Measures of central tendency and spread
Two school basketball teams have the same mean number of points per game. Do they play equally well? A single number doesn't tell the whole story: we need to know where the "centre" of the data is and whether the data are tightly grouped or widely spread out.
Mean, median and mode
Change the data and see where the mean (pink line) and the median (green line) fall:
The ages of Noa's cousins are: $8$, $10$, $12$, $10$ and $10$ years old. Calculate the mean of their ages.
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…
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…
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 $44$.
- a) Order the prices from lowest to highest.
- b) Calculate the median of the prices. Remember that with an even number of data values there are two middle values.
- c) What is the range? Interpret it: is there a large difference in price between websites?
Mean from a frequency table
When the data are given in a frequency table, each value counts as many times as its frequency indicates.
The dot plot shows the goals scored by a team in its recent matches. a) How many matches has the team played? b) In how many matches did it score no goals? c) What is…
The dot plot shows the goals scored by a team in its recent matches.
- a) How many matches has the team played?
- b) In how many matches did it score no goals?
- c) What is the mode of the number of goals? Is there only one mode?
- d) What is the range?
The graph shows the maximum temperatures recorded in a city during a spring week. a) Calculate the mean maximum temperature for the week, rounding to one decimal place.…
The graph shows the maximum temperatures recorded in a city during a spring week.
- a) Calculate the mean maximum temperature for the week, rounding to one decimal place.
- b) Calculate the temperature range for the week (the difference between the maximum and minimum).
- c) Which day of the week was furthest from the mean? Justify your answer.
The range: how spread out the data are
Both teams have the same mean (20 points), but Team A always scores something similar: range $= 22 - 18 = 4$. Team B is very inconsistent: range $= 35 - 5 = 30$. If you want to sign someone reliable, a small range is a good sign.
The table shows the number of books read this term by the students in a class: | No. of books | $0$ | $1$ | $2$ | $3$ | |---|---|---|---|---| | No. of students | $4$ |…
The table shows the number of books read this term by the students in a class:
| No. of books | $0$ | $1$ | $2$ | $3$ |
|---|---|---|---|---|
| No. of students | $4$ | $9$ | $11$ | $6$ |
- a) What is the mode of the number of books read?
- b) How many students are there in the class?
- c) How many students have read more than $1$ book?
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 $60\%$. Iker's…
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 $60\%$. Iker's assignments were $7$, $8$ and $6$, and in the exam he got $7,5$.
- a) Calculate the mean of the three assignments.
- b) Calculate his final term mark by applying the percentages.
- c) Would he have passed if the exam counted for $50\%$ and the assignments for the remaining $50\%$? Check your answer.
State whether the following statements are true or false, reasoning your answer or providing a counterexample: a) In a survey about a class's favourite colour, neither…
State whether the following statements are true or false, reasoning your answer or providing a counterexample:
- a) In a survey about a class's favourite colour, neither the mean nor the median can be calculated.
- b) If we add $2$ to all data values in a distribution, the mean increases by $2$ but the range does not change.
- c) The mode is always the largest value in the distribution.
- d) To calculate the median, it does not matter if the data are unordered.