Exercises: Statistical parameters

To carry out a study on the consumption habits of teenagers on a social network, the profiles of $50$ students from a secondary school were analysed. Classify the following statistical variables obtained in the study as qualitative or quantitative (indicating whether they are discrete or continuous):

• a) Brand of mobile phone used.

• b) Number of "likes" received on the latest post.

• c) Total time (in minutes) spent using the app during the day.

• d) Main topic of the videos watched (sports, comedy, music, etc.).

A group of friends recorded the number of unread WhatsApp messages they have at the end of the day. The data obtained is:

Calculate the mode of this distribution and briefly explain what this value represents in the context of the exercise.

The following table shows the number of books read by the students in a $2º$ ESO class during the first term:

Calculate the arithmetic mean of the number of books read per student.

Seven students timed how long it takes them (in seconds) to solve a Rubik's Cube. The recorded times are:

Calculate the median of these times. Remember to order the data first.

In a survey about the preferred ice cream flavour at the school canteen, the results shown in the following graph were obtained:

From the bar chart, identify the mode and calculate the range of the frequencies of the flavours.

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

The height (in centimetres) of the six starting players of a school basketball team was measured:

Calculate the median height. As there is an even number of values, remember how to proceed with the two central values.

A young person saves different amounts of money weekly. Over the last $5$ weeks, his savings have been:

• a) Calculate the mean of his savings.

• b) Calculate the median.

• c) Which of the two measures do you think best represents the "normal" savings of this young person? Justify your answer by mentioning the effect of the value $54€$.

During a spring week, the minimum temperature recorded in a city was $14\ ^\circ\text{C}$ (Monday) and the maximum was $22\ ^\circ\text{C}$ (Saturday). Calculate the range of temperatures during that period.

The maximum temperatures recorded during a week in April in Seville were: $22$, $25$, $19$, $24$, $27$, $26$ and $21$ °C. Calculate the range of the temperatures for that week.

The ages of the members of a robotics club are: $12$, $13$, $12$, $14$, $13$, $12$ and $15$ years old. Calculate the mean, the median and the range of the ages.

The marks in Salma's last $5$ English exams are: $6$, $7$, $8$, $5$ and $9$.

a) Calculate the mean mark.

b) Calculate the deviation of each mark from the mean (in absolute value) and the mean deviation.

In two $2.º$ ESO classes, the time taken by the students to get to school was recorded. In $2.º$ A, the times range from $5$ to $40$ minutes, and in $2.º$ B, they range from $15$ to $25$ minutes.

a) Calculate the range of the times for each class.

b) In which class are the times most similar to each other? Why?

Lucas has recorded the goals scored by his futsal team in the last $6$ matches of the school league: $1$, $5$, $0$, $4$, $2$, $6$.

To analyze the team's performance, Lucas writes in his notebook:

"Since there are $6$ data values, the two middle values are $0$ and $4$. Therefore, the median is $(0 + 4) / 2 = 2$ goals. Furthermore, the range is $6 - 1 = 5$ goals because $0$ does not count as a goal scored."

a) Find and explain with reasoning the two conceptual mistakes Lucas made.

b) Correct his work by accurately calculating the arithmetic mean, the median, and the range of the team's goals.

Write a set of $5$ integers (representing the number of books read during the past school year by $5$ students in $2.º$ ESO) that simultaneously satisfies all of these conditions:

1. The mode is $3$.

2. The median is $4$.

3. The arithmetic mean is exactly $5$.

4. The range of the data is $6$.

Explain in detail the reasoning you followed to deduce each of the five values and justify whether there is a unique possible combination of numbers or if there could be several different solutions.

A group of friends recorded the number of hours per week they spend playing online video games. The data obtained is as follows:

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

Calculate the measures of central tendency: the arithmetic mean ($\bar{x}$), the median ($Me$), and the mode ($Mo$).

A health app has recorded the daily steps (in thousands) of a teenager over the last two weeks:

$8, 10, 12, 8, 9, 11, 15, 8, 10, 12, 14, 8, 9, 10$

Represent these data in a frequency table and calculate:

1. a) The range of the distribution.
2. b) The mean deviation ($DM$) from the arithmetic mean.

In a basketball training session, the number of three-pointers scored by each player in 25 attempts was recorded. The results are shown in the following graph:

From the bar chart:

1. a) Draw up the absolute frequency table.
2. b) Calculate the mean, median, and mode of the three-pointers scored.

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

A study was carried out on the battery life (in hours) of two different smartphone models, $A$ and $B$, conducting 10 tests with each.

Model $A$: $14, 15, 16, 14, 15, 16, 15, 14, 15, 16$

Model $B$: $10, 20, 15, 12, 18, 11, 19, 15, 14, 16$

1. a) Calculate the mean of both models. What do you notice?
2. b) Calculate the range and mean deviation of each model.
3. c) If you had to buy one based on battery life reliability, which one would you choose and why?

Below is a box plot representing the marks obtained by the students of a class in a resit exam:

Observe the graph and identify the following measures of position:

1. a) Minimum value and maximum value.
2. b) First quartile ($Q_1$), median ($Me$), and third quartile ($Q_3$).
3. c) What percentage of students obtained a mark higher than $8$?

Aisha has taken 4 Mathematics exams this term and her marks were: $6,5$; $8,2$; $7,4$ and $9,1$.

1. a) What is her current mean mark?
2. b) If there is still one final exam left, what mark must Aisha get so that her final mean is exactly $8,0$?

In a survey about the favourite colour of Year 9 students, the following results were obtained:

1. a) What type of statistical variable is being studied?
2. b) Calculate, if possible, the mean, the median, and the mode. If any parameter cannot be calculated, explain why.

Calculate the arithmetic mean and mode of the following distribution of data grouped in intervals, which correspond to the weight (in kg) of the members of a judo team:

The goals scored by a futsal team in their last $5$ matches were: $2$, $5$, $1$, $4$, and $3$.

a) Calculate the mean number of goals per match.

b) Calculate the variance and standard deviation, rounding to two decimal places.

Two basketball players scored in their last $6$ matches:

• Player A: $7$, $8$, $8$, $9$, $6$, $10$ points

• Player B: $2$, $14$, $5$, $11$, $9$, $7$ points

a) Check that both players have the same mean number of points.

b) Calculate the range of points scored by each player.

c) The coach needs a player with consistent performance for the next match. Which one would you choose? Why?

Two athletes train for the $100$ metres sprint and record their times with a smartwatch. Their last $5$ times, in seconds, are:

• Mei: $13,2$; $13,0$; $13,1$; $13,2$; $13,0$

• Bea: $12,6$; $13,8$; $12,9$; $13,4$; $12,8$

a) Calculate the mean and range of times for each athlete.

b) Calculate the standard deviation for each athlete, rounding to two decimal places.

c) The coach must choose one of the two for the final and is looking for consistency. Who would you choose? Justify your answer.

The marks of $11$ students in an exam were: $5$, $7$, $3$, $8$, $6$, $9$, $4$, $7$, $5$, $6$, and $10$.

a) Order the marks and calculate the median.

b) Calculate the first and third quartiles.

c) Between which values is approximately the central $50\%$ of the marks concentrated?

The dot plots show the points scored by two players in their last $6$ matches.

a) Calculate the mean number of points for each player.

b) Calculate the range for each player.

c) Looking at the graphs and calculations, which of the two is more consistent? Explain what that means.

To study the dispersion of his marks ($5$, $6$, $7$, and $8$), a student subtracts the mean from each value and gets: $-1,5$; $-0,5$; $0,5$ and $1,5$. Since the sum of those differences is $0$, he claims: "The standard deviation is $0$, so my marks do not spread at all."

a) Explain the mistake he made in his reasoning.

b) Correctly calculate the standard deviation of the four marks, rounding to two decimal places.

Two basketball players on a cadet team, Lucía and Marta, scored the following points in their last $5$ games:

• Lucía: $14$, $15$, $16$, $17$, $18$

• Marta: $8$, $12$, $16$, $20$, $24$

a) Quickly verify that both players have exactly the same arithmetic mean of points.

b) Without calculating the mean deviation ($DM$) yet, observe the distribution of the data around the mean and reasonably estimate which of the two players will have a larger $DM$. Approximately what do you think the $DM$ of each will be?

c) Accurately calculate the range and the mean deviation ($DM$) for both players.

d) Compare your results with your previous estimation and explain which of the two players is more consistent. Why is the mean alone not enough to judge an athlete's consistency in sports?

Based on meteorological data recorded by AEMET during a spring week, the following daily maximum temperatures (in $^\circ\text{C}$) were obtained in two Spanish cities:

• Santander: $15$, $17$, $16$, $18$, $14$

• Córdoba: $12$, $29$, $15$, $30$, $24$

a) Calculate the arithmetic mean, median, and range of the maximum temperatures for each of the two cities.

b) Calculate the mean deviation ($DM$) of the temperatures for both locations.

c) Interpret the obtained parameters: if a person has to pack a suitcase to travel that week and is only informed of the average temperature, in which of the two cities are they more likely to experience unexpected cold or heat? Justify your answer by indicating which statistical dispersion parameter best describes this situation.

A well-known streamer is analysing the views on their last 5 videos. The mean number of views for the first 4 videos is $12\,500$ people. If they want the total mean of the 5 videos to rise to $15\,000$ views to attract a new sponsor, how many views must they get, as a minimum, on their fifth video? Explain the process, reasoning whether it is possible for the median number of views to be greater than the mean in this case.

In an esports competition, two teams, "Team Alpha" and "Team Beta", obtained the following response times (in milliseconds) in a reflexes test. The data is presented in the following box plots:

Based on the graphs:

1. a) Which team has a lower median response time?
2. b) Which of the two teams is more consistent in their reflexes? Justify your answer using the interquartile range.
3. c) If winning the competition depends on never exceeding $200$ ms, which team has better prospects?

A group of 10 friends has a mean age of exactly $14$ years. When the older brother of one of them joins the group, the mean increases to $15$ years.

1. a) Calculate the age of the older brother.
2. b) If a girl aged $10$ then joins, what will be the new mean of the group of 12 people?
3. c) Is it possible that, after the girl joins, the median of the group remains $14$? Give an example of the ages of the 10 original friends that satisfies this condition.

A survey was conducted on the daily time (in minutes) that teenagers at a high school spend using social media. The data have been grouped in the following table:

Knowing that the mean usage time is $57$ minutes:

1. a) Calculate the value of $x$ (absolute frequency of the interval $[60, 90)$).
2. b) Determine the mode (modal interval) and the median (median interval) of the distribution.
3. c) What percentage of students spend $90$ minutes or more on social media?

Two basketball players, Lucía and María, scored the following points in their last 6 matches:

• Lucía: $18, 22, 20, 19, 21, 20$

• María: $10, 35, 5, 40, 12, 18$

1. a) Calculate the arithmetic mean of points for each player.
2. b) Calculate the range and mean deviation for both players.
3. c) If the coach is looking for a reliable player who ensures a consistent number of points, even if it is not very high, who should he choose? Justify your answer by comparing their dispersion.

The following histogram shows the distribution of marks obtained in a resit exam by 40 students. However, the frequency axis has been partially erased.

Knowing that the total number of students is $40$ and that the heights of the bars are proportional to the frequencies:

1. a) Determine the absolute frequency of each interval.
2. b) Calculate the estimated mean mark using class marks.
3. c) If a mark of $5$ is needed to pass and we assume that marks are spread uniformly within each interval, what estimated percentage of students failed the exam?

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

There is a set of 5 positive integers. It is known that:

• The mean is $10$.

• The median is $12$.

• The mode is $15$ and is unique.

• The range is $12$.

Find the 5 numbers that form this set. Explain the logical steps you followed to work them out.

In an app development company, the salaries of the 10 employees are as follows: eight employees earn $1\,500\ €$ per month, the deputy director earns $4\,000\ €$, and the director earns $7\,000\ €$.

1. a) Calculate the mean, median, and mode of the salaries.
2. b) The director claims: "The mean salary in our company is $2\,300\ €$, we are very generous." An employee responds: "Most of us only earn $1\,500\ €$." Explain, using the calculated parameters, why both are telling the truth but giving such different impressions.
3. c) Which of the three statistical parameters best represents the reality of what a typical worker at that company earns? Justify your choice.

Iker compares two transport apps to go to school. On his last $8$ trips, the waiting times in minutes were:

• App A: $4$, $6$, $5$, $7$, $5$, $6$, $4$, $7$

• App B: $2$, $9$, $3$, $10$, $4$, $8$, $3$, $5$

a) Calculate the mean waiting time for each app.

b) Calculate the standard deviation of each, rounding to two decimal places.

c) If Iker has exactly $10$ minutes available each morning, which app would you recommend to him? Reason your answer using the mean and dispersion.

The graphs show the points scored by two basketball teams in their last $6$ matches.

a) Calculate the mean number of points for each team.

b) Calculate the range and standard deviation for each team, rounding to two decimal places.

c) A commentator states: "Both teams perform equally well, because they have the same mean." Do you agree? Argue your answer using the measures of dispersion.

The heights of $5$ friends are: $160$, $165$, $170$, $168$, and $162$ cm.

a) Calculate the mean and range of the heights.

b) A sixth friend joins the group and the mean does not change. How tall is the new friend?

c) Write down two possible heights for the new friend that would have increased the range, and calculate the new range in each case.

The number of books read this term by the $20$ students in a class is given by this table:

a) Calculate the mean number of books read.

b) Calculate the variance and standard deviation, rounding to two decimal places.

State whether these statements are true or false, giving reasons and providing examples where appropriate:

a) If two data sets have the same mean, they have the same dispersion.

b) The standard deviation cannot be negative.

c) If all the values in a data set are equal, its standard deviation is $0$.

d) If we add a value equal to the mean to a data set, the range does not change.

The table shows the monthly mean temperatures (in °C) for the past year in two coastal cities:

a) Calculate the annual mean temperature of each city.

b) Calculate the temperature range for each city.

c) A family wants to move to a city "without extreme temperatures". Which one would you recommend to them? Justify your answer using the calculated data.

In a small software development company in Spain, $8$ people work. Their monthly net salaries (in euros) are as follows:

The last salary corresponds to the CEO and founder of the company. Management wants to hire a new programmer and draft a job offer indicating the "representative salary" of the workforce.

a) Calculate the arithmetic mean, median, and mode of the employees' salaries.

b) Management proposes publishing the arithmetic mean in the offer, while the union representative demands publishing the median. Compare both methods and critically justify which of the two more accurately reflects the typical salary of the majority of workers.

c) If the director decides to raise her salary next month to $8\,500\text{ \euro}$ while the other salaries remain unchanged, reason without calculating which parameter (between the mean and the median) will change and which will remain unchanged. What fundamental property of the median does this situation demonstrate?

In a clinical trial conducted in a Spanish hospital, response times (in milliseconds) to a stimulus were measured in a group of $5$ patients, yielding the following records:

a) Calculate the arithmetic mean $\bar{x}$, range, and mean deviation ($DM$) of this initial group of patients.

b) A sixth patient is to be added to the trial. What exact value must their time register for the group's arithmetic mean to remain unchanged at $220\text{ ms}$? Justify how you deduce it.

c) If we add that sixth patient with the value obtained in part b), reason whether the new mean deviation will increase, decrease, or remain the same compared to the initial one before doing the calculation. Then, calculate the new $DM$ to verify your prediction.

d) Suppose now that, starting from the original $5$ patients, we want to add a sixth patient such that the range of the times increases to exactly $35\text{ ms}$. Deduce the two possible values that this sixth patient's time could have for this to happen.

Unit Challenge. How has the maximum temperature in your province varied over the last two weeks? Become a climate analyst to prepare a real meteorological report.

1. Data collection: Access the AEMET website (or official meteorological historical data) and record the daily maximum temperature ($T$) of your locality over the last $14$ days. Precisely define the population, the sample, and the type of statistical variable.

2. Organization: In a spreadsheet (or GeoGebra), build a complete frequency table: absolute frequencies, relative frequencies, percentages, and cumulative frequencies.

3. Statistical analysis: Generate the most representative chart (bar chart or frequency polygon) and calculate the measures of central tendency and dispersion: mean ($\bar{x}$), median ($Me$), mode ($Mo$), and range. Express decimals with a comma (for example, $18,5$ °C).

4. Final report: Write a brief reasoned report (half a page) where you analyze whether the mean is representative in the presence of extreme values and propose a practical energy-saving measure for households based on the data obtained.

Assessment criteria:

• Rigor in defining concepts and accuracy in the frequency table.

• Correct calculation of statistical parameters ($\bar{x}$, $Me$, $Mo$, range) and relevance of the chart.

• Critical thinking ability to interpret the results and justify the final proposal.