Twój koszyk jest pusty
Calculating the average
twenty books each this year.
I haven´t read a book this year.
I ´ve read forty.
You see, on avarage, we have read
Example 1
Zack measured five consecutive steps: \latex{ 82 } \latex{ cm }, \latex{ 78 } \latex{ cm }, \latex{ 85 } \latex{ cm }, \latex{ 74 } \latex{ cm }, and \latex{ 76 } \latex{ cm }. How many steps does it take him to arrive at school if it is \latex{ 650 } \latex{ m } from his home?
Solution
Calculate the average length of Zack's steps, as you can assume that each step is around the average. The average length of Zack’s steps is:
\latex{\frac{82 + 78 + 85 + 74 + 76}{5} = \frac{395}{5} = 79 \; ({cm})}
The number of steps needed: \latex{65, 000 \div 79 = 822.78}.
It takes Zack about \latex{ 823 } steps to get to school from home.
It takes Zack about \latex{ 823 } steps to get to school from home.
\latex{ 5. }
\latex{ 4. }
\latex{ 3. }
\latex{ 2. }
\latex{ 1. }
step
average
length of steps
\latex{\text{average} = \frac{\text{sum of values}}{\text{number of values}}} |
Data sets can usually be characterised by their average.
Example 2
The heights of the players of a volleyball team are \latex{ 178 } \latex{ cm }, \latex{ 184 } \latex{ cm }, \latex{ 180 } \latex{ cm }, \latex{ 181 } \latex{ cm },
\latex{ 210 } \latex{ cm }, and \latex{ 182 } \latex{ cm }. What is the average height of the team?
\latex{ 210 } \latex{ cm }, and \latex{ 182 } \latex{ cm }. What is the average height of the team?
Solution
\latex{\text{average} = \frac{178 + 184 + 180 + 181 + 210 + 182}{6} = \frac{1115}{6}= 185.83\; ({cm})}
Five team members are shorter than the team's average height, and only one is taller. Therefore, in this case, the average does not describe the heights of the volleyball players accurately.
The average of a data set can be altered significantly by an outstandingly small or large value.
Example 3
This term, Pamela's maths scores were \latex{ 90, 80, 95, 70, 93, 94, 96. }
How many points does she need to get on her last paper to obtain a grade point average of not less than \latex{ 88 ?}
Solution
Including the last paper, Pamela wrote a total of eight maths papers.
The sum of the scores is therefore \latex{8\times88=704} points.
The total score of the previous papers: 90 + 80 +95 + 70 + 93 + 94 + 96 = 618.
So Pamela needs to get at least \latex{704-618=86} points for the last paper to get an average of at least \latex{ 88 } points.
The sum of the scores is therefore \latex{8\times88=704} points.
The total score of the previous papers: 90 + 80 +95 + 70 + 93 + 94 + 96 = 618.
So Pamela needs to get at least \latex{704-618=86} points for the last paper to get an average of at least \latex{ 88 } points.
The average is not always among the data values.
Example 4
Write down \latex{ 7 } consecutive even numbers starting with \latex{ 254 }. What is the mean of the numbers?
Solution 1
The mean of the numbers:
\latex{(254 + 256 + 258 + 260 + 262 + 264 + 266) \div 7 = 1820\div 7 = 260}.
\latex{(254 + 256 + 258 + 260 + 262 + 264 + 266) \div 7 = 1820\div 7 = 260}.
Solution 2
The median of the data set is \latex{ 260 }.
Numbers following\latex{\;\;\;\;260}: \latex{ 260 + 2}; \latex{ 260 + 4}; \latex{ 260 + 6}
Numbers preceding \latex{260}: \latex{260 - 2}; \latex{ 260 - 4}; \latex{ 260 - 6}
\latex{\begin{rcases}\end{rcases}}
their sum: \latex{7 \times260}
The mean of the numbers is \latex{(7 \times 260)\div7 = 260}, which is the median of the data set.
If the difference between consecutive numbers is constant and the number of terms is odd, their mean is the median.
Exercises
{{exercise_number}}. At a figure skating competition, three skaters received the following points for their performances:
Skater \latex{ 1: 4.8;\, 5.1;\, 5.2;\, 4.8;\, 4.9;\, 5.1;\, 5.0;\, 4.5;\, 4.8;\, 5.1 }
Skater \latex{ 2: } \latex{ 4.9;\, 4.8;\, 4.9;\, 4.9;\, 5.1;\, 5.0;\, 5.1;\, 5.0;\, 5.5;\, 4.8 }
Skater \latex{ 3: } \latex{ 5.2;\, 5.4;\, 5.3;\, 5.0;\, 4.8;\, 4.9;\, 5.2;\, 5.1;\, 4.9;\, 5.0 }
- Determine the final ranking of the skaters in the competition based on their average scores. (The higher the average, the better the position of the skater in the final ranking is.)
- Calculate their average scores without taking into account their worst and best scores. Does this change the final ranking?
{{exercise_number}}. Zoe weighed her backpack every morning before going to school. Here are the results of the measurements:
day | Monday | Tuesday | Wednesday | Thursday | Friday |
mass (\latex{kg}) | \latex{ 7 } | \latex{ 5.5 } | \latex{ 6.5 } | \latex{ 7 } | \latex{ 5 } |
Show the data on a bar chart. What is the average mass of Zoe's backpack? Measure your backpack every day for a week and calculate the average mass of your backpack.
{{exercise_number}}. The number of spectators at the home games of a soccer team was recorded. At the end of the championship, these are the statistics about the number of spectators at each home game:
game | \latex{ 1 } | \latex{2} | \latex{3} | \latex{4} | \latex{5} | \latex{6} | \latex{7} | \latex{8} | \latex{9} |
number of fans | \latex{ 2,456 } | \latex{ 1,674 } | \latex{ 3,451 } | \latex{ 2,785 } | \latex{ 1,592 } | \latex{ 2,003 } | \latex{ 1,976 } | \latex{ 2,184 } | \latex{ 1,963 } |
Show the data on a bar chart. How many spectators attended the soccer games on average?
{{exercise_number}}. Calculate the average number of pages in the science, literature, history, maths, and music textbooks.
{{exercise_number}}. A dice was rolled \latex{ 20 } times. The number of occurrences of the outcomes is shown in the following frequency table. Calculate the average of the throws. Perform a similar experiment.
outcome | \latex{ 1 } | \latex{ 2 } | \latex{ 3 } | \latex{ 4 } | \latex{ 5 } | \latex{ 6 } |
frequency | \latex{ 4 } | \latex{ 3 } | \latex{ 3 } | \latex{ 1 } | \latex{ 5 } | \latex{ 4 } |
{{exercise_number}}. Calculate the average number of children in your classmates' families. First, create a frequency table and then calculate the average.
{{exercise_number}}. The average of two numbers is \latex{5\frac{7}{8}}. One of the numbers is –3.25. What is the other number?
{{exercise_number}}. The average length of the sides of a triangle is \latex{ 4.8 } \latex{ cm }. What is its perimeter?
{{exercise_number}}. If there are \latex{ 0.75 } images per page on average in a book that contains \latex{ 46 8} pages, how many images are there in the book in total?
{{exercise_number}}. Agnes prepares for the maths competition by completing the worksheets from previous years. Her scores in the previous five worksheets were \latex{ 104}; \latex{95}; \latex{89}; \latex{108} and \latex{ 103 } points. How many points must she obtain in the sixth worksheet to have an average score of \latex{ 100 } points?
{{exercise_number}}. A basketball player scored \latex{ 15 } points on average during the first \latex{ 8 } games of a tournament. How many points does he have to score in the ninth game to have an average of \latex{ 16 } points per game?
{{exercise_number}}. The average age of the bands performing at a talent show cannot be less than \latex{ 14 } years and more than \latex{ 35 } years. How old can the fourth band member be if the other members are \latex{ 12, 21, } and \latex{ 20 } years old?
{{exercise_number}}. \latex{ 5 } consecutive numbers divisible by \latex{ 4 } starting with \latex{ 4,324 } are written down. What is their mean?
{{exercise_number}}. \latex{ 35 } consecutive odd numbers are written down. Their median is \latex{ 2,789 }. What is the mean of the numbers?
{{exercise_number}}. Mary bought \latex{ 1 } \latex{ kg } of bananas every weekday starting from Monday. On Friday, she realised she had paid ¢\latex{ 22 } more each day than the previous one. What was the average price of the bananas from Monday to Friday if Mary paid €\latex{2.9 } for \latex{ 1 } \latex{ kg } on Tuesday?
{{exercise_number}}. The mean of seven consecutive numbers divisible by \latex{ 3 } is \latex{ 486 }. What is the smallest number in the sequence?
{{exercise_number}}. Peter recorded the temperature at \latex{ 6 } AM from Monday to Sunday. He noticed that each morning, the temperature was \latex{ 3 °C } lower than the previous morning. The average temperature in the morning during the week was \latex{ 0 °C }. What was the lowest temperature measured by Peter?
{{exercise_number}}. There are \latex{ 10, 11 } and \latex{ 12 }-\latex{ year }-old children at a summer camp. The chart shows the distribution of the ages of the children. (→)
- Make a table of the ages and gender of the participants based on the diagram.
- What is the average age of the boys and girls, respectively?
- What per cent of the children at the summer camp are boys?
- Make a pie chart showing the ratio of boys and girls at the camp.
\latex{ 10 }
\latex{ 9 }
\latex{ 8 }
\latex{ 7}
\latex{ 6}
\latex{ 5}
\latex{ 4}
\latex{ 3}
\latex{ 2}
\latex{ 1}
\latex{ 10}
\latex{ 11}
\latex{ 12}
person
girl
boy
age (\latex{ year })
Quiz
What is wrong with the following sentence? 'There are \latex{ 1.8 } children in an average family.'