Exercises: Statistics with technology and critical reading

The figure shows a spreadsheet with the ages of the members of a book club. In cells B11 and B12 the sheet shows the results of the formulas $=\text{AVERAGE}(B2:B9)$ and $=\text{MEDIAN}(B2:B9)$, respectively. The top bar shows the formula of cell B11, which is the selected cell.

1. a) What does the formula in cell B11, $=\text{AVERAGE}(B2:B9)$, calculate?

1. b) And the one in cell B12, $=\text{MEDIAN}(B2:B9)$?

1. c) Calculate both values by hand from the ages in B2 to B9 and check that they match the ones the sheet shows in B11 and B12.

A television channel showed this bar chart about the audience of two programmes, as shown in the figure.

1. a) At first glance, how many times more audience does programme B seem to have than A?

1. b) Read the actual values from the bars and calculate the true proportion between both audiences.

1. c) What trick does the graph use to exaggerate the difference? How should the vertical axis be designed so as not to be misleading?

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

A statistical calculator indicates that the mean of the $10$ marks of a group is $6.4$ and the median is $6.5$.

1. a) If the teacher decides to give $0.5$ extra marks to all students, what will the new mean and the new median be? Reason your answer without knowing the individual marks.

1. b) If instead they multiply all the marks by $1.1$, what happens to the mean and the median?

1. c) The sum of the $10$ marks was $64$. After checking, a mark of $4$ is corrected to $7$. What is the new mean?

For each of these data sets, choose the most suitable chart (bar chart, pie chart, line graph, frequency polygon or histogram) and justify your choice:

1. a) The percentage of time a teenager spends sleeping, studying, on leisure and on screens.

1. b) The month-by-month evolution of the average temperature of a city over a year.

1. c) The maths marks of a class of $28$ students.

1. d) The number of pets that the students in a year group have at home.

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

Lucas is using a spreadsheet to calculate the average mark of his five Mathematics exams. He has entered the marks in cells $B2$ to $B6$: $6$, $7$, $8$, $7$, and $7$. To organize himself, in cell $B7$ he calculates the sum of all the marks using the formula $=\text{SUM}(B2:B6)$, obtaining $35$. Then, in cell $B8$, he writes $=\text{AVERAGE}(B2:B7)$ hoping to obtain his final average mark, but the screen displays $11.67$, an impossible mark since it is out of $10$.

1. a) Identify the mistake Lucas made when defining the formula's range in $B8$ and explain with reasoning why the spreadsheet returns $11.67$.
2. b) Write the correct formula he should have entered in $B8$ to calculate the mean directly with the $\text{AVERAGE}$ function.
3. c) If Lucas wanted to use the sum value he already calculated in $B7$, what formula with basic arithmetic operations (without using the word $\text{AVERAGE}$) could he write in $B8$?

A municipality's newsletter publishes a bar chart to show how the number of registered members at the municipal sports centre has grown. In 2022 there were $500$ registered people, and in 2023 $520$ were recorded. However, the graphic designer decided to truncate the vertical axis and start it at $490$ instead of $0$. Thus, the 2022 bar has a graphical height of $10$ mm ($500 - 490$) and the 2023 bar measures $30$ mm ($520 - 490$).

1. a) Estimate the visual impression a resident gets by looking at the chart at first glance without reading the numbers on the axis: by how many times does the number of users appear to have multiplied?
2. b) Check the reality by calculating the real percentage increase in registrations from 2022 to 2023.
3. c) Critically assess whether the chart is misleading and explain what mandatory geometric modification the vertical axis should have so as not to distort the reader's perception.

The figure shows a spreadsheet with the points per match scored by a basketball player, as shown below.

1. a) What value will cell B10 show if it contains the formula $=\text{AVERAGE}(B2:B8)$? Calculate it.

1. b) Cell B11 contains $=\text{MAX}(B2:B8) - \text{MIN}(B2:B8)$. Which statistical parameter does it calculate and what is its value?

1. c) Write in your own words the formula that should be entered to obtain the number of matches in which she scored more than $15$ points.

A soft drinks brand published this pie chart about consumer preferences, as shown in the figure.

1. a) Add up the percentages of all the sectors. What result do you get? Is this possible in a pie chart?

1. b) Explain what might have happened in the survey to obtain these data and why a pie chart is not suitable to present them.

1. c) What type of chart would you choose to represent these percentages correctly?

The school council of a secondary school with $850$ students wants to decide which new extracurricular activity to offer. A student surveys $40$ students outside the gym and concludes: "$70\ \%$ of students want more sport, so we need to set up a climbing club".

1. a) What is the sample size? What is the population?

1. b) What flaw is there in the way the sample was chosen? How does it affect the conclusion?

1. c) Propose a better way to select the $40$ survey respondents and explain why the conclusion would be more reliable.

1. d) What percentage of the population does the sample represent? Express it to two decimal places.

A coach uses an app to compare the minutes played by two basketball teams across $10$ matches. The app shows these results:

• North Team: mean $= 18.4$ min, standard deviation $= 7.9$ min.

• South Team: mean $= 17.8$ min, standard deviation $= 2.1$ min.

1. a) Which team distributes the minutes more evenly among its players? Which parameter tells you this?

1. b) If the North coach wants a specific player to play a lot, is it predictable that they will manage it? What about in the South?

1. c) The North team's mean is slightly higher. Does that mean that all of its players play more than those in the South? Reason your answer.

A small local IT services company has seven employees on its staff. In a spreadsheet, their monthly net salaries (in euros) have been recorded in cells $B2$ to $B8$: $1\,250$, $1\,300$, $1\,300$, $1\,350$, $1\,400$, $1\,450$, and $5\,950$ (the latter corresponds to the company's management). To publish a job vacancy, the communications department must decide which statistical parameter to include as the team's "typical salary".

1. a) Calculate the result that the formula $=\text{AVERAGE}(B2:B8)$ would provide and the one that $=\text{MEDIAN}(B2:B8)$ would yield.
2. b) Choose which of the two methods is more representative to honestly inform someone applying for an entry-level position at the company, and justify your choice by analyzing the impact of the highest salary.
3. c) If the company ultimately decides to publish the mean value instead of the median, what persuasive intent do you think it is pursuing, and which rule of critical data reading is being violated?

In a school spreadsheet, cells $B2$ to $B6$ show the weekly hours that five students from a class spend doing sports. By mistake, the data item corresponding to cell $B6$ has been deleted. For the other four cells, the values are known: $3$, $4$, $6$, and $8$ hours.

1. a) Knowing that cell $B7$ contains the formula $=\text{AVERAGE}(B2:B6)$ and returns exactly a value of $6$ hours, reconstruct the numerical value that must appear in the empty cell $B6$.
2. b) With the data item for $B6$ now recovered, calculate the value that the formula $=\text{MEDIAN}(B2:B6)$ would return.
3. c) Imagine that the student in cell $B6$ competes at a professional level and actually trains $25$ hours per week instead of the value you found. Without recalculating everything, explain whether the median value would change compared to the previous section and justify why the median is a more robust parameter against outliers than the mean.

Decision-making task. An online newspaper publishes the headline "Teenagers' spending on video games soars!" alongside the graph in the figure, as shown below.

1. a) Read the actual values for annual spending from the graph and calculate the true percentage increase between the first and last year.

1. b) Why does the graph give a much more alarming impression than the data? Point out at least two design decisions that cause this.

1. c) Rewrite the headline honestly using the data.

1. d) What questions would you ask about the survey sample before accepting even that moderate increase?

To calculate the standard deviation of the data $4$, $6$, $7$, $7$, $9$, $11$ using a spreadsheet, a table is constructed with auxiliary columns:

Data $x_i$	$x_i - \bar{x}$	$(x_i - \bar{x})^2$
$4$
$6$
$7$
$7$
$9$
$11$

1. a) Calculate the mean $\bar{x}$ and complete the table in your exercise book.

1. b) What spreadsheet formula would give the value of the cell in the second column for data $4$? And for the third column?

1. c) Calculate the mean of the squares in the third column (the variance) and, using a calculator, find the square root to obtain the standard deviation.

1. d) The sum of the second column is $0$. Is this a coincidence or would it happen with any data set? Justify your answer.

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

An athletics club needs to sign a relay runner and has the times of two candidates in their last $8$ $100\ \text{m}$ races:

• Candidate 1: mean $= 12.40\ \text{s}$, standard deviation $= 0.45\ \text{s}$.

• Candidate 2: mean $= 12.55\ \text{s}$, standard deviation $= 0.10\ \text{s}$.

1. a) The coach needs a runner who will most reliably beat $12.9\ \text{s}$ in the championship, because a bad time will eliminate the team. Who should they sign? Reason your answer using the mean and standard deviation.

1. b) If instead they were looking for the runner capable of achieving the best time of the day, would your choice change? Why?

1. c) Calculate the coefficient of variation for each candidate, $CV = \frac{\sigma}{\bar{x}}$, and explain what information it adds compared to the standard deviation.

State whether each statement about the use of technology in statistics is true or false and justify your answer:

1. a) "The mean calculated using a spreadsheet from data grouped into intervals (using class midpoints) is exactly the same as the one calculated using the original data."

1. b) "If a statistical chart is produced with computer software, it cannot be misleading."

1. c) "A scientific calculator can provide the standard deviation even if we do not understand the formula, but interpreting it remains our job."

1. d) "To compare the dispersion of two data sets with very different means, it is better to use the standard deviation than the coefficient of variation."

A news website publishes the following headline: "Health alert: eating watermelon in summer triggers hospital admissions for heatstroke". To justify this, they display the following table with genuine data from a district over six months, entered in a spreadsheet in columns $A$ (Month), $B$ (Tons of watermelon sold), and $C$ (Heatstroke cases treated):

• May: $15$ t of watermelon and $2$ heatstrokes.

• June: $60$ t of watermelon and $16$ heatstrokes.

• July: $150$ t of watermelon and $45$ heatstrokes.

• August: $170$ t of watermelon and $52$ heatstrokes.

• September: $45$ t of watermelon and $8$ heatstrokes.

• October: $10$ t of watermelon and $0$ heatstrokes.

1. a) Model the observed relationship between both variables: describe whether there is a positive, negative, or zero correlation between watermelon consumption and heatstroke cases, justifying it using the data from the table.
2. b) Apply the critical reading criterion "correlation does not imply causation" to debunk the article's headline. What is the underlying confounding variable that explains why both data series rise and fall together?
3. c) In cell $C7$, there is a $0$ corresponding to the month of October. If someone were to delete that number, leaving the cell completely empty (blank), how would this affect the result returned by the formula $=\text{AVERAGE}(C2:C7)$? Explain the technical difference between how a spreadsheet processes an empty cell and a cell with the number $0$ when calculating the mean.

A soft drink brand presents an advertising campaign under the slogan: "$80\,\%$ of consumers prefer our brand over the leading competitor". When reviewing the technical report supporting the advertisement, a consumers' association discovers three questionable methodological decisions:

1. The total sample consisted of $20$ customers selected as they were leaving a concert sponsored by the brand itself (of which $16$ said "Yes" to their soft drink and $4$ said "No").

2. The advertisement includes a pie chart where the "Yes" sector is drawn with a central angle of $324^\circ$ and the "No" sector with $36^\circ$.

3. The brand claims that if they manage to convince $5$ more people to vote favorably in a second survey with $25$ people in total ($21$ affirmative votes out of $25$), the approval percentage will have grown by exactly $25\,\%$ compared to the initial sample.

1. a) Detect the error made in the construction of the pie chart: calculate the central angle that would actually correspond to $80\,\%$ and explain what misleading visual effect is caused by the angle drawn by the brand.
2. b) Formulate a reasoned critique of the sample size and the selection procedure: why can these results not be extrapolated to the whole population of consumers in Ireland?
3. c) Correct the calculation error in the technical report: calculate the actual percentage of support if there were $21$ favorable opinions out of $25$ respondents, and explain why it is false that support has increased by $25\,\%$, clarifying the difference between percentage points and percentage increase.

Topic challenge. Is the weather over these two weeks representative in your town or city? Become a data analyst and produce a critical meteorological report.

1. Data collection and classification: Go to the Met Éireann website and record the daily maximum temperature of your county over the last $15$ days. Identify the type of statistical variable with reasons and create its complete frequency table in a spreadsheet (or GeoGebra).

2. Parameters and chart: Using spreadsheet formulas, obtain the mean ($\bar{x}$), the median ($Me$), the range and the standard deviation ($\sigma$). With the same tool, generate the most appropriate statistical chart, properly labelling the axes and units.

3. Critical reading and conclusions: Write a brief report (one page) analysing whether the mean faithfully describes thermal behaviour or whether a high standard deviation (for example, greater than $3.5$ °C) indicates too much dispersion. Reflect: could a media outlet manipulate the perception of the climate by using only one of these parameters?

Evaluation criteria:

• Rigour in classifying the variable and correct creation of the frequency table.

• Precision in the calculated parameters and technical quality of the chart in a spreadsheet or GeoGebra.

• Depth of critical reading regarding the representativeness of the measures and dispersion.