Frequency tables
You have asked 20 classmates in your Second Year class how many pets they have, and you have 20 jumbled numbers in your copy. Like this, it makes no sense: the first step is to sort and count, that is, to build a frequency table.
Populations, samples and types of data
A study is to be carried out to find out the favorite book of the $1\,200$ students of a secondary school in Cork. To do so, $80$ students are chosen at random and asked…
A study is to be carried out to find out the favorite book of the $1\,200$ students of a secondary school in Cork. To do so, $80$ students are chosen at random and asked the title of the book they have liked the most this year.
Identify the following elements of this statistical study:
-
Population
-
Sample
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Individual
Classify the following statistical variables as qualitative or quantitative. For those that are quantitative, state whether they are discrete or continuous: a) Your…
Classify the following statistical variables as qualitative or quantitative. For those that are quantitative, state whether they are discrete or continuous:
- a) Your classmates' favourite colour.
- b) The number of WhatsApp messages you receive per day.
- c) The exact height of the players on a basketball team.
- d) The phone model used by teachers at the school.
- e) The number of siblings of 2nd Year students.
Absolute, relative and cumulative frequency
A group of friends recorded the number of times they went to the cinema in the last month. The results obtained are: $2, 1, 0, 3, 2, 1, 1, 0, 2, 2, 1, 0, 3, 2, 1$…
A group of friends recorded the number of times they went to the cinema in the last month. The results obtained are:
$2, 1, 0, 3, 2, 1, 1, 0, 2, 2, 1, 0, 3, 2, 1$
Organise these data into a frequency table including the frequency ($n_i$) for each value of the variable.
The following table shows the number of pets that the families in a housing development have. Complete the column of cumulative absolute frequencies ($N_i$): | Nº of…
The following table shows the number of pets that the families in a housing development have. Complete the column of cumulative absolute frequencies ($N_i$):
| Nº of pets ($x_i$) | Absolute frequency ($n_i$) | Cumulative frequency ($N_i$) |
|---|---|---|
| $0$ | $8$ | |
| $1$ | $12$ | |
| $2$ | $7$ | |
| $3$ | $3$ | |
| Total | $30$ | --- |
Hint. The cumulative frequency of each row is the sum of its absolute frequency and the cumulative frequency of the previous row. The first row carries over directly: $N_1 = n_1$. The last cumulative frequency must match the total.
In a survey on the number of pets owned by families in a Year 9 class, the following incomplete table was produced. Complete it in your exercise book knowing that the…
In a survey on the number of pets owned by families in a Year 9 class, the following incomplete table was produced. Complete it in your exercise book knowing that the total number of students is $30$.
| Pets ($x_i$) | Abs. frequency ($n_i$) | Cum. absolute frequency ($N_i$) | Cum. relative frequency ($H_i$) |
|---|---|---|---|
| 0 | 12 | ||
| 1 | 9 | ||
| 2 | 26 | ||
| 3 | 4 |
What percentage of students have at most one pet?
Finding missing values
In a table with intervals, the same rules apply. Use $f_i = \dfrac{n_i}{N}$, $F_i = \dfrac{N_i}{N}$ and $n_i = N_i - N_{i-1}$. If two intervals have equal relative frequencies, they have equal absolute frequencies.
Mode and mean from a frequency table
The mode is the value with the greatest frequency $n_i$. The mean is $\bar{x} = \dfrac{\sum x_i n_i}{N}$: multiply each value by its frequency, add, and divide by $N$. This lets you find an unknown frequency.
A youth clothing shop wants to analyse the number of items of clothing each customer buys. It is known that the cumulative relative frequency ($H_i$) for buying $2$…
A youth clothing shop wants to analyse the number of items of clothing each customer buys. It is known that the cumulative relative frequency ($H_i$) for buying $2$ items is $0.75$. If $160$ customers have been analysed:
- a) How many customers bought $2$ items or fewer?
- b) If the frequency of buying just $1$ item is $60$ customers, how many customers bought exactly $2$ items?
A survey on the number of books read in the past year was carried out among a group of people. It is known that the mode is $3$ books and that the arithmetic mean is…
A survey on the number of books read in the past year was carried out among a group of people. It is known that the mode is $3$ books and that the arithmetic mean is exactly $2.5$ books. Using the data in 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 | $20$ |
Analyze the following frequency table corresponding to the number of absences of a class of $25$ students during one month. Find the two logical consistency errors in…
Analyze the following frequency table corresponding to the number of absences of a class of $25$ students during one month. Find the two logical consistency errors in the cumulative frequency columns ($N_i$ and $F_i$) and justify why they are errors based on the properties of frequency tables.
| Absences ($x_i$) | $n_i$ | $f_i$ | $N_i$ | $F_i$ |
|---|---|---|---|---|
| $0$ | $10$ | $0.40$ | $10$ | $0.40$ |
| $1$ | $6$ | $0.24$ | $16$ | $0.64$ |
| $2$ | $5$ | $0.20$ | $14$ | $0.84$ |
| $3$ | $4$ | $0.16$ | $25$ | $1.05$ |
Data grouped into intervals
When the variable is continuous (time, height…) with many different values, the data are grouped into intervals of the form $[a, b)$ (includes $a$, does not include $b$).
Cumulative frequency graphs
Plot each cumulative frequency $N_i$ against the upper end of its interval and join the points. The graph never goes down.
Pie charts to tables
A pie chart shows each category as an angle out of $360^\circ$. So $n_i = \dfrac{\text{angle}}{360} \times N$.
A study has been carried out on the distance (in km) that $40$ people travel to get to their workplace. The data have been grouped into the following intervals:…
A study has been carried out on the distance (in km) that $40$ people travel to get to their workplace. The data have been grouped into the following intervals: $[0, 4)$, $[4, 8)$, $[8, 12)$ and $[12, 16)$.
Given that the absolute frequencies are $14$, $10$, $8$ and $8$ respectively:
- a) Construct the frequency table including the class mark ($c_i$) of each interval.
- b) How many people travel $8$ km or more?
You can do it on paper and send a photo with your answer.
Students in a class were asked how many hours on average they spend studying per day. Some data in the table have been lost. Complete it knowing that the total number of…
Students in a class were asked how many hours on average they spend studying per day. Some data in the table have been lost. Complete it knowing that the total number of students is $25$.
| Hours of study ($x_i$) | 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 | $25$ | $1$ |
From the cumulative frequency polygon of the grades of a mathematics exam, reconstruct the absolute frequency table. Knowing that the total number of students is $30$,…
From the cumulative frequency polygon of the grades of a mathematics exam, reconstruct the absolute frequency table. Knowing that the total number of students is $30$, determine how many students obtained a grade greater than or equal to $5$ but strictly 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 grade and the second is the cumulative absolute frequency $N_i$.
Two-way tables
When two variables are cross-tabulated at the same time (for example, preferred music and gender), a two-way table is used: the rows are the values of one variable and the columns those of the other.
A survey was carried out among secondary school students about their favourite sport. The results are shown in this two-way table, but some data are missing. Complete…
A survey was carried out among secondary school students about their favourite sport. The results are shown in this two-way table, but some data are missing. Complete it.
| Football | Basketball | Tennis | TOTAL | |
|---|---|---|---|---|
| Girls | $45$ | $20$ | $95$ | |
| Boys | $25$ | $15$ | ||
| TOTAL | $105$ | $55$ |
Using the table, calculate the percentage of (total) students who prefer basketball.
A study is to be carried out on the use of social networks. To do this, a sample of 2nd Year students are asked about their favourite network (TikTok or Instagram) and…
A study is to be carried out on the use of social networks. To do this, a sample of 2nd Year students are asked about their favourite network (TikTok or Instagram) and whether they post content daily (Yes or No).
Complete the table knowing that:
-
A total of $80$ students were surveyed.
-
$45$ 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 students who prefer Instagram post content daily?
A survey on the use of three streaming platforms (A, B, and C) was conducted among 2nd Year students. This incomplete two-way table relates the preferred platform to the…
A survey on the use of three streaming platforms (A, B, and C) was conducted among 2nd Year students. This incomplete two-way table relates the preferred platform to the primary 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 surveyed students is $69$, complete the table and calculate what percentage of the total students mainly uses a Smartphone to watch Platform B.