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 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?

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?

$40$ 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.

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.

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?

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 $30$ 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?

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.

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.

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.

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.