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 $25$ meter 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.

The $30$ students in a class were asked how many times they visited the local sports center 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 store compares the prices of the same sneakers across $8$ different websites and finds these values, in pesos: $2\,520$, $2\,280$, $2\,700$, $2\,340$, $3\,000$, $2\,160$, $2\,460$ and $2\,640$.

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 grades from the last math exam of a class.

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

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

1. a) Calculate the mean of the three assignments.
2. b) Calculate his final term grade 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 favorite type of movie 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.

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, justifying your answer or providing a counterexample:

1. a) In a survey about a class's favorite color, 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.

Martina has achieved these grades in her first four math tests: $5.5$; $6$; $7$ and $6.5$. Only one test remains to finish the term, and she wants her mean grade to be exactly $6.5$.

1. a) What grade does she need to get on 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 of secondary 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?