an equation.

A missing frequency

Missing frequency $=$ total frequency $-$ sum of the other frequencies.

Check by adding all the frequencies, the one found included: the sum must be the total.

Example 6: A missing frequency from the total

Mei Ling asked the 40 students in her class how many cups of bubble tea they bought last week. The table shows her results.

Number of cups	Frequency
0	9
1	12
2	$x$
3	6
4	3
Total	40

(a) Find the value of $x$ .

(b) What percentage of the students bought at least 2 cups?

Solution

(a) The other frequencies add up to $9 + 12 + 6 + 3 = 30$ .

$x = 40 - 30 = 10$

Check: $9 + 12 + 10 + 6 + 3 = 40$ .

(b) At least 2 cups means 2, 3 or 4 cups: $10 + 6 + 3 = 19$ students.

$\frac{19}{40} \times 100\% = 47.5\%$

(a) $x = 10$ (b) 47.5% of the students bought at least 2 cups.

Example 7: Two linked frequencies

Ravi recorded the number of goals his school team scored in each of its 30 football matches this year. The number of matches in which the team scored 2 goals is twice the number in which it scored 1 goal.

Number of goals	Number of matches
0	4
1	$x$
2	$2x$
3	5
4	3
Total	30

(a) Find the value of $x$ .

(b) In how many matches did the team score more than 1 goal?

Solution

(a) The frequencies add up to 30:

$4 + x + 2x + 5 + 3 = 30$

$3x + 12 = 30$ (collect the terms in $x$ and the numbers)

$3x = 18$ (subtract 12 from both sides)

$x = 6$ (divide both sides by 3)

So the team scored 1 goal in 6 matches and 2 goals in $2 \times 6 = 12$ matches.

Check: $4 + 6 + 12 + 5 + 3 = 30$ .

(b) More than 1 goal means 2, 3 or 4 goals: $12 + 5 + 3 = 20$ .

(a) $x = 6$ (b) The team scored more than 1 goal in 20 matches.

Grouping Data into Given Classes

Test marks out of 100 can take many different values. A frequency table with a row for every mark would be very long, and most of its frequencies would be 0 or 1. Instead, the marks are put into equal groups such as 40–49, 50–59, 60–69, and so on. Each group is called a class (a different meaning from a class of students).

The classes are given in the question. Each mark belongs to exactly one class: 49 goes in 40–49, and 50 goes in 50–59. Tally each mark into its class just as before.

A table of grouped data

Tally each value into the one class that contains it, then count and add the frequencies as before.

The table tells us how many values are in each class, but not what they are: from it we can say how many students scored 70 to 79, but not how many scored more than 75.

Example 8: Tallying marks into given classes

The marks of 30 students in a Mathematics test, out of 100, were:

67, 45, 72, 83, 60, 58, 79, 91, 64, 50, 70, 69, 53, 88, 62, 75, 49, 65, 80, 77, 59, 68, 42, 74, 96, 61, 86, 56, 71, 66

(a) Using the classes 40–49, 50–59, 60–69, 70–79, 80–89 and 90–99, make a tally and frequency table.

(b) Which class has the greatest frequency?

(d) The pass mark is 50. What percentage of the students passed?

(e) Can you tell from your table how many students scored more than 75? Explain.

Solution

(a) Go through the marks in order: 67 goes in 60–69, 45 in 40–49, 72 in 70–79, and so on. The marks 50 and 59 both go in 50–59; 60 and 69 both go in 60–69.

Mark	Tally	Frequency
40–49		3
50–59		5
60–69		9
70–79		7
80–89		4
90–99		2
Total		30

Check: $3 + 5 + 9 + 7 + 4 + 2 = 30$ , the number of students.

(b) The greatest frequency is 9, for the class 60–69.

(d) The students who failed scored 40–49: 3 students. So $30 - 3 = 27$ passed, and $\frac{27}{30} \times 100\% = 90\%$ .

(e) No. The table shows 7 students in 70–79, but not their marks, so it cannot say how many of them scored more than 75. (The raw data show 8 students with more than 75: 79 and 77, four in 80–89 and two in 90–99.)

(b) 60–69 (c) 13 students (d) 90% (e) No: the table does not show the marks within the class 70–79.

Common Mistakes

Drawing the fifth tally stroke upright, or putting six strokes in a bundle. A bundle is four upright strokes with the fifth drawn across them.
Leaving out a value whose frequency is 0, or listing numerical values out of order.
Trusting the total alone. Two wrong counts can cancel out and still give the right total; count each row again from the raw data.
Adding the values (0, 1, 2, 3) instead of the frequencies to find how many people were asked, or adding the frequencies when the question asks for a total of the values, such as the number of books the class read (Example 4) or the number of people living in the block (Practice 6). The total of the values is each value times its frequency, added.
Reading "fewer than 2" as including 2, or "at least 2" as leaving 2 out.
Putting a mark on the edge of a class in the wrong class: 50 belongs to 50–59, not to 40–49.

Practice

Try each question before you open its solution.

Practice 1: Counting tally marks

Some students were asked their favourite sport. The tally marks show their answers. How many students chose each sport, and how many students were asked?

Sport	Tally
Football
Basketball
Swimming

ShowHide solution

Football: 3 bundles of five and 2 strokes, $3 \times 5 + 2 = 17$ .

Basketball: 2 bundles of five and 1 stroke, $2 \times 5 + 1 = 11$ .

Swimming: 1 bundle of five and 3 strokes, $5 + 3 = 8$ .

Students asked: $17 + 11 + 8 = 36$ .

Football 17, basketball 11 and swimming 8: 36 students were asked.

Practice 2: Drinks at the canteen

24 students were asked which drink they like best at the school canteen drinks stall: Milo (M), teh (T), kopi (K) or bandung (B). Their answers were:

M, T, K, M, B, M, T, M, K, T, M, B, K, M, T, M, B, T, K, M, M, B, T, K

(a) Make a tally and frequency table.

(b) Which drink is the most popular?

ShowHide solution

(a) Go through the list once, drawing a stroke in the right row for each answer.

Drink	Tally	Frequency
Milo		9
Teh		6
Kopi		5
Bandung		4
Total		24

Check: $9 + 6 + 5 + 4 = 24$ , the number of students.

(b) The greatest frequency is 9, for Milo.

(b) Milo (c) $\frac{1}{6}$ of the students chose bandung.

Practice 3: T-shirt sizes

Ravi recorded the T-shirt size each of 20 students in his CCA needs for Sports Day:

M, L, S, M, XL, M, L, M, S, L, M, XL, L, M, S, L, M, XL, M, L

(a) Make a tally and frequency table.

(b) Which size is the most common?

ShowHide solution

(a) T-shirt sizes are categories with an order, so the rows are listed in that order: S, M, L, XL.

Size	Tally	Frequency
S		3
M		8
L		6
XL		3
Total		20

Check: $3 + 8 + 6 + 3 = 20$ , the number of students.

(b) The greatest frequency is 8, for size M.

(b) M (c) 45% of the students need size L or larger.

Practice 4: Library loans

Siti recorded the number of library books each of 24 students borrowed last month:

2, 1, 0, 3, 2, 5, 1, 2, 3, 0, 1, 2, 5, 3, 2, 1, 0, 2, 3, 1, 5, 2, 3, 1

(a) Make a tally and frequency table for the numbers of books 0 to 5.

(b) How many students borrowed at least 3 books?

ShowHide solution

(a) No student borrowed 4 books, but 4 lies between 0 and 5, so its row is kept with a frequency of 0.

Number of books	Tally	Frequency
0		3
1		6
2		7
3		5
4		0
5		3
Total		24

Check: $3 + 6 + 7 + 5 + 0 + 3 = 24$ .

(b) At least 3 means 3, 4 or 5 books: $5 + 0 + 3 = 8$ .

(b) 8 students (c) 12.5% of the students borrowed no books.

Practice 5: Bubble tea flavours

A bubble tea stall recorded the number of cups of each flavour it sold one morning.

Flavour	Number of cups
Brown sugar	45
Taro	30
Matcha	25
Original	50

(a) How many cups were sold that morning?

(b) What percentage of the cups sold were taro?

(d) What fraction of the cups sold were not original?

ShowHide solution

(a) $45 + 30 + 25 + 50 = 150$

(b) $\frac{30}{150} \times 100\% = 20\%$

(d) Not original: $150 - 50 = 100$ cups, so the fraction is $\frac{100}{150} = \frac{2}{3}$ .

(a) 150 cups (b) 20% (c) 2 : 1 (d) $\frac{2}{3}$

Practice 6: People in a household

The table shows the number of people living in each of the 40 households in a block of flats.

Number of people	Number of households
1	4
2	7
3	10
4	12
5	5
6	2

(a) How many households have at least 4 people?

(b) How many households have fewer than 3 people?

(d) How many people live in the block altogether?

(e) How many households have at most 3 people?

ShowHide solution

(a) At least 4 means 4, 5 or 6 people: $12 + 5 + 2 = 19$ .

(b) Fewer than 3 means 1 or 2 people: $4 + 7 = 11$ .

(d) Multiply each number of people by its number of households and add:

$1 \times 4 + 2 \times 7 + 3 \times 10 + 4 \times 12 + 5 \times 5 + 6 \times 2$

$= 4 + 14 + 30 + 48 + 25 + 12 = 133$

(e) At most 3 means 3 or fewer: 1, 2 or 3 people. $4 + 7 + 10 = 21$ .

(a) 19 households (b) 11 households (c) 17.5% (d) 133 people (e) 21 households

Practice 7: Lunch at the canteen

Hui Min asked the 40 students in her class on how many school days last week (Monday to Friday) they bought lunch at the school canteen.

Number of days	Frequency
0	2
1	5
2	$y$
3	9
4	8
5	6
Total	40

(a) Find the value of $y$ .

(b) What percentage of the students bought lunch at the canteen on at least 4 days?

ShowHide solution

(a) The other frequencies add up to $2 + 5 + 9 + 8 + 6 = 30$ , so $y = 40 - 30 = 10$ .

Check: $2 + 5 + 10 + 9 + 8 + 6 = 40$ .

(b) At least 4 days means 4 or 5 days: $8 + 6 = 14$ students, and $\frac{14}{40} \times 100\% = 35\%$ .

(a) $y = 10$ (b) 35% of the students bought lunch at the canteen on at least 4 days.

Practice 8: Quiz scores

50 students took a quiz of 5 questions. The number of students who scored 3 is three times the number who scored 1.

Score	Number of students
0	2
1	$x$
2	9
3	$3x$
4	14
5	5
Total	50

(a) Find the value of $x$ .

(b) Which score is the most common?

ShowHide solution

(a) The frequencies add up to 50:

$2 + x + 9 + 3x + 14 + 5 = 50$

$4x + 30 = 50$

$4x = 20$ (subtract 30 from both sides)

$x = 5$ (divide both sides by 4)

So 5 students scored 1 and $3 \times 5 = 15$ scored 3. Check: $2 + 5 + 9 + 15 + 14 + 5 = 50$ .

(b) The greatest frequency is 15, for the score 3.

(a) $x = 5$ (b) 3 (c) 68%

Practice 9: Push-ups in one minute

During a PE lesson, Mr Tan counted the number of push-ups each of 25 students did in one minute:

23, 35, 18, 27, 41, 30, 26, 12, 38, 29, 33, 45, 20, 24, 37, 19, 31, 48, 22, 36, 15, 28, 34, 40, 25

(a) Using the classes 10–19, 20–29, 30–39 and 40–49, make a tally and frequency table.

(b) Which class has the greatest frequency?

ShowHide solution

(a) Go through the numbers in order, putting each in its class: 23 in 20–29, 35 in 30–39, 18 in 10–19, and so on. 20 goes in 20–29, and 30 and 40 begin the next two classes.

Number of push-ups	Tally	Frequency
10–19		4
20–29		9
30–39		8
40–49		4
Total		25

Check: $4 + 9 + 8 + 4 = 25$ .

(b) The greatest frequency is 9, for 20–29.

(b) 20–29 (c) 48% of the students did at least 30 push-ups.

Practice 10: A grouped table with a missing frequency

The table shows the marks, out of 100, of 40 students in a Science test.

Mark	Number of students
30–39	3
40–49	$x$
50–59	11
60–69	9
70–79	6
80–89	2
Total	40

(a) Find the value of $x$ .

(b) The pass mark is 50. What percentage of the students passed?

ShowHide solution

(a) The other frequencies add up to $3 + 11 + 9 + 6 + 2 = 31$ , so $x = 40 - 31 = 9$ .

(b) The students who failed scored 30–39 or 40–49: $3 + 9 = 12$ . So $40 - 12 = 28$ passed, and $\frac{28}{40} \times 100\% = 70\%$ .

(c) The 2 students in 80–89 certainly scored 75 or more. The table does not show the marks of the 6 students in 70–79: anywhere from none of them to all 6 may have scored 75 or more. So the number is somewhere from 2 to 8, and the table cannot say whether it is exactly 6.

(a) $x = 9$ (b) 70% (c) No: the table does not show the marks within 70–79, so the number who scored 75 or more could be anything from 2 to 8.

Practice 11: Siblings in a class

Mrs Goh recorded the number of siblings of each of the 40 students in her class. 15% of the students have no siblings.

Number of siblings	Number of students
0	$a$
1	16
2	$b$
3	4
4	2
Total	40

(a) Find the values of $a$ and $b$ .

(b) Find the ratio of the number of students with at least 2 siblings to the number with fewer than 2 siblings.

ShowHide solution

(a) 15% of 40 students: $\frac{15}{100} \times 40 = 6$ , so $a = 6$ .

The other frequencies add up to $6 + 16 + 4 + 2 = 28$ , so $b = 40 - 28 = 12$ .

Check: $6 + 16 + 12 + 4 + 2 = 40$ .

(b) At least 2 siblings means 2, 3 or 4: $12 + 4 + 2 = 18$ .

Fewer than 2 siblings means 0 or 1: $6 + 16 = 22$ . (Check: $18 + 22 = 40$ .)

Ratio $= 18 : 22 = 9 : 11$ (divide both terms by 2).

(a) $a = 6$ and $b = 12$ (b) 9 : 11

Key Points

Tally marks are drawn in bundles of five: four upright strokes and a fifth across them. Count the bundles in fives and add the strokes left over.
The frequency of a value or category is the number of times it occurs. A frequency table lists each value or category with its frequency, and the total at the foot.
To make a tally and frequency table, list every value (in order, a value with frequency 0 included), go through the raw data once, tallying and ticking off each item, then count the tallies.
The total frequency must equal the number of data items; count each row again too, since two wrong counts can cancel out.
The most common value or category has the greatest frequency.
At least 2 means 2 or more; at most 2 means 2 or fewer; more than 2 means 3 or more; fewer than 2 means 1 or fewer.
Percentage $= \frac{\text{frequency}}{\text{total frequency}} \times 100\%$ ; a ratio of two frequencies is written in its simplest form.
The total of the values (books read, people in the block) is each value times its frequency, added.
A missing frequency is the total minus the sum of the others; two linked unknowns are found from an equation.
Numerical data with many different values, such as test marks, are tallied into given classes such as 40–49 and 50–59; each value goes in exactly one class, and the table no longer shows the single values.