Frequency tables
At recess, Yaritza asked 20 classmates how many times they went to the movies last month. She has a list of 20 jumbled numbers, and as it stands you can't see anything in it. How do you turn that list into information you can read at a glance? The first step in any study is to sort and count: make a frequency table.
Population, sample, and variable
Before counting, you need to know who you are asking and what you are asking.
There are three kinds of variables:
| Type | What it is | Example |
|---|---|---|
| Qualitative | A quality, not a number | favorite sport, color |
| Discrete quantitative | A number you count (separate values) | siblings, goals, "likes" |
| Continuous quantitative | A number you measure (any value within a range) | height, time, weight |
Why use a sample? Asking the whole population is almost always too expensive or impossible. If the sample is chosen well, it represents the whole population.
To find out opinions on the use of cell phones in the classroom, a survey was conducted among $45$ students at a school with a total of $600$ students. State the key…
To find out opinions on the use of cell phones in the classroom, a survey was conducted among $45$ students at a school with a total of $600$ students. State the key elements of this statistical study:
- a) The population under study.
- b) The selected sample.
- c) The individual.
Classify the following statistical variables as qualitative, discrete quantitative or continuous quantitative based on data from a music application: a) The musical…
Classify the following statistical variables as qualitative, discrete quantitative or continuous quantitative based on data from a music application:
- a) The musical genre of the songs (Pop, Rock, Trap...).
- b) The number of "likes" on a playlist.
- c) The duration of a song in seconds.
- d) The name of the artist or band.
- e) The storage space occupied by the app on the phone (in MB).
A study is to be conducted on the type of content most watched on a popular streaming platform by teenagers in a city. To do so, $150$ students in $3^\text{o}$ of…
A study is to be conducted on the type of content most watched on a popular streaming platform by teenagers in a city. To do so, $150$ students in $3^\text{o}$ of secondary school from three randomly selected secondary schools are surveyed. Identify the following elements of the statistical study:
- a) Population.
- b) Sample.
- c) Individual.
- d) Type of statistical variable.
Stratified sampling
Sometimes the population has different groups (strata), like the grades in a school. For the sample to represent them well, you take from each group an amount proportional to its size. This is called stratified random sampling with proportional representation.
It is organized in a table with the population, the sample, and the relative frequency of each stratum in the sample ($h_i = f_i / N$).
A school wants to conduct a representative survey on the use of digital textbooks. The school has $400$ students in the 1st year of secondary, $350$ in the 2nd year of…
A school wants to conduct a representative survey on the use of digital textbooks. The school has $400$ students in the 1st year of secondary, $350$ in the 2nd year of secondary, $300$ in the 3rd year of secondary, and $250$ in the 4th year of secondary. They want to draw a sample of $130$ students using stratified random sampling with proportional allocation.
Draw the table for the total population and, from it, create the frequency table for the sample indicating how many students from each grade should be surveyed. What is the relative frequency of the 3rd year of secondary students in the sample compared to that in the total population?
Frequencies
A group of friends write down the number of times they have been to the cinema in the last month. The data obtained are:…
A group of friends write down the number of times they have been to the cinema in the last month. The data obtained are:
$2, 1, 0, 3, 2, 1, 1, 0, 2, 2, 1, 4, 1, 0, 2, 1, 1, 2, 3, 2$.
Organise these data into a frequency table that includes:
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The tally (using tally marks).
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The absolute frequency ($f_i$).
In a 3rd ESO class, students were asked about their favorite sport. Complete the missing values in the following frequency table: | Sport | Absolute frequency ($f_i$) |…
In a 3rd ESO class, students were asked about their favorite sport. Complete the missing values in the following frequency table:
| Sport | Absolute frequency ($f_i$) | Relative frequency ($h_i$) | Percentage (%) |
|---|---|---|---|
| Basketball | $6$ | $0.24$ | $24\%$ |
| Football | $10$ | ||
| Tennis | $4$ | $0.16$ | |
| Swimming | $0.20$ | $20\%$ | |
| Total | $25$ | $1.00$ | $100\%$ |
A group of $25$ young people recorded the number of times they went to the cinema in the last quarter, obtaining the following data:…
A group of $25$ young people recorded the number of times they went to the cinema in the last quarter, obtaining the following data:
$2, 1, 0, 3, 2, 1, 2, 0, 1, 4, 2, 1, 0, 2, 1, 2, 3, 2, 0, 1, 2, 1, 0, 2, 1$
Organise this information into a frequency table that includes: absolute frequency ($n_i$), relative frequency ($h_i$) and percentage frequency ($\%$).
Completing tables with gaps
Often some information is missing. It is solved with two ideas: $h_i = f_i/N$ and the fact that the columns add up to $N$ and 1.
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If you know a pair $f_i$ and $h_i$, you find the total: $N = \dfrac{f_i}{h_i}$.
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With $N$ and $h_i$: $f_i = h_i \times N$.
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What is missing to complete the total is found by subtracting.
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A percent is converted to $h_i$ by dividing by 100 (35% = 0.35).
A survey was conducted on the number of pets families in a neighbourhood have. Due to an error, some data in the table were erased. Knowing that a total of $40$ families…
A survey was conducted on the number of pets families in a neighbourhood have. Due to an error, some data in the table were erased. Knowing that a total of $40$ families were surveyed, calculate the missing values:
| No. of pets | Families ($f_i$) | Relative Frequency ($h_i$) |
|---|---|---|
| $0$ | $12$ | $0.30$ |
| $1$ | $0.45$ | |
| $2$ | $7$ | |
| $3$ | $3$ | $0.075$ |
| Total | $40$ | $1.00$ |
Fill in the blanks in the following frequency table for a discrete quantitative variable: | Data value ($x_i$) | Abs. frequency ($n_i$) | Rel. frequency ($h_i$) |…
Fill in the blanks in the following frequency table for a discrete quantitative variable:
| Data value ($x_i$) | Abs. frequency ($n_i$) | Rel. frequency ($h_i$) | Percentage frequency ($\%$) |
|---|---|---|---|
| $1$ | $8$ | $0.20$ | |
| $2$ | $0.35$ | ||
| $3$ | $12$ | ||
| $4$ | $15\%$ | ||
| Total | $N=$ | $1.00$ | $100\%$ |
A survey was conducted among office employees about the number of languages they speak. We know that $40\%$ speak only one language, $35\%$ speak two languages, $20\%$…
A survey was conducted among office employees about the number of languages they speak. We know that $40\%$ speak only one language, $35\%$ speak two languages, $20\%$ speak three languages and the rest speak four languages.
If a total of $60$ people work in the office, construct the complete frequency table (absolute, relative and percentage). How many people speak exactly four languages?
You can do it on paper and send a photo with your answer.
Pie chart: the degrees
To draw a pie chart, each category takes up a sector. Since a full turn measures $360°$, the angle of each sector is
$$\text{angle} = h_i \times 360° = \frac{f_i}{N} \times 360°$$Why? If a category is half of the data, it gets half of the circle.
In a Physical Education class, students were asked about their favorite sport outside school hours. The results are: Soccer: $12$ students. Basketball: $8$ students.…
In a Physical Education class, students were asked about their favorite sport outside school hours. The results are:
-
Soccer: $12$ students.
-
Basketball: $8$ students.
-
Swimming: $6$ students.
-
Tennis: $4$ students.
Create a table showing the absolute frequencies and calculate the angles required to represent these data in a circle graph (central angle size).
Cumulative frequencies
A study has been conducted on the number of books read by members of a book club during the summer. Look at the table and answer the questions using cumulative…
A study has been conducted on the number of books read by members of a book club during the summer. Look at the table and answer the questions using cumulative frequencies:
| Books read ($x_i$) | Students ($f_i$) | Cumulative frequency ($F_i$) |
|---|---|---|
| $0$ | $2$ | $2$ |
| $1$ | $5$ | $7$ |
| $2$ | $8$ | $15$ |
| $3$ | $6$ | $21$ |
| $4$ | $4$ | $25$ |
- a) How many students have read exactly $2$ books?
- b) How many students have read $2$ books or fewer?
- c) What is the total number of students participating in the study?
A courier company studies the weight of the parcels it sends daily. It has organised the information in the following table of cumulative relative frequencies ($H_i$): |…
A courier company studies the weight of the parcels it sends daily. It has organised the information in the following table of cumulative relative frequencies ($H_i$):
| Weight (kg) | $f_i$ | $h_i$ | $H_i$ |
|---|---|---|---|
| $[0, 2)$ | $20$ | $0.40$ | $0.40$ |
| $[2, 4)$ | $15$ | $0.30$ | $0.70$ |
| $[4, 6)$ | $10$ | $0.20$ | $0.90$ |
| $[6, 8)$ | $5$ | $0.10$ | $1.00$ |
Answer the following questions based on the table:
- a) What percentage of packages weigh less than $4$ kg?
- b) What relative frequency is represented by packages weighing $6$ kg or more?
- c) What is the total number of packages analysed in this study?
In a survey on the number of pets that families in a neighbourhood have, the results shown in the following absolute frequency table have been obtained: | No. of pets…
In a survey on the number of pets that families in a neighbourhood have, the results shown in the following absolute frequency table have been obtained:
| No. of pets ($x_i$) | No. of families ($n_i$) |
|---|---|
| $0$ | $12$ |
| $1$ | $18$ |
| $2$ | $14$ |
| $3$ | $6$ |
Calculate the column of cumulative absolute frequencies ($N_i$) and the cumulative relative frequencies ($H_i$). What percentage of families have at most $2$ pets?
Data grouped into intervals
With continuous variables (or discrete ones with many different values), the data are grouped into intervals or classes, usually of the same width. They are written $[10, 20)$: it includes 10 but not 20.
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Width: the difference between the endpoints, $b - a$.
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Class mark $x_i$: the midpoint of the interval, $\dfrac{a+b}{2}$. It represents all the data values in the interval in calculations.
| Time on the bus (min) | Mark $x_i$ | $f_i$ | $F_i$ |
|---|---|---|---|
| $[0, 10)$ | 5 | 3 | 3 |
| $[10, 20)$ | 15 | 8 | 11 |
| $[20, 30)$ | 25 | 6 | 17 |
| $[30, 40]$ | 35 | 3 | 20 |
The last interval is closed with a bracket to include the maximum value.
For a study on the weight of students' backpacks, the data have been grouped into intervals of $1$ kg. Calculate the class mark ($x_i$) for each of the intervals in the…
For a study on the weight of students' backpacks, the data have been grouped into intervals of $1$ kg. Calculate the class mark ($x_i$) for each of the intervals in the table:
| Weight (kg) | Class mark ($x_i$) | Frequency ($f_i$) |
|---|---|---|
| $[2, 3)$ | $4$ | |
| $[3, 4)$ | $12$ | |
| $[4, 5)$ | $9$ | |
| $[5, 6)$ | $5$ |
The time, in minutes, taken by $20$ students to get from home to school has been measured. The data are:…
The time, in minutes, taken by $20$ students to get from home to school has been measured. The data are:
$5, 12, 18, 24, 7, 15, 30, 10, 22, 14, 8, 11, 28, 5, 19, 13, 16, 25, 12, 9$
Construct a frequency table by grouping the data into $5$ intervals of equal width, starting at $5$ and ending at $30$. Include in the table the class marks ($x_i$) and the absolute frequencies ($n_i$).
You can do it on paper and send a photo with your answer.
A nutritionist analyses the amount of sugar (in grams) per $100\text{ ml}$ in $40$ different soft drinks. The results have been grouped in the following table: | Amount…
A nutritionist analyses the amount of sugar (in grams) per $100\text{ ml}$ in $40$ different soft drinks. The results have been grouped in the following table:
| Amount of sugar ($g$) | Class mark ($x_i$) | Abs. frequency ($n_i$) |
|---|---|---|
| $[0, 4)$ | $5$ | |
| $[4, 8)$ | $12$ | |
| $[8, 12)$ | $18$ | |
| $[12, 16]$ | $5$ |
- a) Complete the column of class marks.
- b) Calculate the column of relative frequencies ($h_i$).
- c) State how many soft drinks have $8$ grams of sugar or more.
Reconstructing tables with cumulative frequencies
When the individual frequencies are missing, they are recovered by subtracting consecutive cumulative frequencies:
$$f_i = F_i - F_{i-1} \qquad\qquad N = \dfrac{F_i}{H_i}$$Why? $F_i$ already contains everything before it; when you subtract $F_{i-1}$, only what the value $x_i$ contributes is left.
In a video game store, a study was conducted on the number of peripherals (controllers, keyboards, mice) that customers purchase in a single visit. However, part of the…
In a video game store, a study was conducted on the number of peripherals (controllers, keyboards, mice) that customers purchase in a single visit. However, part of the information in the frequency table has been lost. We know that the study was carried out on a total of $N = 80$ customers.
Complete the missing data in the table, knowing that the cumulative relative frequency of buying 3 peripherals or fewer is $0.875$.
| Peripherals ($x_i$) | Absolute freq. ($f_i$) | Relative freq. ($h_i$) | Cumulative absolute freq. ($F_i$) | Cumulative relative freq. ($H_i$) |
|---|---|---|---|---|
| 1 | 24 | |||
| 2 | ||||
| 3 | 16 | |||
| 4 | ||||
| 5 | 6 |
You are given the following frequency table for a discrete quantitative variable $X$ representing the number of times a group of students went to the cinema this month.…
You are given the following frequency table for a discrete quantitative variable $X$ representing the number of times a group of students went to the cinema this month. However, the absolute frequencies $f_2$ and $f_3$ have been erased.
| $x_i$ | $f_i$ | $F_i$ | $H_i$ |
|---|---|---|---|
| 0 | 5 | 5 | $0.125$ |
| 1 | $0.400$ | ||
| 2 | 31 | ||
| 3 | 9 | $1.000$ |
Calculate the total number of data values ($N$) and the missing frequencies to complete the table. Justify each step using the properties of cumulative and relative frequencies.
To organise an esports tournament, the organisation analyses the weekly hours that young members of a club spend training. The following cumulative relative frequency…
To organise an esports tournament, the organisation analyses the weekly hours that young members of a club spend training. The following cumulative relative frequency table ($H_i$) shows the distribution, but the data for the middle categories are missing.
Knowing that there is the same number of young people who train "From 5 to 10 hours", "From 10 to 15 hours", and "From 15 to 20 hours", and that the club has a total of $120$ members, calculate exactly how many members train in each of the intervals in the table.
| Training hours | Cumulative relative freq. ($H_i$) |
|---|---|
| $[0, 5)$ | $0.15$ |
| $[5, 10)$ | |
| $[10, 15)$ | $0.65$ |
| $[15, 20)$ | |
| $[20, 25]$ | $1.00$ |
Intervals, marks, and the mean
Two more tools for harder tables:
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If all the intervals have the same width $a$, consecutive class marks are exactly $a$ apart. With two consecutive marks you get the width, and with the first lower limit you can rebuild all the intervals: upper limit = lower limit + width.
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The mean from the table is $\bar{x} = \sum x_i \cdot h_i$ (with intervals, $x_i$ is the class mark). If some relative frequencies are missing, you set up an equation with this equality and with $\sum h_i = 1$.
A technology laboratory is testing the lifespan in hours of a new drone battery. They have grouped the results in a continuous variable frequency table, but they have…
A technology laboratory is testing the lifespan in hours of a new drone battery. They have grouped the results in a continuous variable frequency table, but they have only provided us with the class marks ($x_i$) and some frequencies.
It is known that all intervals have the same width and that the lower limit of the first interval is $1.2$ hours. Reconstruct the complete table (intervals, class marks and frequencies), knowing that the cumulative absolute frequency of the third interval is $42$ and that the total number of drones tested is $N = 50$.
| Class mark ($x_i$) | Abs. frequency ($f_i$) | Cumulative abs. frequency ($F_i$) |
|---|---|---|
| $1.45$ | 8 | |
| $1.95$ | 14 | |
| $2.45$ | ||
| $2.95$ | ||
| $3.45$ | 3 |
A survey has been conducted on the number of pets in $200$ families. The results are presented in the following incomplete relative frequency table ($h_i$). | No. of…
A survey has been conducted on the number of pets in $200$ families. The results are presented in the following incomplete relative frequency table ($h_i$).
| No. of pets | $h_i$ |
|---|---|
| 0 | $0.35$ |
| 1 | $x$ |
| 2 | $y$ |
| 3 | $0.12$ |
| 4 or more | $0.08$ |
Calculate the values of $x$ and $y$, knowing that the mean number of pets is exactly $1.25$ (for the interval "4 or more", use the value $4$ as the class mark). Once found, construct the complete absolute frequency table.
You can do it on paper and send a photo with your answer.
When the data change
If a data value is corrected or new data arrive, you have to redo the table: absolute frequencies are moved or added, and $N$, $h_i$, and $F_i$ are recalculated.
In a study on water consumption per household in a housing estate, the following cumulative frequency data have been obtained, but the individual absolute frequencies…
In a study on water consumption per household in a housing estate, the following cumulative frequency data have been obtained, but the individual absolute frequencies have been omitted.
| Consumption ($m^3$) | Cumulative abs. frequency ($F_i$) |
|---|---|
| $[0, 10)$ | $12$ |
| $[10, 20)$ | $30$ |
| $[20, 30)$ | $65$ |
| $[30, 40)$ | $90$ |
| $[40, 50]$ | $100$ |
It has been decided that households consuming more than $30 m^3$ will pay a surcharge. However, after a review, it is discovered that $10$ households in the interval $[20, 30)$ actually consumed $32 m^3$. Calculate the new corrected percentage relative frequency table ($\%$) and determine what percentage of the total number of households will ultimately have to pay the surcharge.
A local municipality analyzes the age of the users of a sports center. The data for the first $160$ users are summarized in this cumulative frequency table: | Age |…
A local municipality analyzes the age of the users of a sports center. The data for the first $160$ users are summarized in this cumulative frequency table:
| Age | $F_i$ |
|---|---|
| $[0, 18)$ | $40$ |
| $[18, 35)$ | $95$ |
| $[35, 60)$ | $140$ |
| $[60, 85]$ | $160$ |
If during the afternoon $40$ new users register, all of them belonging to the age group $[18, 35)$, how does the relative frequency table change? Calculate the resulting percentage relative frequency table ($\%$) after the new registrations and comment on how this change has affected the relative importance of the group over $60$ years old.
Back to the beginning: Yaritza's list of 20 numbers becomes, with tally marks, a table where you can read, for example, that 40% of her classmates went to the movies exactly once.