Frequency Tables
Emma asked 20 classmates how many times they went to the movies last month. She got a messy list: 0, 1, 1, 2, 0, 3, 1, 2, 2, 1, 4, 1, 0, 2, 1, 3, 1, 2, 0, 1. Nothing jumps out. The first step of any statistics project is to sort and count, and the tool for that is a frequency table.
Populations, samples, and variables
Before making a table, decide who you are studying and what you are measuring.
To find out the opinion on the use of cell phones in the classroom, a survey has been carried out among $45$ students of a middle school that has a total of $600$…
To find out the opinion on the use of cell phones in the classroom, a survey has been carried out among $45$ students of a middle school that has a total of $600$ students. State the fundamental 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 the data from a music app: a) The musical genre of…
Classify the following statistical variables as qualitative, discrete quantitative or continuous quantitative based on the data from a music app:
- a) The musical genre of the songs (Pop, Rock, Trap...).
- b) The number of "likes" a playlist has.
- c) The duration of a song in seconds.
- d) The name of the artist or band.
- e) The storage capacity occupied by the app on the cell phone (in MB).
A study is to be carried out on the type of content most watched on a well-known streaming platform by teenagers in a city. To do this, $150$ students in grade…
A study is to be carried out on the type of content most watched on a well-known streaming platform by teenagers in a city. To do this, $150$ students in grade $3^\text{o}$ from three different randomly chosen middle schools are asked. Identify the following elements of the statistical study:
- a) Population.
- b) Sample.
- c) Individual.
- d) Type of statistical variable.
Frequency, relative frequency, and percent
To count without losing your place, go through the data once and make a tally mark for each value. Group the marks in fives: four bars and a slash across them.
Try your own data in the widget and watch the mean and the spread change.
A group of friends records the number of times they have been to the cinema in the last month. The data obtained are:…
A group of friends records the number of times they have been to the cinema in the last month. The data obtained are:
$2, 1, 0, 3, 2, 1, 1, 0, 2, 2, 1, 4, 1, 0, 2, 1, 1, 2, 3, 2$.
Organize 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 an 8th grade class, students were asked about their favorite sport. Complete the missing values in the following frequency table: | Sport | Absolute frequency ($f_i$)…
In an 8th grade 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 | $6$ | $0,24$ | $24\%$ |
| Soccer | $10$ | ||
| Tennis | $4$ | $0,16$ | |
| Swimming | $0,20$ | $20\%$ | |
| Total | $25$ | $1,00$ | $100\%$ |
A group of $25$ young people has recorded the number of times they have been to the cinema in the last quarter, obtaining the following data:…
A group of $25$ young people has recorded the number of times they have been to the cinema in the last quarter, obtaining the following data:
$2, 1, 0, 3, 2, 1, 2, 0, 1, 4, 2, 1, 0, 2, 1, 2, 3, 2, 0, 1, 2, 1, 0, 2, 1$
Organize this information into a frequency table that includes: absolute frequency ($n_i$), relative frequency ($h_i$) and percentage frequency ($\%$).
Finding missing values
Tables often come with holes. The three quantities $f$, $h$, and $N$ are tied together, so one hole can usually be filled from the others.
A survey was carried out on the number of pets owned by families in a neighborhood. By mistake, some data has been deleted from the table. Given that a total of $40$…
A survey was carried out on the number of pets owned by families in a neighborhood. By mistake, some data has been deleted from the table. Given 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 gaps in the following frequency table for a discrete quantitative variable: | Data ($x_i$) | Absolute freq. ($n_i$) | Relative freq. ($h_i$) | Percentage…
Fill in the gaps in the following frequency table for a discrete quantitative variable:
| Data ($x_i$) | Absolute freq. ($n_i$) | Relative freq. ($h_i$) | Percentage freq. ($\%$) |
|---|---|---|---|
| $1$ | $8$ | $0,20$ | |
| $2$ | $0,35$ | ||
| $3$ | $12$ | ||
| $4$ | $15\%$ | ||
| Total | $N=$ | $1,00$ | $100\%$ |
A survey has been carried out among the employees of an office on the number of languages they speak. We know that $40\%$ speak only one language, $35\%$ speak two…
A survey has been carried out among the employees of an office on 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?
Cumulative frequency
Sometimes the question is "How many went at most twice?" The cumulative frequency answers that.
A study has been carried out on the number of books read by the students of a reading club during the summer. Look at the table and answer the questions using the…
A study has been carried out on the number of books read by the students of a reading club during the summer. Look at the table and answer the questions using the 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?
In a survey on the number of pets families have in a neighborhood, the results shown in the following absolute frequency table have been obtained: | No. of pets ($x_i$)…
In a survey on the number of pets families have in a neighborhood, 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 for cumulative absolute frequencies ($N_i$) and cumulative relative frequencies ($H_i$). What percentage of families have at most $2$ pets?
A courier company is studying the weight of the parcels it sends daily. It has organised the information in the following cumulative relative frequency table ($H_i$): |…
A courier company is studying the weight of the parcels it sends daily. It has organised the information in the following cumulative relative frequency table ($H_i$):
| Weight (lb) | $f_i$ | $h_i$ | $H_i$ |
|---|---|---|---|
| $[0, 2)$ | $20$ | $0,40$ | $0,40$ |
| $[2, 4)$ | $15$ | $0,30$ | $0,70$ |
| $[4, 6)$ | $10$ | $0,20$ | $0,90$ |
| $[6, 8)$ | $5$ | $0,10$ | $1,00$ |
Answer the following questions using the table:
- a) What percentage of parcels weigh less than $4$ pounds?
- b) What relative frequency do the parcels weighing $6$ pounds or more represent?
- c) What is the total number of parcels analyzed in this study?
Grouped data and midpoints
When a variable is continuous (or discrete with many different values), a tally of every value is useless. Instead we group the data into intervals of equal width.
An interval like $[10, 20)$ includes 10 but not 20. The square bracket means "included" and the round bracket means "not included". The last interval of a table is often closed, like $[12, 16]$, so it includes the largest value.
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The midpoint of an interval is the number halfway between its ends: $\dfrac{a + b}{2}$. Some problems call it the class mark and write it $x_i$. It stands in for all the data in that interval.
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The width of an interval is $b - a$.
For a study on the weight of students' backpacks, the data have been grouped into intervals of $1$ pounds. Calculate the class mark ($x_i$) for each of the intervals in…
For a study on the weight of students' backpacks, the data have been grouped into intervals of $1$ pounds. Calculate the class mark ($x_i$) for each of the intervals in the table:
| Weight (lb) | Class mark ($x_i$) | Frequency ($f_i$) |
|---|---|---|
| $[2, 3)$ | $4$ | |
| $[3, 4)$ | $12$ | |
| $[4, 5)$ | $9$ | |
| $[5, 6)$ | $5$ |
The time, in minutes, it takes $20$ students to get from their house to school has been measured. The data are:…
The time, in minutes, it takes $20$ students to get from their house 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 grouping the data into $5$ intervals of equal width, starting at $5$ and ending at $30$. Include the class marks ($x_i$) and the absolute frequencies ($n_i$) in the table.
A nutritionist analyzes the amount of sugar (in grams) per $100\text{ ml}$ of $40$ different soft drinks. The results have been grouped in the following table: | Amount…
A nutritionist analyzes the amount of sugar (in grams) per $100\text{ ml}$ of $40$ different soft drinks. The results have been grouped in the following table:
| Amount of sugar ($g$) | Class mark ($x_i$) | Absolute freq. ($n_i$) |
|---|---|---|
| $[0, 4)$ | $5$ | |
| $[4, 8)$ | $12$ | |
| $[8, 12)$ | $18$ | |
| $[12, 16]$ | $5$ |
- a) Complete the class marks column.
- b) Calculate the relative frequencies column ($h_i$).
- c) State how many soft drinks have $8$ grams of sugar or more.
Angles for a circle graph
A frequency table can become a circle graph (pie chart). A full circle is $360^\circ$, and each category gets a share of it equal to its relative frequency.
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.
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Tennis: $4$ students.
Create a table showing the absolute frequencies and calculate the degrees needed to represent these data in a pie chart (size of the central angle).
Rebuilding a table from clues
Harder problems give only some of the entries plus clues in words. Use the toolbox below, one clue at a time.
Sometimes the table gives cumulative relative frequencies $H_i$ with some of them missing, plus a clue that two groups have the same size. Turn each $H$ into a count, then use the clue.
The following frequency table is available for a discrete quantitative variable $X$ representing the number of times a group of students has been to the cinema this…
The following frequency table is available for a discrete quantitative variable $X$ representing the number of times a group of students has been 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 | |||
| 2 | 31 | ||
| 3 | 9 | $1,000$ |
Calculate the total number of data ($N$) and the missing frequencies to complete the table. Justify each step using the properties of cumulative and relative frequencies.
A video game store has carried out a study on the number of peripherals (controllers, keyboards, mice) that customers buy in a single visit. However, some of the…
A video game store has carried out a study on the number of peripherals (controllers, keyboards, mice) that customers buy in a single visit. However, some 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 absolute frequency of buying 2 peripherals is twice that of buying 4, and 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 |
To organize an e-sports tournament, the organization analyzes the weekly hours that young people in a club dedicate to training. The following cumulative relative…
To organize an e-sports tournament, the organization analyzes the weekly hours that young people in a club dedicate to training. The following cumulative relative frequency table ($H_i$) shows the distribution, but the data for the central categories are missing.
If we know that there is the same number of young people who train "From 5 to 10 hours" as young people who train "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. ($H_i$) |
|---|---|
| $[0, 5)$ | $0,15$ |
| $[5, 10)$ | |
| $[10, 15)$ | $0,65$ |
| $[15, 20)$ | |
| $[20, 25]$ | $1,00$ |
When the data change
Real data get corrected or extended. To update a table, work with the frequencies $f$ first, and then recompute the cumulative and relative columns.
In a study on water consumption per household in a neighborhood, the following cumulative frequency data have been obtained, but the individual absolute frequencies have…
In a study on water consumption per household in a neighborhood, the following cumulative frequency data have been obtained, but the individual absolute frequencies have been omitted.
| Consumption ($m^3$) | Cumulative absolute freq. ($F_i$) |
|---|---|
| $[0, 10)$ | $12$ |
| $[10, 20)$ | $30$ |
| $[20, 30)$ | $65$ |
| $[30, 40)$ | $90$ |
| $[40, 50]$ | $100$ |
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 households will finally have to pay the surcharge.
A local council 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 | $F_i$ |…
A local council 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 $40$ new users register during the afternoon, all of them belonging to the $[18, 35)$ age group, 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.
The mean from a table
A frequency table has enough information to find the mean without going back to the raw list.
When a table gives only midpoints, rebuild the intervals first.
A survey has been carried out 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 carried out 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.
A technology laboratory is testing the battery life in hours of a new drone battery. They have grouped the results into a continuous variable frequency table, but they…
A technology laboratory is testing the battery life in hours of a new drone battery. They have grouped the results into 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 ($x_i$) | Absolute freq. ($f_i$) | Cumulative absolute freq. ($F_i$) |
|---|---|---|
| $1,45$ | 8 | |
| $1,95$ | 14 | |
| $2,45$ | ||
| $2,95$ | ||
| $3,45$ | 3 |
Stratified samples
If a school has more 7th graders than 10th graders, a random sample of the whole school might happen to include very few 10th graders. A stratified random sample avoids this. Divide the population into groups called strata and sample from each one in proportion to its size.
An educational center wants to carry out a representative survey on the use of digital books. The school has $400$ students in 7th grade, $350$ in 8th grade, $300$ in…
An educational center wants to carry out a representative survey on the use of digital books. The school has $400$ students in 7th grade, $350$ in 8th grade, $300$ in 9th grade, and $250$ in 10th grade. They want to extract a sample of $130$ students using stratified random sampling with proportional allocation.
Draw the table of the total population and, from it, create the frequency table of the sample indicating how many students from each grade level should be surveyed. What is the relative frequency of 9th grade students in the sample compared to that of the total population?
Back to Emma's messy list: in five rows she can now see that most classmates went once (40%), that 85% went at most twice, and that nobody went more than four times.