Frequency Tables
You asked 20 classmates how many pets they have, and now you have 20 numbers scattered across your notebook page. Nobody can read a mess like that. The first step is to sort and count: build a frequency table. By the end of this page you will know how to answer questions like "How many students have at most one pet?" in a few seconds.
Populations, Samples, and Variables
Before you make a table, you need to know who you are asking and what you are measuring.
The thing you record about each individual is called a variable. There are two main kinds:
| Type of variable | What it records | Examples |
|---|---|---|
| Categorical (qualitative) | A category or label | favorite color, phone brand, type of music |
| Numerical (quantitative), discrete | A number you get by counting (whole numbers) | pets at home, siblings, text messages today |
| Numerical (quantitative), continuous | A number you get by measuring (any value in a range) | height, time in a race, weight |
A study is to be carried out to find out the favorite book of the $1\,200$ students at a middle school in Chicago. To do this, $80$ students are chosen at random and…
A study is to be carried out to find out the favorite book of the $1\,200$ students at a middle school in Chicago. To do this, $80$ students are chosen at random and asked the title of the book they liked the most this year.
State what the following elements of this statistical study are:
-
Population
-
Sample
-
Individual
Classify the following statistical variables as qualitative or quantitative. For the quantitative ones, state whether they are discrete or continuous: a) The favorite…
Classify the following statistical variables as qualitative or quantitative. For the quantitative ones, state whether they are discrete or continuous:
- a) The favorite color of your classmates.
- b) The number of WhatsApp messages you receive a day.
- c) The exact height of the players on a basketball team.
- d) The cell phone model used by the teachers at the school.
- e) The number of siblings of the 7th grade students.
Absolute and Relative Frequency
A frequency table lists each value of the variable and how often it appears.
To build a table: list each value once, make a tally mark for each data value, count the tallies to get $n_i$, then divide each count by $N$.
Try changing the data below and watch how the counts change.
A group of friends has recorded the number of times they have been to the movies in the last month. The results obtained are:…
A group of friends has recorded the number of times they have been to the movies in the last month. The results obtained are:
$2, 1, 0, 3, 2, 1, 1, 0, 2, 2, 1, 0, 3, 2, 1$
Organize these data into a frequency table that includes the absolute frequency ($n_i$) for each value of the variable.
In a survey on the favorite social network of $20$ teenagers, the following absolute frequency results have been obtained: | Social Network | Absolute frequency ($n_i$)…
In a survey on the favorite social network of $20$ teenagers, the following absolute frequency results have been obtained:
| Social Network | Absolute frequency ($n_i$) |
|---|---|
| TikTok | $10$ |
| $6$ | |
| YouTube | $3$ |
| Other | $1$ |
| Total | $20$ |
Calculate the relative frequency ($f_i$) and the percentage ($\%$) for each category.
A group of 20 teenagers has noted down how many hours a week they spend watching series on streaming platforms. The data obtained is as follows:…
A group of 20 teenagers has noted down how many hours a week they spend watching series on streaming platforms. The data obtained is as follows:
$5, 8, 12, 5, 10, 15, 8, 8, 10, 12, 15, 5, 8, 10, 12, 8, 10, 12, 10, 8$.
Organise this data into a frequency table that includes:
-
Absolute frequency ($n_i$).
-
Relative frequency ($f_i$).
-
Percentage ($\%$).
From Percents to Counts and Pie Charts
Relative frequency works in both directions. If you know $N$ and $f_i$, you can find the count:
$$n_i = f_i \times N$$When a pie chart shows percents, multiply each by $N$ to get counts. When it shows angles, first divide by $360^\circ$ to get $f_i$, then multiply by $N$.
For a study on eating habits, the following qualitative data has been collected on the type of fruit eaten for a snack by $25$ children at a camp: Apple, Banana, Pear,…
For a study on eating habits, the following qualitative data has been collected on the type of fruit eaten for a snack by $25$ children at a camp:
Apple, Banana, Pear, Apple, Orange, Banana, Apple, Pear, Banana, Apple, Orange, Apple, Pear, Apple, Banana, Orange, Apple, Banana, Pear, Apple, Orange, Apple, Banana, Apple, Pear.
Construct a complete frequency table that includes: Variable ($x_i$), Absolute frequency ($n_i$), Relative frequency ($f_i$) and Percentage ($\%$).
In a local video game championship, the age of the participants has been recorded. Look at the relative frequency table and answer: | Age | Relative frequency ($f_i$) |…
In a local video game championship, the age of the participants has been recorded. Look at the relative frequency table and answer:
| Age | Relative frequency ($f_i$) |
|---|---|
| 12 years | $0,15$ |
| 13 years | $0,35$ |
| 14 years | $0,30$ |
| 15 years | $0,20$ |
If a total of $120$ people participated in the championship:
- a) How many participants are exactly $13$ years old?
- b) How many participants are older than $13$ years?
Look at the following pie chart which represents the preferred type of music of the $200$ students in a school: Construct the absolute and relative frequency table…
Look at the following pie chart which represents the preferred type of music of the $200$ students in a school:
Construct the absolute and relative frequency table corresponding to this graph.
Cumulative Frequency
Sometimes you want to know "how many have at most 1 pet?" That needs a running total.
Notice the pattern: at most uses $N_i$ directly. More than uses $N - N_i$.
The following table shows the number of pets owned by families in a neighborhood. Complete the cumulative absolute frequency column ($N_i$): | No. of Pets ($x_i$) |…
The following table shows the number of pets owned by families in a neighborhood. Complete the cumulative absolute frequency column ($N_i$):
| No. of Pets ($x_i$) | Absolute frequency ($n_i$) | Cumulative frequency ($N_i$) |
|---|---|---|
| $0$ | $8$ | |
| $1$ | $12$ | |
| $2$ | $7$ | |
| $3$ | $3$ | |
| Total | $30$ | --- |
A youth clothing store wants to analyze the number of items each customer buys. It is known that the cumulative relative frequency ($H_i$) for the purchase of $2$ items…
A youth clothing store wants to analyze the number of items each customer buys. It is known that the cumulative relative frequency ($H_i$) for the purchase of $2$ items is $0,75$. If $160$ customers have been analyzed:
- a) How many customers bought $2$ or fewer items?
- b) If the absolute frequency of buying $1$ single item is $60$ customers, how many customers bought exactly $2$ items?
Analyse the following frequency table corresponding to the number of absences of a class of $25$ students during a month. Find the two logical consistency errors that…
Analyse the following frequency table corresponding to the number of absences of a class of $25$ students during a month. Find the two logical consistency errors that exist in the cumulative frequency column and justify why they are errors based on the properties of frequency tables.
| Absences ($x_i$) | $n_i$ | $f_i$ | $N_i$ | $F_i$ |
|---|---|---|---|---|
| $0$ | $10$ | $0,40$ | $10$ | $0,40$ |
| $1$ | $6$ | $0,24$ | $16$ | $0,64$ |
| $2$ | $5$ | $0,20$ | $14$ | $0,84$ |
| $3$ | $4$ | $0,16$ | $25$ | $1,05$ |
Finding Missing Values
Real problems often give you a table with holes. The relationships you already know let you fill them in.
The students in a class were asked how many hours on average they spend studying a day. Some data from the table has been lost. Complete it knowing that the total number…
The students in a class were asked how many hours on average they spend studying a day. Some data from the table has been lost. Complete it knowing that the total number of students is $25$.
| Hours of study ($x_i$) | Absolute frequency ($n_i$) | Relative frequency ($f_i$) |
|---|---|---|
| $1$ h | $5$ | $0,20$ |
| $2$ h | $0,40$ | |
| $3$ h | $6$ | |
| $4$ h | $4$ | $0,16$ |
| Total | $25$ | $1$ |
In a survey on the number of pets owned by the families of a 7th grade class, the following incomplete table has been drawn up. Complete it in your notebook knowing that…
In a survey on the number of pets owned by the families of a 7th grade class, the following incomplete table has been drawn up. Complete it in your notebook knowing that the total number of students is $30$.
| Pets ($x_i$) | Absolute freq. ($n_i$) | Cumulative absolute freq. ($N_i$) | Cumulative relative freq. ($H_i$) |
|---|---|---|---|
| 0 | 12 | ||
| 1 | 9 | ||
| 2 | 26 | ||
| 3 | 4 |
What percentage of students has at most one pet?
A survey has been carried out on the number of books read in the last year by a group of people. It is known that the mode is $2$ books and that the arithmetic mean is…
A survey has been carried out on the number of books read in the last year by a group of people. It is known that the mode is $2$ books and that the arithmetic mean is exactly $2,5$ books. With the data from the incomplete table, find the values of $x$ and $y$ (absolute frequencies).
| Books read ($x_i$) | $n_i$ |
|---|---|
| $1$ | $5$ |
| $2$ | $x$ |
| $3$ | $y$ |
| $4$ | $3$ |
| TOTAL | $20$ |
Grouped Data and Class Midpoints
When a variable is continuous (time, height) or has many different values, listing every value would make a huge table. Instead, you group the data into intervals called classes.
How to choose classes:
-
Find the range: largest value minus smallest value.
-
Pick a nice number of classes (usually 4 to 8) and make them all the same width: width $\approx$ range $\div$ number of classes.
-
Write each class as $[a, b)$, which includes $a$ but not $b$. The last class is closed, $[a, b]$, so the largest value fits.
A study has been carried out on the distance (in miles) that $40$ people travel to get to their workplace. The data has been grouped into the following intervals:…
A study has been carried out on the distance (in miles) that $40$ people travel to get to their workplace. The data has been grouped into the following intervals: $[0, 4)$, $[4, 8)$, $[8, 12)$ and $[12, 16)$.
Knowing that the absolute frequencies are $14, 10, 8$ and $8$ respectively:
- a) Construct the frequency table including the class midpoint ($c_i$) of each interval.
- b) How many people travel $8$ miles or more?
A city council wants to analyze the age of the young people who use the new bike lane. It is known that the variable is continuous quantitative and the data has been…
A city council wants to analyze the age of the young people who use the new bike lane. It is known that the variable is continuous quantitative and the data has been grouped into five intervals of constant width. Complete the table knowing that:
-
The total age range is from $10$ to $30$ years.
-
The absolute frequency of the second interval is twice that of the first.
-
$N_3 = 45$
-
$f_4 = 0,25$
-
$n_5 = 5$
-
The total sample is $N = 60$.
Also state the class midpoints ($x_i$) for each interval.
A Physical Education teacher records the time (in seconds) that his students take to run a $100$-foot sprint. The results are grouped in a frequency table with intervals…
A Physical Education teacher records the time (in seconds) that his students take to run a $100$-foot sprint. The results are grouped in a frequency table with intervals of $2$ seconds in width.
Knowing that:
-
The first interval starts at $12$ seconds.
-
$f_1 = 15\%$
-
$f_2 = 35\%$
-
$N_3 = 21$ students.
-
$F_3 = 0,7$ (cumulative relative frequency).
-
The rest of the students are distributed equally in the last two intervals up to a maximum time of $22$ seconds.
Calculate the total number of students ($N$) and complete the absolute frequency table.
Rebuilding a Table from Clues
Some tough problems hand you a few facts about a table (a relative frequency here, a cumulative total there) and ask you to rebuild all of it. Use this plan:
-
Find $N$ (the total).
-
Turn every fact into a count: $n_i = f_i \cdot N$, $N_i = F_i \cdot N$.
-
Use subtraction on cumulative totals: $n_i = N_i - N_{i-1}$.
-
Fill in the rest, then check that the $n_i$ add up to $N$ and the last $N_i$ equals $N$.
A cumulative frequency polygon is just a graph of the $N_i$: each point sits at the upper end of a class. The number of data in a range is the difference of the heights at its two ends.
In a study on the mobile data consumption of a group of teenagers, the following frequency table has been started. However, some data has been lost. Knowing that the…
In a study on the mobile data consumption of a group of teenagers, the following frequency table has been started. However, some data has been lost. Knowing that the sample size is $N = 80$ individuals and that the cumulative relative frequency of the third interval is $F_3 = 0,825$, complete the table for absolute ($n_i$), relative ($f_i$), cumulative absolute ($N_i$) and cumulative relative ($F_i$) frequencies.
| Consumption (GB) | $n_i$ | $f_i$ | $N_i$ | $F_i$ |
|---|---|---|---|---|
| $[0, 5)$ | $12$ | |||
| $[5, 10)$ | $0,35$ | |||
| $[10, 15)$ | ||||
| $[15, 20]$ |
In an e-sports club, the weekly training hours of its 40 members have been recorded. They have been organised into four class intervals: $[0, 4), [4, 8), [8, 12)$ and…
In an e-sports club, the weekly training hours of its 40 members have been recorded. They have been organised into four class intervals: $[0, 4), [4, 8), [8, 12)$ and $[12, 16]$.
The following data is known:
-
The relative frequencies of the first two intervals are identical: $f_1 = f_2$.
-
The cumulative relative frequency of the third interval is $F_3 = 0,9$.
-
The last interval has an absolute frequency of $n_4 = 4$.
Deduce the complete table of absolute and relative frequencies. How many players train at least $8$ hours a week?
From the following cumulative frequency polygon of the marks of a mathematics exam, reconstruct the absolute frequency table. Knowing that the total number of students…
From the following cumulative frequency polygon of the marks of a mathematics exam, reconstruct the absolute frequency table. Knowing that the total number of students is $30$, determine how many students have obtained a mark greater than or equal to $5$ but less than $9$.
Note: The points on the graph are $(1, 3), (3, 7), (5, 15), (7, 24), (9, 28), (10, 30)$, where the first coordinate is the mark and the second is the cumulative absolute frequency $N_i$.
Two-Way Tables
A two-way table records two variables at once (for example, favorite music and grade level). The rows are the values of one variable, the columns are the values of the other, and each cell counts the individuals who fit both.
A survey has been carried out among the students of a middle school about their favorite sport. The results are shown in this two-way table, but some data is missing.…
A survey has been carried out among the students of a middle school about their favorite sport. The results are shown in this two-way table, but some data is missing. Complete it.
| Soccer | Basketball | Tennis | TOTAL | |
|---|---|---|---|---|
| Girls | $45$ | $20$ | $95$ | |
| Boys | $25$ | $15$ | ||
| TOTAL | $105$ | $55$ |
From the table, calculate the percentage of students (total) who prefer basketball.
A study on the use of social media is to be carried out. To do this, a sample of 7th grade students is asked about their favorite network (TikTok or Instagram) and…
A study on the use of social media is to be carried out. To do this, a sample of 7th grade students is asked about their favorite network (TikTok or Instagram) and whether they post content daily (Yes or No).
Complete the table knowing that:
-
$80$ students were surveyed in total.
-
$45$ students prefer Instagram.
-
Of those who prefer TikTok, $15$ do not post daily.
-
In total, $50$ students in the sample post content daily.
| Posts daily (Yes) | Posts daily (No) | TOTAL | |
|---|---|---|---|
| TikTok | $15$ | ||
| $45$ | |||
| TOTAL | $50$ | $80$ |
What percentage of the students who prefer Instagram post content daily?
A survey has been carried out on the use of three streaming platforms (A, B and C) among 7th grade students. This incomplete two-way table is presented, which relates…
A survey has been carried out on the use of three streaming platforms (A, B and C) among 7th grade students. This incomplete two-way table is presented, which relates the preferred platform to the main device used.
| Tablet | Smartphone | PC | TOTAL | |
|---|---|---|---|---|
| Platform A | $8$ | $5$ | ||
| Platform B | $14$ | $22$ | ||
| Platform C | $6$ | $7$ | $25$ | |
| TOTAL | $18$ | $35$ |
Knowing that the total number of students surveyed is $75$, complete the table and calculate what percentage of the total number of students mainly uses a Smartphone to watch Platform B.
Back to the start: your 20 messy numbers about pets become a clean table in three steps (tally, count, divide by $N$), and from it you can answer in seconds that $14$ of the $20$ classmates, or $70\%$, have at most one pet.