Single-variable descriptive statistics

In a study on the use of social networks at a secondary school, the following variables were identified. Classify each of them as qualitative, discrete quantitative, or continuous quantitative:

1. a) Favourite social network (TikTok, Instagram, Twitter, etc.).
2. b) Number of followers on the main account.
3. c) Daily time (in minutes) spent using a mobile phone.
4. d) Mobile phone brand (Apple, Samsung, Xiaomi, etc.).
5. e) Numerical mark obtained in the last Mathematics exam.

A local council wants to know teenagers' opinions on the city's new skate parks. To do this, it randomly selects $200$ young people aged between $14$ and $18$ who attend youth centres in different neighbourhoods.

State the population under study and the chosen sample. Do you think the sample is representative if only youth centres are surveyed?

A group of $20$ 1st Baccalaureate students were asked how many books they have read for pleasure over the past year. The data obtained are as follows:

$2, 1, 0, 3, 2, 1, 1, 4, 2, 0, 1, 2, 3, 1, 2, 0, 1, 2, 1, 5$

Organise these data into a frequency table including the absolute frequency ($n_i$), relative frequency ($f_i$) and cumulative absolute frequency ($N_i$).

Look at the following chart representing the favourite sport of a group of students in your year:

From the bar chart:

1. a) What is the mode of this distribution?
2. b) How many students in total prefer basketball or swimming?
3. c) If $30$ students were surveyed in total, how many chose swimming (knowing that the bar reaches halfway between 4 and 8)?

You can do it on paper and send a photo with your answer.

A survey was conducted on the number of siblings the students in a class have. The cumulative absolute frequency table ($N_i$) is as follows:

Answer the following questions, giving reasons for your answers:

1. a) How many students are only children?
2. b) How many students have exactly $2$ siblings?
3. c) What is the total number of students in the class?

A music band wants to know which format their fans prefer for listening to their songs. They have made a pie chart with the results of $400$ surveys:

If $10 \%$ of the fans prefer the physical format, $30 \%$ prefer digital download and $60 \%$ prefer streaming:

1. a) Calculate the absolute frequency of fans who prefer each format (knowing that $N=400$).

1. b) What is the relative frequency ($f_i$) of the streaming format?

The figure shows the histogram of marks in a mathematics exam scored out of $50$.

1. a) How many students scored fewer than $20$ marks?

1. b) Which interval has the most students?

1. c) How many students took the exam in total?

You can do it on paper and send a photo with your answer.

The figure shows the hours of sleep reported by students in a class.

1. a) Construct the absolute frequency table from the graph.

1. b) What percentage of students sleep $8$ hours or more?

1. c) What is the mode of the distribution?

You can do it on paper and send a photo with your answer.

A group of researchers has carried out a survey of 40 high school students about the number of hours per week they spend watching content on streaming platforms. The data collected are as follows:

$12, 15, 10, 8, 12, 14, 20, 18, 15, 12, 10, 9, 14, 15, 12, 18, 22, 15, 12, 10,$

$14, 16, 12, 15, 18, 12, 10, 14, 15, 12, 16, 14, 12, 15, 10, 18, 12, 14, 15, 12$

1. a) Classify the statistical variable studied.

1. b) Make a frequency table that includes absolute, relative, cumulative and cumulative relative frequencies.

1. c) Represent the data using a bar chart (hours per week on the horizontal axis and absolute frequency on the vertical axis).

You can do it on paper and send a photo with your answer.

A study was conducted on the number of children in a sample of families in a neighbourhood. Some data in the frequency table have been lost:

1. a) Find the sample size ($N$).
2. b) Calculate the values of $a, b, c$ and $d$ to complete the table.
3. c) Determine the mode and median for the number of children.

A town hall analyzes household water consumption using a relative frequency histogram. Due to the nature of the data, the intervals have different widths: $[0, 10)$, $[10, 20)$, and $[20, 40)$ monthly cubic meters. The heights of the histogram rectangles (frequency density) are $0.03$, $0.04$, and $0.015$, respectively.

1. a) Calculate the relative frequency of each interval.
2. b) What is the average consumption per household in this municipality?
3. c) Determine the value of the $80$th percentile and interpret its meaning in the context of water conservation.

You can do it on paper and submit a photo with your answer.

A retail chain wants to show off its sales and produces two graphs with the same quarterly data: in panel A the vertical axis starts at €0, and in panel B it is truncated.

1. a) Calculate the actual percentage change in sales between the first and third quarters.

1. b) What visual effect does the truncated axis in panel B produce? How does it differ from the impression given by panel A?

1. c) Which of the two graphs would you use in an honest report for shareholders? And which version might a competitor use to alarm the market? Justify both answers.

The marks obtained by $7$ friends on an Economics project are:

$6.5; \quad 8.0; \quad 4.5; \quad 7.0; \quad 9.5; \quad 6.5; \quad 5.0$

Calculate the arithmetic mean ($\bar{x}$) and the median ($Me$) of the marks.

In an e-sports competition, the times (in seconds) taken by participants to complete the first level have been recorded. As this is a continuous variable, the data have been grouped into intervals:

Calculate the class midpoints ($x_i$) of each interval and determine the total sample size ($N$).

A telephone company measured the duration (in minutes) of 50 calls made during the morning. As it is a continuous variable, the data have been grouped into the following intervals:

1. a) Calculate the class midpoint of each interval.
2. b) Calculate the arithmetic mean and standard deviation of the call durations.
3. c) What percentage of calls last 10 minutes or more?

The ages of the members of a hiking club are grouped in the table: $[10, 20)$: $5$; $[20, 30)$: $12$; $[30, 40)$: $18$; $[40, 50)$: $10$; $[50, 60]$: $5$.

1. a) Construct the complete table with class midpoints and cumulative frequencies.

1. b) Draw the histogram and the frequency polygon.

1. c) Calculate the mean age using the class midpoints, and state the modal class and the class interval containing the median.

You can do it on paper and send a photo with your answer.

The monthly electricity consumption (in kWh) of $40$ households is grouped as follows: $[100, 150)$: $4$; $[150, 200)$: $9$; $[200, 250)$: $14$; $[250, 300)$: $8$; $[300, 350]$: $5$.

1. a) Calculate the mean monthly consumption using the class midpoints.

1. b) State the modal class and the median class, and estimate the median assuming that the data are uniformly distributed within their class interval.

1. c) The electricity company claims that "half of the households consume less than $225$ kWh". Is this consistent with the data? Justify your answer using your estimate of the median.

A study was carried out on the weekly leisure spending of students at a secondary school. The data are presented grouped into equal-width intervals in the following table, but the absolute frequencies for two middle intervals ($n_2$ and $n_3$) are missing:

Given that the sample size is $N = 60$ and the arithmetic mean of spending is $\bar{x} = 44$ €, calculate the values of $n_2$ and $n_3$. Once the table is complete, determine the exact value of the median using linear interpolation.

An entrance examination for a vocational training course was held, and the marks obtained by the candidates are summarised in the following cumulative frequency table:

1. a) How many students took the exam?
2. b) Calculate the median, lower quartile, and upper quartile. Explain the meaning of the upper quartile in this context.
3. c) Calculate the 40th percentile ($P_{40}$) and interpret it.

From the following box plot showing the reaction time (in milliseconds) of a group of people to a visual stimulus:

1. a) Identify the values corresponding to the minimum, maximum, $Q_1$, $Q_2$ (median), and $Q_3$, knowing that the scale is linear and the edges of the box are at $210$ and $290$.
2. b) Calculate the interquartile range and interpret it.
3. c) Can it be stated that $50\%$ of the population has a reaction time greater than the median? Justify your answer.

The figure shows the box-and-whisker plot of the reaction times of a group of simulator pilots.

1. a) Read the minimum, maximum, quartiles, and median from the graph, and calculate the range and interquartile range.

1. b) Between which two values is the central $50 \%$ of the data concentrated? What percentage of the data lies below $Q_1$?

1. c) In view of the position of the median inside the box and the length of the whiskers, does the distribution appear symmetric or skewed? In which direction?

The waiting times (in minutes) of $15$ patients at a health centre were: $5, 7, 8, 9, 10, 10, 11, 12, 12, 13, 14, 15, 16, 18, 42$.

1. a) Calculate the median and the lower and upper quartiles.

1. b) Draw the corresponding box-and-whisker plot, indicating the values used.

1. c) Apply the $1.5$ times interquartile range rule to decide whether there are any outliers, and interpret what that data point might represent in the real context of the health centre.

You can do it on paper and send a photo with your answer.

Two basketball players, Aoife and Darragh, have scored the following numbers of points in their last 8 games:

• Aoife: $18, 22, 15, 20, 25, 14, 19, 23$

• Darragh: $20, 21, 19, 22, 20, 20, 21, 17$

The following dot plot shows the data for both players (each dot is a game):

1. a) Calculate the mean and the range of points for each player.
2. b) Calculate the standard deviation of both.
3. c) Use the coefficient of variation to determine which of the two players is more consistent in their scoring. Justify your answer.

We want to compare the consistency of set menu prices in two different areas of a city. The data obtained (in euros) are:

• Area A: $12, 13, 12, 15, 14, 12, 16, 11$

• Area B: $9, 10, 18, 15, 11, 14, 10, 21$

1. a) Calculate the mean and standard deviation of the prices in each area.
2. b) In which area are the prices more dispersed in absolute terms (standard deviation)?
3. c) In which area are the prices more heterogeneous in relative terms? Use the coefficient of variation to compare both distributions.

A nutritionist analyzes the sugar content (in grams per 100 ml) of two brands of "zero" soft drinks, A and B. The results are summarized in the following box plots. Read the values from the scale of the horizontal axis.

Looking at the graphs:

1. a) Which of the two brands has a more spread-out distribution in the middle $50\%$ of its data? Justify your answer by calculating the interquartile range of each brand from the scale of the graph.

1. b) If brand A has a mean of $0.5$ g and brand B a mean of $0.8$ g, and both have identical standard deviations, which of the two is more "reliable" or consistent in its production? Use the appropriate statistical parameter to compare both brands.

In an archery competition, the scores of two archers, Aoife and Niamh, are recorded after 40 shots. Aoife has a mean score of $8.2$ with a standard deviation of $1.3$. Niamh has the following cumulative frequencies for her scores: $25\%$ of her shots were $7$ points or fewer, $50\%$ were $8$ points or fewer, and $75\%$ were $9$ points or fewer.

1. a) Assuming Niamh's scores follow an approximately normal distribution (for which $\sigma \approx \frac{IQR}{1.35}$), determine with reasoning which of the two archers is more consistent.
2. b) If winning a tournament requires at least $30\%$ of shots to score $10$, and we know that Aoife obtained $10$ on $12$ occasions, does Aoife meet the requirement? Can we know if Niamh meets it from her quartile data?

An investor analyses the daily return of two crypto-assets over a month ($30$ days). Asset $X$ has a mean return of $2\%$ and a variance of $9\%$. Asset $Y$ has a daily return that can be summarised as follows: for $10$ days it lost $1\%$, for $15$ days it gained $3\%$, and for $5$ days it gained $5\%$.

1. a) Calculate the mean return and standard deviation of asset $Y$.
2. b) Using the coefficient of variation, determine which of the two assets is more volatile (less predictable).
3. c) If the investor decides to create a combined portfolio with $60\%$ of the capital in asset $X$ and $40\%$ in asset $Y$, what would be the expected mean return of the portfolio?

Two shops in the same neighbourhood recorded their daily sales (in euros) over $40$ days, grouped into intervals:

• Shop A: $[500, 700)$: $6$ days; $[700, 900)$: $14$ days; $[900, 1,100)$: $12$ days; $[1.100, 1,300)$: $8$ days.

• Shop B: $[500, 700)$: $14$ days; $[700, 900)$: $4$ days; $[900, 1,100)$: $4$ days; $[1.100, 1,300)$: $18$ days.

1. a) Using the class midpoints, calculate the mean daily sales for each shop.

1. b) Calculate the standard deviation of each distribution (using class midpoints) and its coefficient of variation.

1. c) An investor wants to buy one of the two shops and is concerned about income stability. Which one would you recommend to them? Justify your answer using the calculated parameters.

In a small company, the mean monthly salary of the 10 employees is $1450$ € with a standard deviation of $120$ €. Due to good annual results, management decides to apply a linear pay rise to all workers consisting of a fixed amount of $50$ € and a variable amount of $3\%$ on the current basic salary ($y_i = 1.03 \times x_i + 50$).

1. a) Calculate what the new mean salary of the workforce will be.
2. b) Calculate the new standard deviation.
3. c) Has the coefficient of variation changed? Justify mathematically how these transformations affect the relative dispersion.

In a prestigious esports academy, the analyst team is studying the reaction time (in milliseconds) of 50 professional players. Upon reviewing the final report, they realise that a coffee stain has smudged part of the frequency table records. Only the following data remain: the arithmetic mean of the distribution is $\bar{x} = 182$ ms, the standard deviation is $\sigma = 12$ ms, and it is known that reaction time follows an approximately symmetric distribution. If a new player is added whose reaction time is $160$ ms, reason mathematically how the coefficient of variation of the original sample will change, and determine the exact value of the new variance after including this data point.

An urban bike rental service is reviewing its costs. The monthly maintenance cost $C$ per bicycle depends on the number of kilometres travelled $x$ according to the linear function $C(x) = 0.15x + 40$. Last month, the mean mileage of its fleet of 25 bicycles was $1200$ km with a variance of $14400$ km$^2$.

1. a) Calculate the mean monthly maintenance cost and its standard deviation.
2. b) Due to an increase in spare parts prices, the company decides to apply a fixed extra charge of $10$ € per bicycle, and additionally, the variable cost per kilometre increases by $20\%$. What will be the new coefficient of variation of the maintenance costs?

A set of 10 numerical data points has a mean of $\bar{x} = 15$ and a variance of $s^2 = 20$. Due to a transcription error, one of the data points which was $12$ was recorded as $22$.

1. a) Calculate the true mean of the dataset after correcting the error.
2. b) Calculate the exact value of the true variance of the set. (Hint: Use the relationship between the variance and the sum of the squares of the data, $\sum x_i^2$).