Frequency tables
You asked 20 classmates in your school how many times they went to the cinema last month and you have a list of 20 numbers in no particular order. You cannot see any pattern like that. Is a trip to the cinema once a month typical, or rare? The first step in any study is to sort and count: to make a frequency table.
Population, sample and variables
Before making a table you need to know what you are studying and who you are asking.
To find out opinions about the use of mobile phones in the classroom, a survey has been carried out on $45$ students of a school that has a total of $600$ students.…
To find out opinions about the use of mobile phones in the classroom, a survey has been carried out on $45$ students of a school that has a total of $600$ students. State what the fundamental elements of this statistical study are:
- 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 carried out on the most-watched type of content on a well-known streaming platform among teenagers in a city. To do this, $150$ students in 3rd Year…
A study is to be carried out on the most-watched type of content on a well-known streaming platform among teenagers in a city. To do this, $150$ students in 3rd Year from three different randomly chosen secondary schools are asked. Identify the following elements of the statistical study:
- a) Population.
- b) Sample.
- c) Individual.
- d) Type of statistical variable.
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:
-
The tally (using tally marks).
-
The absolute frequency ($f_i$).
In a 3rd Year class, students were asked about their favourite sport. Complete the missing values in the following frequency table: | Sport | Absolute frequency ($f_i$)…
In a 3rd Year class, students were asked about their favourite 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 | $25$ | $1.00$ | $100\%$ |
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 | $N=$ | $1.00$ | $100\%$ |
Finding missing values
Often part of a table is missing. The relationships between the columns let you rebuild it: $h_i = \dfrac{f_i}{N}$, so $f_i = h_i \times N$ and $N = \dfrac{f_i}{h_i}$.
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 | $40$ | $1.00$ |
In a Physical Education class, students were asked about their favourite sport outside school hours. The results are: Football: $12$ students. Basketball: $8$ students.…
In a Physical Education class, students were asked about their favourite sport outside school hours. The results are:
-
Football: $12$ 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 pie chart (central angle size).
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.
Data grouped into intervals
With continuous variables (or discrete variables with many different values) the data are grouped into intervals or classes, usually of equal width. They are written like this: $[10, 15)$ includes 10 but not 15.
- Mid-interval value $x_i$: the midpoint of the interval. It represents all the data values in that interval in calculations.
If you are given only the mid-interval values, the width is the difference between two consecutive ones, and the lower limit of an interval is its mid-interval value minus half the width.
- Width: the difference between the endpoints.
| Time (min) | Mid-interval value $x_i$ | $f_i$ |
|---|---|---|
| $[0, 10)$ | 5 | 3 |
| $[10, 20)$ | 15 | 8 |
| $[20, 30)$ | 25 | 6 |
| $[30, 40)$ | 35 | 3 |
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 ($f_i$) |
|---|---|---|
| $[2, 3)$ | $4$ | |
| $[3, 4)$ | $12$ | |
| $[4, 5)$ | $9$ | |
| $[5, 6)$ | $5$ |
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?
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.
Back to the start: your 20 numbers become a table, and the table shows that most people went to the cinema once or twice, so once a month is typical.