One-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 ($N_i$).

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 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.

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?

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.

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?

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 ($y_i = 1,03 \cdot x_i + 50$).

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.

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 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.

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, $\sum x_i^2$).

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.