Exercises: Measures of central tendency and spread

The ages of Noa's cousins are: $8$, $10$, $12$, $10$ and $10$ years old. Calculate the mean of their ages.

The points scored by a player in her last $7$ matches are: $9$, $14$, $6$, $11$, $8$, $12$ and $10$.

1. a) Order the data from smallest to largest.
2. b) What is the median?

The shoe sizes worn by the $11$ players on a futsal team are: $38$, $39$, $40$, $39$, $41$, $39$, $40$, $42$, $39$, $38$ and $39$. What is the mode? What does this datum mean for the kit manager?

A weather app recorded these minimum temperatures (in °C) during a week in January: $2$, $-1$, $0$, $-3$, $1$, $4$ and $-2$. Calculate the range of the minimum temperatures for that week.

The dot plot shows the goals scored by a team in its recent matches.

1. a) How many matches has the team played?
2. b) In how many matches did it score no goals?
3. c) What is the mode of the number of goals? Is there only one mode?
4. d) What is the range?

In a school chess tournament, Álex achieved these points in the $5$ rounds: $3$, $5$, $4$, $5$ and $3$. Calculate his mean points per round.

The table shows the number of books read this term by the students in a class:

1. a) What is the mode of the number of books read?
2. b) How many students are there in the class?
3. c) How many students have read more than $1$ book?

The times (in seconds) of $6$ swimmers in a $25$ metre race are: $31$, $28$, $35$, $29$, $33$ and $30$. Let's calculate their median step by step:

1. a) Order the times from smallest to largest.
2. b) Since there is an even number of data values, what are the two middle values?
3. c) The median is the mean of those two values: calculate it.

To calculate the median and the range of the number of books read during the past year by a group of $5$ friends ($4$, $1$, $5$, $2$, and $8$), Cian wrote the following in his notebook:

"The middle value of the list just as they gave it to us is $5$, so the median is $Me = 5$. The range is $8 - 4 = 4$, because I subtracted the first value from the last one".

1. a) Find the two mistakes made by Cian and explain in your own words why his reasoning is not correct in either case.
2. b) Correct Cian's calculations and determine the correct values of the median and the range.

Met Éireann has recorded the following maximum temperatures (in $^\circ\text{C}$) during a week in April in Galway: $14$, $16$, $15$, $19$, $17$, $13$, and $18$.

1. a) Before doing any written numerical calculations, observe the data and estimate between which two integer values the arithmetic mean must lie. Do you think it will be closer to $14\text{ }^\circ\text{C}$ or to $17\text{ }^\circ\text{C}$? Justify your estimate.
2. b) Calculate precisely the arithmetic mean, the median, and the range of the recorded maximum temperatures.
3. c) Check whether the exact value of the mean confirms your initial estimate and argue whether the result obtained is reasonable considering the range of the week.

The $30$ students in a class were asked how many times they visited the local sports centre in the last month. The results are:

1. a) Calculate the arithmetic mean of the number of visits.
2. b) What is the mode? What does it mean in this context?

An online shop compares the prices of the same trainers across $8$ different websites and finds these values, in euros: $42$, $38$, $45$, $39$, $50$, $36$, $41$ and $44$.

1. a) Order the prices from lowest to highest.
2. b) Calculate the median of the prices. Remember that with an even number of data values there are two middle values.
3. c) What is the range? Interpret it: is there a large difference in price between websites?

In a school target-shooting competition with a ball, each participant shoots $10$ times and the hits are recorded. The table collects the results of all participants:

1. a) How many participants were there in total?
2. b) What is the mode of the number of hits?
3. c) Calculate the range of the distribution.
4. d) Calculate the mean number of hits per participant, rounding to one decimal place.

The graph shows the maximum temperatures recorded in a city during a spring week.

1. a) Calculate the mean maximum temperature for the week, rounding to one decimal place.
2. b) Calculate the temperature range for the week (the difference between the maximum and minimum).
3. c) Which day of the week was furthest from the mean? Justify your answer.

The bar chart shows the marks from the last maths exam of a class.

1. a) How many students took the exam?
2. b) What is the mode of the marks?
3. c) Calculate the median of the marks.
4. d) Calculate the arithmetic mean, rounding to two decimal places. Does the class pass "on average"?

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

In a subject, the term mark is calculated as follows: the mean of the three assignments counts for $40\%$ of the final mark and the exam counts for $60\%$. Cian's assignments were $7$, $8$ and $6$, and in the exam he got $7.5$.

1. a) Calculate the mean of the three assignments.
2. b) Calculate his final term mark by applying the percentages.
3. c) Would he have passed if the exam counted for $50\%$ and the assignments for the remaining $50\%$? Check your answer.

The ages of the $6$ members of a group of friends are: $11$, $12$, $11$, $13$, $12$ and $12$ years old.

1. a) Calculate the mean and the median of the ages.
2. b) The older brother of one of them, who is $24$ years old, starts hanging out with the group. Calculate the new mean and the new median.
3. c) Which of the two measures changed more? Which describes the "typical age" of the group better after the change? Reason your answer.

$40$ students were asked for their favourite type of film and the results were: adventure, $12$ votes; comedy, $12$ votes; animation, $9$ votes; science fiction, $5$ votes; documentary, $2$ votes.

1. a) What is the mode of this variable?
2. b) Does it make sense to calculate the mean or the median of these data? Explain why.
3. c) When a variable has two values with the same maximum frequency, we say it is bimodal. Is this the case here? Justify your answer.

In a school basketball tournament in Cork, your school team has played $5$ games. The following statistical parameters about their scores are known:

• The arithmetic mean of points is exactly $46$.

• The mode is $52$ points (and it is the only value that repeats).

• The median is $46$ points.

• The range of the scores is $14$ points.

1. a) Deduce the exact points scored by the team in each of the $5$ games and write them in order from least to greatest. Explain step by step how you deduced each value from the statistical clues.
2. b) If in a sixth game the team scores $58$ points, what will happen to the range and to the season mean? Reason whether they will increase, decrease, or remain the same without needing to calculate their new numerical values.

The environmental department of a municipality conducted a survey of $20$ families in a neighbourhood about the number of times they visit the green glass recycling container over the course of a month. The responses were:

$1$, $4$, $2$, $3$, $2$, $1$, $5$, $2$, $3$, $4$, $2$, $6$, $3$, $2$, $1$, $2$, $4$, $3$, $2$, and $8$.

1. a) Create an absolute frequency table to organise the data and calculate the mode, median, and range of the visits.
2. b) Using the frequency table, calculate the arithmetic mean of visits to the green container per family and interpret the meaning of the result.
3. c) In the municipal newsletter, a councillor published: "Most families in the neighbourhood visit the glass recycling container $8$ times a month, which is why the container fills up quickly". Use the calculated measures of central tendency to argue why the councillor's statement is wrong and what statistical concept she has confused with reality.

The daily screen time hours of $9$ students in a class are: $2$, $3$, $3$, $4$, $4$, $5$, $5$, $6$ and $12$.

1. a) Calculate the mean and the median of these data.
2. b) One of the data values is much larger than the others. Which of the two measures above is affected more by it? Reason your answer.
3. c) If the data value of $12$ hours was a survey error and was actually $2$ hours, recalculate the mean and the median and explain what has changed.

State whether the following statements are true or false, reasoning your answer or providing a counterexample:

1. a) In a survey about a class's favourite colour, neither the mean nor the median can be calculated.
2. b) If we add $2$ to all data values in a distribution, the mean increases by $2$ but the range does not change.
3. c) The mode is always the largest value in the distribution.
4. d) To calculate the median, it does not matter if the data are unordered.

Aoife has achieved these marks in her first four maths tests: $5.5$; $6$; $7$ and $6.5$. Only one test remains to finish the term, and she wants her mean mark to be exactly $6.5$.

1. a) What mark does she need to get in the final test?
2. b) What if she were happy with a mean of $6$?
3. c) Could she reach a mean of $7$? Justify your answer.

In a local basketball league, the table shows the goals scored by a team in the matches played, but one frequency has been rubbed out:

Given that the mean number of goals per match is exactly $1.4$:

1. a) Set up the equation that allows you to find $x$ and solve it.
2. b) How many matches has the team played in total?
3. c) With the complete data, what is the mode of the distribution?

To calculate the mean number of siblings in her class, Aitana has this table:

Aitana does the following calculation: $(0 + 1 + 2 + 3) : 4 = 1.5$ and claims: "the mean is $1.5$ siblings".

1. a) Explain what mistake Aitana has made.
2. b) Calculate the mean correctly.
3. c) Aitana also claims that "the median is $1.5$ because it is between $1$ and $2$". Is she right? Calculate the median by ordering the $30$ data values and check it.

Two handball teams scored these points in their last $5$ matches:

• Team Alpha: $60$, $62$, $58$, $61$, $59$.

• Team Beta: $45$, $80$, $52$, $70$, $53$.

1. a) Calculate the mean number of points for each team.
2. b) Calculate the range for each team.
3. c) If you were the opposing coach, which team seems more consistent to you? And more unpredictable? Reason your answers using the calculated measures.

In the school charity run, the $12$ runners from the 1st Year team took these times (in minutes) to complete the $5$ km course: $24$, $26$, $25$, $31$, $28$, $26$, $30$, $27$, $25$, $26$, $29$ and $33$.

1. a) Calculate the mean, median, mode and range of the times.
2. b) The coach says a time is "good" if it is below the team mean. How many runners achieved a "good" time?
3. c) Next year, the slowest runner promises to improve their time by $5$ minutes. If they succeed and the others repeat their time, how do the mean and median change? Work it out.

The ages of the $5$ members of a book club are: $10$, $14$, $15$, $21$ and $23$ years old.

1. a) Calculate the current mean and median.
2. b) A new member joins and the median becomes $16$ years old. How old is the new member?
3. c) Instead of that, if the club wanted the mean to increase by exactly $1$ year when a sixth person joins, how old should this person be?

A tech start-up in Galway has a staff of $8$ people whose gross monthly salaries are: five junior developers who earn $1250\text{ €}$ each, two senior developers who earn $1750\text{ €}$ each, and the executive director, who earns $6250\text{ €}$. To post a job opening, the Human Resources department discusses how to communicate the typical salary and considers two options:

• Method A: Report the mean salary (arithmetic mean).

• Method B: Report the median salary (median).

1. a) Calculate the mean salary, the median salary, and the salary range of the company.
2. b) Compare both methods and justify which of the two represents the salary of an average worker in that company in a fairer and more realistic way. Explain what distorting effect the director's salary has on the arithmetic mean.
3. c) Imagine that the company decides to dispense with both the mean and the median and solely publish that "the salary range is $5000\text{ €}$". Would this information be sufficient for a candidate to know approximately how much they will earn? Justify what the range provides and why it does not replace measures of central tendency.

Athletics Ireland must select a representative for an international competition in the $100\text{ m}$ hurdles event. The timed performances (in seconds) of two athletes, Aoife and Niamh, in their last $6$ races are analysed:

• Aoife's times: $13.1$; $13.2$; $13.3$; $13.3$; $13.4$, and $13.5$.

• Niamh's times: $12.6$; $12.8$; $13.1$; $13.5$; $13.8$, and $14.0$.

1. a) Calculate the arithmetic mean, median, and range of the times for each of the two athletes.
2. b) If a sports analyst looked exclusively at the mean and the median, could they notice any difference between the overall performance of Aoife and that of Niamh? Explain what decisive information the range provides in this situation regarding the profile of each runner.
3. c) The technical staff needs to choose one athlete for a team event where a consistent mark with no surprises of disqualification or delay is required, and another athlete for an individual event where a medal is only possible if they manage to run under $13\text{ s}$. Using the calculated statistical parameters, justify which athlete you would send to each event.
4. d) If in a seventh race Aoife trips over a hurdle and records an outlier time of $16.8\text{ s}$, reason which of her statistical measures (the mean or the median) will undergo a greater variation and which will remain more protected, without needing to recalculate the values.

Topic challenge. How do screens influence our environment? You are going to lead a real statistical investigation to analyze this habit in your community.

1. Data collection: Ask $15$ people (family members or classmates) how many hours a day they spend on their mobile phone (accept decimals, e.g. $1.5$ h or $2.0$ h) or collect the maximum temperatures of the last $15$ days from the Met Éireann website. Clearly identify the population, the sample, the individual, and the type of variable.

2. Tabulation and calculation: Enter the data into a spreadsheet (Excel, Calc, or Google Sheets). Create the frequency table (absolute, relative, and percentages). Calculate the measures of central tendency (mean $\bar{x}$, median, and mode) and dispersion (range).

3. Representation: Build the most appropriate statistical chart (bar chart or line graph), paying attention to titles, scales, and legends.

4. Final report: Write a brief report with your proposal: analyze whether the data are homogeneous or dispersed and provide a reasoned conclusion on the healthy use of free time.

Assessment criteria:

• Correct identification of key concepts and precise creation of the frequency table.

• Exact calculation of measures of central tendency and dispersion using a spreadsheet.

• Clarity and rigor in the digital graphical representation.

• Critical justification of the conclusions based on the obtained parameters.