Exercises: One-dimensional descriptive statistics

In a study on the use of social networks in a high school, the following variables have been identified. Classify each of them as qualitative, discrete quantitative, or continuous quantitative:

1. a) Favorite social network (TikTok, Instagram, Twitter, etc.).
2. b) Number of followers on the main account.
3. c) Daily time (in minutes) spent using the cell phone.
4. d) Brand of the mobile device (Apple, Samsung, Xiaomi, etc.).
5. e) Numerical score obtained in the last Math exam.

A city council wants to know the opinion of teenagers about the new skate parks in the city. To do this, it randomly selects $200$ young people between $14$ and $18$ years old who attend the youth centers in the different neighborhoods.

Indicate what the population under study is and what the chosen sample is. Do you think the sample is representative if they only ask in the youth centers?

A group of $20$ 11th-grade students have been asked how many books they have read for pleasure during the last 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 that includes the absolute frequency ($n_i$), the relative frequency ($f_i$) and the cumulative absolute frequency ($N_i$).

Observe the following graph representing the favorite sport of a group of students in your grade:

From the bar chart:

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

The grades obtained by $7$ friends in an Economics assignment 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 grades.

In a video game competition, the times (in seconds) taken by participants to complete the first level have been recorded. As it is a continuous variable, the data has been grouped into intervals:

Time (s)	Frequency ($n_i$)
$[30, 40)$	$12$
$[40, 50)$	$25$
$[50, 60)$	$18$
$[60, 70)$	$5$

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

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

No. of siblings ($x_i$)	$N_i$
$0$	$6$
$1$	$18$
$2$	$26$
$3$	$30$

Answer the following questions, justifying 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 in which format their fans prefer to listen to their songs. They have created a pie chart with the results of $400$ surveys:

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

1. a) Calculate the absolute frequency of fans who prefer each format (knowing that $N=400$).
2. b) What is the relative frequency ($f_i$) of the streaming format?

A group of researchers has carried out a survey of 40 high school juniors on the number of weekly hours 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.
2. b) Create a frequency table that includes absolute, relative, cumulative, and relative cumulative frequencies.
3. c) Represent the data using a bar chart.

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

Duration (min)	$[0, 5)$	$[5, 10)$	$[10, 15)$	$[15, 20)$	$[20, 25)$
No. of calls	$12$	$18$	$10$	$7$	$3$

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

An entrance exam for a vocational program has been held and the scores obtained by the candidates are summarized in the following cumulative frequency table:

Score ($x_i$)	$3$	$4$	$5$	$6$	$7$	$8$	$9$
Cumulative Freq. ($N_i$)	$4$	$10$	$25$	$42$	$56$	$64$	$70$

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

Two basketball players, Ashley and Brandon, have scored the following amounts of points in their last 8 games:

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

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

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.

From the following box plot representing 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 ends 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 the answer.

In a small company, the mean salary of the 10 employees is $\$1450$ per month with a standard deviation of $\$120$. Due to the good results of the year, the management decides to apply a linear increase to all workers consisting of a fixed part of $\$50$ and a variable part of $3\%$ on the current base 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.

A study has been carried out on the number of children in a sample of families in a neighborhood. Some data from the frequency table have been lost:

No. of children ($x_i$)	$0$	$1$	$2$	$3$	$4$
Absolute freq. ($n_i$)	$8$	$a$	$12$	$b$	$2$
Relative freq. ($f_i$)	$0,20$	$0,35$	$c$	$d$	$0,05$

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 the median of the number of children.

We want to compare the homogeneity of the prices of lunch specials in two different areas of a city. The data obtained (in dollars) 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 the 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.

In a prestigious esports academy, the team of analysts studies the reaction time (in milliseconds) of 50 professional players. When reviewing the final report, they realize that a coffee stain has erased part of the records in the frequency table. Only the following data are preserved: the arithmetic mean of the distribution is $\bar{x} = 182$ ms, the standard deviation is $\sigma = 12$ ms, and it is known that the reaction time follows an approximately symmetric distribution. If it is decided to add a new player whose reaction time is $160$ ms, reason mathematically how the coefficient of variation of the original sample will vary and determine the exact value of the new variance after the addition of these data.

A courier company uses electric vans for its deliveries. The monthly maintenance cost $C$ per van depends on the number of miles driven $x$ according to the linear function $C(x) = 0,15x + 40$. In the last month, the mean mileage of its fleet of 25 vans was $1200$ miles with a variance of $14400$ miles$^2$.

1. a) Calculate the mean monthly maintenance cost and its standard deviation.
2. b) Due to a rise in the price of components, the company decides to apply a fixed surcharge of an extra $\$10$ to each van and, in addition, the variable cost per mile increases by $20\%$. What will be the new coefficient of variation of the maintenance costs?

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

Spending ($\$$)	$x_i$ (Midpoint)	$n_i$
$[0, 20)$	$10$	$8$
$[20, 40)$	$30$	$n_2$
$[40, 60)$	$50$	$n_3$
$[60, 80)$	$70$	$12$

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

A nutritionist analyzes the sugar content (in grams per 100 fl oz) of two brands of "zero" soft drinks, A and B. The results are summarized in the following box plots. Observing the graphs:

1. a) Which of the two brands presents a more dispersed distribution in the central $50\%$ of its data? Justify your answer by calculating the interquartile range visually.
2. b) If brand A presents 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 constant in its production? Use the appropriate statistical parameter to compare both brands.

In an archery competition, the scores of two archers, Emily and Sophia, are recorded after 40 shots. Emily has a mean score of $8,2$ with a standard deviation of $1,3$. Sophia presents the following cumulative frequencies of her scores: $25\%$ of her shots were $7$ points or less, $50\%$ were $8$ points or less, and $75\%$ were $9$ points or less.

1. a) Assuming Sophia'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 that at least $30\%$ of the shots have a score of $10$, and we know that Emily has obtained a $10$ on $12$ occasions, does Emily meet the requirement? Can it be known if Sophia meets it with the data from her quartiles?

A city council analyzes the water consumption per household using a relative frequency histogram. Due to the nature of the data, the intervals have different widths: $[0, 10)$, $[10, 20)$, $[20, 40)$, and $[40, 70)$ cubic feet per month. The heights of the rectangles of the histogram (frequency density) are $0,03$; $0,04$; $0,015$; and $0,0033$ respectively.

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

An investor analyzes 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 the 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?

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

1. a) Calculate the true mean of the data set 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, $\sum x_i^2$).