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Mathematics 9.

Table of contents
Combinatorics, sets
Algebra and arithmetics
Functions
Triangles, quadrilaterals, polygons
Equations, inequalities, simultaneous equations
Congruent transformations
Statistics
1. The representation of data
2. The description of data
The description of data
The “middle value” and the “deviation” of data describing the data set resulting after statistical data collection can be calculated in several ways. It is always the content of the data and the aim that helps decide which calculation method is worth using. Care should be taken because applying an inadequate value can give a misleading result.
Mode
In a quiz show one answer should be chosen out of the four possible answers. The audience is allowed to give assistance to the player. The question was as follows: “Which colour was named after Titian (Italian: Tiziano Vecellio)?” The votes for the possible answers are shown in Figure 10.
The player did believe in the audience, thus he chose the data appearing most frequently, i.e. red, and he gave the correct answer.
yellow
red
green
blue
Figure 10
\latex{ 0 }
\latex{ 10 }
\latex{ 20 }
\latex{ 30 }
\latex{ 40 }
\latex{ 50 }
\latex{ 45 }%
\latex{ 40 }%
\latex{ 8 }%
\latex{ 7 }%
DEFINITION: The data appearing most frequently in a set of data is called the mode of the sample.
mode
If in a set of data more data appear equal times, and the rest of the data appear fewer times, then the set of data has more modes.
Advantages:
  • in view of the mode we can guess the data with a better chance;
  • it can easily be determined.
Disadvantages:
  • it highlights one data point and does not provide information about the rest;
  • it cannot be used if all data appear only once or twice.
A sample can have several modes. E.g. if the sample is \latex{ 2;\, 2;\, 3;\, 3;\, 4, } then number \latex{ 2 } and \latex{ 3 } are the modes.
Example 1
The following table shows the number of tornadoes in the USA between \latex{ 1970 } and \latex{ 1979 }. Let us find the mode of the number of tornadoes.
Year
Tornado
\latex{ 1970 }
\latex{ 1971 }
\latex{ 1972 }
\latex{ 1973 }
\latex{ 1974 }
\latex{ 1975 }
\latex{ 1976 }
\latex{ 1977 }
\latex{ 1978 }
\latex{ 1979 }
\latex{ 788 }
\latex{ 947 }
\latex{ 835 }
\latex{ 920 }
\latex{ 947 }
\latex{ 1102 }
\latex{ 741 }
\latex{ 888 }
\latex{ 653 }
\latex{ 852 }
Solution

Only \latex{ 947 } appears twice, the rest of the data appear once, so the mode of the number of tornadoes is: \latex{ 947 }.
It is obvious that this value does not characterise the “middle value” of the data well, since it is one of the largest values appearing in the sample.
The “middle value” of the number of tornadoes is better characterised by one tenth of the sum of the \latex{ 10 } numbers, i.e. by their mean:

 

\latex{\frac{653+ 888 +741 +1102 +947 +920+ 835 +947+ 788 +852}{10}=867.3}
Mean (also referred to as average)

Meteorologists give the mean daily temperature as the mean of the temperatures measured at \latex{ 1 } and \latex{ 7 } o'clock in the morning and in the afternoon. Figure 11 shows the values measured in Belgrade on \latex{ 25 } November \latex{ 2000 }. The mean daily temperature was:

 

\latex{\frac{1+4+12+7}{4}=6(C^{\circ})}.
 
The daily range of temperatures is the difference of the largest value and the smallest value: \latex{12-1=11 (C^{\circ})}. If we calculate the deviation of each data from the mean, then we get the table in Figure 12. The sum of the deviations from the mean: \latex{-5-2+6+1=0}.
Time
Temperature
\latex{ 1 } am
\latex{ 7 } am
\latex{ 1 } pm
\latex{ 7 } pm
\latex{ 1 } \latex{^{\circ}C}
\latex{ 4 } \latex{^{\circ}C}
\latex{ 12 } \latex{^{\circ}C}
\latex{ 7 } \latex{^{\circ}C}
The sum of the deviations from the mean of the values greater than the mean is the same as the sum of the deviations from the mean of the values less than the mean. Based on this we can determine the following characteristics of the mean (average) of the data:
Deviation
Temperature
\latex{1-6=-5}
\latex{4-6=-2}
\latex{12-6=6}
\latex{7-6=1}
\latex{ 1 } \latex{^{\circ}C}
\latex{ 4 } \latex{^{\circ}C}
\latex{ 12 } \latex{^{\circ}C}
\latex{ 7 } \latex{^{\circ}C}
DEFINITION: If we divide the sum of the data by the number of values, then we get the arithmetic mean or average of the sample.
average, arithmetic mean
The average calculated as the arithmetic mean is one of the most often used characteristics of the “middle value” of the data.
Advantages:
  • the sum of its deviations from the values greater than it is the same as the sum of its deviations from the values less than it.
Disadvantages:
  • an outlier can distort it very much.
DEFINITION: The difference between the greatest and the smallest value appearing among the data is called the sample range.
sample range, range of the sample
Example 2
The salary of the business manager of the Flying Elephants Company is \latex{ 500 } euros, the accounting director earns \latex{ 280 } euros, the two heads of department get \latex{ 100 } and \latex{ 110 } euros, the driver earns 60 euros, three unskilled labourer earn 50 and three other earn \latex{ 40 } euros. Let us give the average salary of the workers of the company.
Solution

The average salary can be calculated by adding up the salary of each worker and then by dividing the sum by the number of workers:

 

\latex{\frac{500+ 280+ 110+ 100+ 60+ 50+ 50+ 50+ 40+ 40+ 40}{11}=120}
 
Therefore the average salary of the workers is 120 euros.
However this average salary does not characterise the payment situation well. A few higher salaries raise the average very much.
We can get a more realistic picture if we order the salaries (no matter whether into ascending or descending order), and we choose the one in the middle. The descending order of the salaries is as follows (euros): \latex{ 500;\, 280;\, 110;\, 100;\, 60;\, 50;\, 50;\, 50;\, 40;\, 40;\, 40 }. Out of the \latex{ 11 } items the sixth one is in the middle, and it is \latex{ 50 }. So according to the workers 50 euros characterise the measure of the salaries better.
Only the management of the company is proud of it, the majority of the workers do not consider it to be realistic.
Median
The heights of the five players of a basketball team being on the court are: \latex{ 185\, cm }, \latex{ 180\, cm }, \latex{ 184\, cm }, \latex{ 210\, cm }, \latex{ 199\, cm }. Let us put the players into ascending order based on height and let us select the one in the middle. The data in the middle is \latex{ 185\, cm } which means that the same number of players are shorter than \latex{ 185\, cm } as taller.
Figure 13
\latex{ 180\, cm }
\latex{ 184\, cm }
\latex{ 185\, cm }
\latex{ 199\, cm }
\latex{ 210\, cm }
DEFINITION: The median of odd number of data \latex{(2n + 1} pieces) is the data in the middle (the one at the \latex{(n + 1)^{th}} place). The median of even number of data (\latex{2n} pieces) is the mean of the two data in the middle (the ones at the \latex{n^{th}} and \latex{(n + 1)^{th}} place).
the median of the data
In the case of many data putting the data into order might be time-consuming.
Advantages:
  • the same number of values are less than the median as greater;
  • the sum of the distances of the median from the data is minimal.
Example 3
Five friends live in one street. Andrew’s place is \latex{ 30\, m }, Ben’s place is \latex{ 84\, m }, Chris’ place is \latex{ 132\, m }, Denis’ place is \latex{ 156\, m } and Ernest’s place is \latex{ 210\, m } away from the end of the street. At whose place shall they meet so that the boys together walk the shortest distance possible?
Solution
At whoever's place they meet the sum of the distances what Andrew and Ernest cover is the distance between their houses.
ANDREW
BEN
CHRIS
DENIS
ERNEST
Figure 14
If they meet at Ben's, Chris' or Denis' place, then the sum of the distances what Ben and Denis cover is the distance between their houses; if they meet at Andrew's or Ernest's place, then the total distance is greater than the previous sum, therefore it is not worth meeting at Andrew's or Ernest's place. If they do not meet at Chris' place, then the distance what Chris covers should be added to the previous two sums, so it is not worth meeting not at Chris' place. Therefore the sum of the distances the five friends cover is the smallest if they meet at the place in the middle, i.e. at Chris' place.
The median of the distances measured from the end of the street is \latex{ 132 }.
In the example we could see that the median is the number for which it is true that the sum of the distances between the number and the data is minimal.
Standard deviation
There are \latex{ 3 } children in the Blue, in the Green and also in the Yellow family. The table contains the ages of the children.
Family name
Age of the children (year)
Blue
Green
Yellow
\latex{ 2 }
\latex{ 12 }
\latex{ 9 }
\latex{ 15 }
\latex{ 16 }
\latex{ 16 }
\latex{ 28 }
\latex{ 17 }
\latex{ 20 }
The mean of the ages of the children in each family:
Blue: \latex{\frac{2+15+28}{3}=15 \;}(years);
Green: \latex{\frac{12+16+17}{3}=15\;}(years);
Yellow: \latex{\frac{9+16+20}{3}=15\;}(years).
We can see that the mean of the ages of the children is the same in all three families. But this mean does not characterise the data equally well for each family. We can say about the children of the Green family that they are about “\latex{ 15 } years old”, whereas we cannot say the same about the Blue family, because there are big differences between the ages, their ages “deviate” very much.
The measure of the “deviation” of the data is shown by the so called standard deviation:
DEFINITION: The standard deviation of the data \latex{x_1, x_2, …, x_n} is
\latex{\sqrt{\frac{(x_1-\overline x)^2+(x_2-\overline x)^2+\dots+(x_n-\overline x)^2}{n}}},
where \latex{\overline x=\frac{x_1+x_2+\dots+x_n}{n}} is the mean of the data.
the standard deviation of data

In the above example the standard deviation of the ages of the children in the case of the Blue family:

\latex{\sqrt{\frac{(2-15)^2+(15-15)^2+(28-15)^2}{3}}\approx 10.61}.
 

The standard deviation of the ages of the children in the case of the Green:

 

\latex{\sqrt{\frac{(12-15)^2+(16-15)^2+(17-15)^2}{3}}\approx 2.16}.
 

The standard deviation of the ages of the children in the case of the Yellow:

 

\latex{\sqrt{\frac{(9-15)^2+(16-15)^2+(20-15)^2}{3}}\approx 4.55}.
 
We can see that the standard deviation of the ages of the children is the smallest in the Green family; the ages are “closest” to the mean in this family. The mean characterises the data better where the standard deviation of the data is smaller.
⯁ ⯁ ⯁
Here three vote-evaluating systems will be described all of which seem to be fair, but there are cases when the three voting systems give three different results.
During the voting the voters have to rank three candidates.
  1. According to the first method, whoever gets the most first ranks, is the winner.
  2. According to the second method \latex{ 3 } points are given for a first rank, \latex{ 2 } points are given for a second rank, \latex{ 1 } point is given for a third rank, and whoever gets the most points is the winner.
  3. According to the third method whoever gets more than half of the first ranks is the winner. If there is no such candidate, then the candidate with the least number of first ranks steps back, and the votes given for this candidate will be transferred to those candidates who were ranked as second.
Let us have a look at the votes of \latex{ 33 } voters for candidates \latex{A, B, C}.
The number of voters who ranked as \latex{ ABC:\, 10 };
The number of voters who ranked as \latex{ ACB:\, 4 };
The number of voters who ranked as \latex{ BAC:\, 2 };
The number of voters who ranked as \latex{ BCA:\, 7 };
The number of voters who ranked as \latex{ CAB:\, 3 };
The number of voters who ranked as \latex{ CBA:\, 7 }​​​​​​​.
Let us evaluate the voting with all three methods.
The number of first ranks: \latex{A:14;\; B: 9;\; C:10}. Candidate A got the most first ranks, so this candidate is the winner.
The points are: \latex{A =14\times 3 + 5 \times 2 + 14 \times1 = 66}; \latex{B = 9 \times3 + 17 \times2 + 7 \times1 = 68}; \latex{C = 10 \times 3 + 11 \times 2 + 12 \times 1 = 64}. Candidate B got the most points, so this candidate is the winner.
No one has \latex{ 51 }% of the first ranks, thus candidate \latex{ B }, who has the least number of first ranks, steps back. Out of those who ranked candidate \latex{ B } as first \latex{ 2 } voters ranked candidate \latex{ A } as second, thus candidate \latex{ A } has \latex{ 16 } votes, \latex{ 7 } voters ranked candidate \latex{ C } as second, thus candidate \latex{ C } has \latex{ 17 } votes. Therefore according to this method candidate \latex{ C } is the winner of the voting.
KENNETH ARROW was a winner of Nobel Memorial Prize in Economics in \latex{ 1972 } for having proven that it is not possible to create a vote-evaluating system which is fair from all points of view.
Exercises
{{exercise_number}}. The graph shows the centimetres of snow having fallen on five consecutive days in February.
  1. Give the sample range.
  2. How many centimetres of snow fell on average daily?
  3. Give the standard deviation of the sample.
snow(\latex{ cm })
Monday
Tuesday
Wednesday
Thursday
Friday
day
\latex{ 15 }
\latex{ 12 }
\latex{ 9 }
\latex{ 6 }
\latex{ 3 }
{{exercise_number}}. In the below diagram the number of stop signs found in the streets in a part of a town is represented.
One
symbol means one stop sign.
Colchicum
Violet
Cowslip
Rose
Tulip
Wildflower
Marsh-marigold
Poppy
street
Number of STOP signs (pieces)
\latex{ 5 }
\latex{ 4 }
\latex{ 3 }
\latex{ 2 }
\latex{ 1 }
Give the mode and the median of the number of stop signs.
{{exercise_number}}. A die is rolled \latex{ 20 } times. The result of the rolls: \latex{ 1 } was rolled \latex{ 3 } times, \latex{ 2 } was rolled \latex{ 3 } times, \latex{ 3 } was rolled \latex{ 6 } times, \latex{ 4 } was rolled \latex{ 2 } times, \latex{ 5 } was rolled \latex{ 1 } time and \latex{ 6 } was rolled \latex{ 5 } times.
  1. Draw the frequency diagram of the rolled numbers.
  2. Give the mean, the median and the mode of the rolled numbers.
{{exercise_number}}. Guess and then check with calculation in which town is the standard deviation of the data the largest. Look up what problems it causes if the deviation of precipitation is large over time.
precipitation (\latex{ mm })
month
\latex{ 6 }th
\latex{ 7 }th
\latex{ 8 }th
\latex{ 6 }th
\latex{ 8 }th
\latex{ 7 }th
\latex{ 6 }th
\latex{ 7 }th
\latex{ 8 }th
Bristol
York
Manchester
\latex{ 100 }
\latex{ 50 }
\latex{ 0 }
\latex{ 38 }
\latex{ 110 }
\latex{ 2 }
\latex{ 70 }
\latex{ 55 }
\latex{ 25 }
\latex{ 58 }
\latex{ 48 }
\latex{ 44 }
{{exercise_number}}. Dan measured how many kilometres he covered by bicycle on seven consecutive days in the summer holiday, and then he represented the data on a column chart.
Give the mean, the standard deviation, the median and the mode of the number of kilometres he cycled daily.
distance covered by bicycle (\latex{ km })
Monday
Tuesday
Wednesday
Thursday
Friday
day
Saturday
Sunday
\latex{ 40 }
\latex{ 30 }
\latex{ 20 }
\latex{ 10 }
\latex{ 22 }
\latex{ 14 }
\latex{ 14 }
\latex{ 35 }
\latex{ 20 }
\latex{ 4 }
\latex{ 14 }
{{exercise_number}}. Consider the following sample:
\latex{ 100;\, 50;\, 50;\, 20;\, 20;\, 20;\, 20;\, 20;\, 20;\, 20;\, 20;\, 15;\, 15;\, 15;\, 15;\, 15;\, 15;\, 15;\, 15;\, 15;\, 15;\, 10;\, 10;\, 10;\, 10 }.
Calculate the mode, the arithmetic mean, the standard deviation and the median of the sample.
{{exercise_number}}. The following table contains the ages of the first \latex{ 30 } customers entering the music shop Saturday morning. Give the mode, the mean, the standard deviation and the median of the sample.
Age (year)
Number of customers
\latex{ 17 }
\latex{ 18 }
\latex{ 19 }
\latex{ 20 }
\latex{ 21 }
\latex{ 22 }
\latex{ 23 }
\latex{ 3 }
\latex{ 2 }
\latex{ 4 }
\latex{ 5 }
\latex{ 7 }
\latex{ 5 }
\latex{ 4 }
{{exercise_number}}. Consider sample \latex{ 2;\, 2;\, 3;\, 3;\, 3;\, 4 }.
  1. How do the mean, the median and the mode of the sample change if we replace one of the \latex{ 2s } by a \latex{ 1 }?
  2. How do the mean, the median and the mode of the sample change if we add an \latex{ 8 } to the list items?
{{exercise_number}}.
  1. In sample \latex{ 3;\,3;\,8;\,8;\,1; } \latex{ a } what should a be equal to so that the mode of the sample is \latex{ 8 }?
  2. In sample \latex{ 8;\, 18;\, b;\, 7;\, 19 } what should \latex{ b } be equal to so that the mean of the sample is \latex{ 15 }?
  3. In sample \latex{ 4;\, 9;\, 10;\, 10;\, c } what should \latex{ c } be equal to so that the median of the sample is \latex{ 9 }?
  4. In sample \latex{ 20;\, 20;\, 16;\, 14;\, 18;\, d } what can \latex{ d } be equal to if the sample range is \latex{ 8 }?
{{exercise_number}}. Women working at the secretariat of a company think that their salaries are lower than of the men working at the same place. If they manage to convince the director about it, then their salaries will be increased. The salaries of the three women working at the secretariat are: \latex{ 250 } euros, \latex{ 100 } euros, \latex{ 100 } euros. The salaries of the three men are equally \latex{ 150 } euros.
  1. Calculate the mean of the salaries.
  2. Calculate the mean of the salaries of the women and of the men separately.
  3. Give the median of the salaries of the women and of the men.
  4. Which “middle value” shall the representative of the women use while fighting for the pay rise, and how shall the director reason if he does not want to increase the salaries?
{{exercise_number}}. The art teacher categorised the students' work into three categories: A – excellent, B – satisfactory, C – not satisfactory. Which “middle value” would help us decide which category most of them were put into?
{{exercise_number}}. The mean of \latex{ 10 } test results in the class is \latex{ 71 } points. What is the sum of the points of the \latex{ 10 } tests?
{{exercise_number}}. The mean of \latex{ 4 } test results in the class is \latex{ 75 } points. The fifth test result was \latex{ 90 } points. What will be the mean of the test results including this test too?
{{exercise_number}}. Sam will get grade A in Mathematics at the end of the year, if the mean of his points for \latex{ 10 } tests with total points of \latex{ 100 } is at least \latex{ 80 }. However the mean of his points is only \latex{ 79 }, so the teacher wants to give him only grade B. Sam is objecting to it, as he is only missing a single point. Is it really so, does he only need one point to get grade A? Explain.
{{exercise_number}}. The mean of the algebra test results in a class of \latex{ 25 } students is \latex{ 82 } points. The mean of the same test results in another class with \latex{ 27 } students is \latex{ 69 } points. Give the mean of the test results of the students of the two classes together?
{{exercise_number}}. The table shows the number of Olympic medals won by Hungary (the art games inclusive) at the new age Olympic Games at which Hungary participated.
Year
Location
Gold medal
Silver medal
Bronze medal
Athens
Paris
St. Louis
London
Stockholm
Paris
Amsterdam
Los Angeles
Berlin
London
Helsinki
Melbourne
Rome
Tokyo
Mexico
Munich
Montreal
Moscow
Seoul
Barcelona
Atlanta
Sydney
Athens
Beijing
London
\latex{ 1896 }
\latex{ 1900 }
\latex{ 1904 }
\latex{ 1908 }
\latex{ 1912 }
\latex{ 1924 }
\latex{ 1928 }
\latex{ 1932 }
\latex{ 1936 }
\latex{ 1948 }
\latex{ 1952 }
\latex{ 1956 }
\latex{ 1960 }
\latex{ 1964 }
\latex{ 1968 }
\latex{ 1972 }
\latex{ 1976 }
\latex{ 1980 }
\latex{ 1988 }
\latex{ 1992 }
\latex{ 1996 }
\latex{ 2000 }
\latex{ 2004 }
\latex{ 2008 }
\latex{ 2012 }
\latex{ 2 }
\latex{ 1 }
\latex{ 2 }
\latex{ 3 }
\latex{ 3 }
\latex{ 2 }
\latex{ 5 }
\latex{ 6 }
\latex{ 10 }
\latex{ 10 }
\latex{ 16 }
\latex{ 9 }
\latex{ 6 }
\latex{ 10 }
\latex{ 10 }
\latex{ 6 }
\latex{ 4 }
\latex{ 7 }
\latex{ 11 }
\latex{ 11 }
\latex{ 7 }
\latex{ 8 }
\latex{ 8 }
\latex{ 3 }
\latex{ 8 }
\latex{ 1 }
\latex{ 2 }
\latex{ 1 }
\latex{ 4 }
\latex{ 2 }
\latex{ 4 }
\latex{ 5 }
\latex{ 5 }
\latex{ 1 }
\latex{ 5 }
\latex{ 10 }
\latex{ 10 }
\latex{ 8 }
\latex{ 7 }
\latex{ 10 }
\latex{ 13 }
\latex{ 5 }
\latex{ 10 }
\latex{ 6 }
\latex{ 12 }
\latex{ 4 }
\latex{ 6 }
\latex{ 6 }
\latex{ 5 }
\latex{ 4 }
\latex{ 3 }
\latex{ 2 }
\latex{ 1 }
\latex{ 2 }
\latex{ 3 }
\latex{ 4 }
\latex{ - }
\latex{ 5 }
\latex{ 5 }
\latex{ 13 }
\latex{ 16 }
\latex{ 7 }
\latex{ 7 }
\latex{ 5 }
\latex{ 12 }
\latex{ 16 }
\latex{ 13 }
\latex{ 15 }
\latex{ 6 }
\latex{ 7 }
\latex{ 10 }
\latex{ 3 }
\latex{ 3 }
\latex{ 2 }
\latex{ 6 }
  1. Represent the data on a diagram which shows the number of medals and the changes in the number of medals. (Use a spreadsheet and diagram creator computer software.)
  2. Represent the number of medals on a diagram which shows the ratio of the gold, the silver and the bronze medals.
  3. Give the mean, the median and the mode of the number of each medal type.
  4. Examine whether the following assumption is true: the Hungarian sportsmen and sportswomen performed better at those Olympic Games in the case of which the time difference was \latex{ 2 } hours at most.
{{exercise_number}}. Agnes was preparing for the national final of a mathematical championship in the following way: she solved the exercises of the finals of the previous \latex{ 5 } years. She solved the exercises of the first four finals and scored \latex{ 95;\, 97;\, 101 } and \latex{ 91 } points respectively. How many points did she score in the case of the fifth final if after writing it the mean score of Agnes became \latex{ 1 } point greater?
{{exercise_number}}. If we increase each of ten numbers by \latex{ 5 }, then which of the following statements are true and which are false?
  1. The mean of the numbers does not change.
  2. The mean of the numbers increases by \latex{ 50 }.
  3. The mean of the numbers increases by \latex{ 10 }.
  4. The mean of the numbers increases by \latex{ 5 }.
  5. The mean of the numbers decreases to its half.
What can we say about each statement if we write mode or median instead of the mean?
{{exercise_number}}. If each of \latex{ 8 } values decreases by half, then which of the following statements are true and which are false?
  1. The mean of the numbers does not change.
  2. The mean of the numbers doubles.
  3. The mean of the numbers decreases to its half.
  4. The mean of the numbers decreases by \latex{\frac{1}{2}}.
  5. The mean of the numbers increases by \latex{ 2 }.
What can we say about each statement if we write mode or median instead of the mean?
{{exercise_number}}. Exactly one of the following four statements is false:
  1. Ann is older than Barbara.
  2. Cissy is younger than Barbara.
  3. The sum of the ages of Barbara and Cissy is twice the age of Ann.
  4. Cissy is older than Ann.
Who is the youngest among Ann, Barbara and Cissy?
{{exercise_number}}. The weights of the players of a basketball team of five are measured so that the mean is calculated after each team member. How many kilograms heavier is the last player than the first player if the mean increased by \latex{ 1 } kilogram each time?
{{exercise_number}}. The statistics of two baseball players, Smith and Jones were as follows before the last day of the season: Jones had \latex{ 200 } hits out of \latex{ 600 }, Smith had \latex{ 199 } hits out of \latex{ 600 }, i.e. Jones was leading. On the last day Jones had \latex{ 8 } hits out of \latex{ 8 }, while Smith had \latex{ 11 } hits out of \latex{ 13 }, i.e. Jones was better again. 
After summarising the results of the season who did finally win this competition between the two of them?
number of students
occasions
{{exercise_number}}. In a survey the participants were asked on how many books they had read that year. The mean was 56 books. If they missed a person who had read \latex{ 68 } books, the mean went down to \latex{ 55 }.
What could have been the largest number given as an answer to the question?
Puzzle
What do you think is the median of the height of the \latex{ 5 } children shown in the figure the same as the mean of their heights?

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