Average, Fashion and Median exercises

Study mode, mean and median with the solved and step-by-step exercises. Clear your doubts and prepare for exams and entrance exams.

Median Exercises

Exercise 1

In a pediatrics office, a doctor saw nine children in one day. He measured and noted the heights of the children as per the consultations.

1st consultation 0.90 m
2nd consultation 1.30 m
3rd consultation 0.85 m
4th consultation 1.05 m
5th consultation 0.98 m
6th consultation 1.35 m
7th consultation 1.12 m
8th consultation 0.99 m
9th consultation 1.15 m

Determine the median heights of children in consultations.

Correct answer: 1.05 m.

The median is a measure of central tendency. To determine the median we must organize the ROL of the data, which is to place them in ascending order.

0.85 m 0.90 m 0.98 m 0.99 m 1.05 m 1.12 m 1.15 m 1.30 m 1.35 m

The median is the central value, in this case, the fifth value: 1.05 m.

Exercise 2

(Enem 2021) The manager of a concessionaire presented the following table at a meeting of directors. It is known that at the end of the meeting, in order to prepare goals and plans for the next year, the administrator will evaluate sales based on the median number of cars sold in the period from January to December.

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Image for question resolution.

What was the median of the data presented?

a) 40.0
b) 42.5
c) 45.0
d) 47.5
e) 50.0

Correct answer: b) 42.5

To determine the median, we need to organize the ROL of data, that is, put them in ascending order.

Image for question resolution.

Since the number of elements is even, we must calculate the simple arithmetic mean between the two central values.

numerator 40 space plus space 45 over denominator 2 end of fraction equals 42 comma 5

Therefore, 42.5 is the median of the data presented.

Exercise 3

(Enem 2015) In a selective for the final of the 100 meters freestyle swimming, in an Olympics, the athletes, in their respective lanes, obtained the following times:

Image for question resolution.

The median time shown in the table is

a) 20.70.
b) 20.77.
c) 20.80.
d) 20.85.
e) 20.90.

Correct answer: a) 20.70.

To determine the median we must assemble the ROL of data, arranging them in ascending order.

Image for question resolution.

If the dataset is odd, the median is the central value. If the dataset number is even, the median will be the arithmetic mean between the central values.

numerator 20 comma 80 space plus space 20 comma 60 over denominator 2 end of fraction equals 20 comma 70

Therefore, the median is 20.70.

Exercise 4

(UNEB 2013) Brazilians willing to pay a daily rate of up to €11 thousand (R$ 30.69 thousand) for a suite are the hot spot in the world's luxury hotel market.

Competing for the finest hotels, the clientele in Brazil occupies the third position in the ranking of reservations by The Leading Hotels of the World (LHW). The seal brings together some of the most sophisticated establishments in the world.

From 2010 to 2011, the local revenue of the light truck grew 16.26%.

Last year, the Brazilian office broke the record of US$ 31 million (R$ 66.96 million) in reserves.
(TOURIST..., 2012, p. B 3).

Image for question resolution.

The median of spending, in millions of reais, of Brazilian tourists with luxury hotels, in 2011, is equal to

a) 3.764
b) 3,846
c) 3.888
d) 3,924
e) 3,996

Correct answer: e) 3,996

The median of the chart data is the arithmetic mean of the central values, in dollars.

numerator 1 comma 5 space plus space 2 comma 2 over denominator 2 end of fraction equals 1 comma 85

The median is $1.85 million. However, the question asks for values ​​in Reais.

The text states that US$ 31 million (of dollars) was equivalent to R$ 66.96 million (of reais).

We need to determine how many reais were worth one dollar. For this, we do the division:

numerator 66 comma 96 over denominator 31 end of fraction equal to 2 comma 16

Thus, 2.16 is the dollar-to-real conversion rate.

1 comma 85 space x space 2 comma 16 space equals space 3 comma 996

In real, Brazilians spent 3.996 million reais.

Average

Exercise 7

The following table shows prices for motorcycle taxi rides to different neighborhoods in the city of Rio de Janeiro and the amount of trips recorded in one day, for each neighborhood.

neighborhoods Price Number of trips
Meier BRL 20.00 3
Mature BRL 30.00 2
Botafogo BRL 35.00 3
Copacabana BRL 40.00 2

Calculate the average price of trips on that day.

Answer: BRL 27.00.

As each price has a different contribution to the average, as the amounts of trips are different for each neighborhood, the average has to be weighted by the amount of trips.

The weighted average is the division between each price multiplied by the respective amounts of trips and the total trips.

numerator left parenthesis 20 space. space 3 right parenthesis space plus space left parenthesis 30 space. space 2 right parenthesis space plus space left parenthesis 35 space. space 2 right parenthesis space plus space left parenthesis 40 space. space 2 right parenthesis on denominator 3 space plus space 2 space plus space 3 space plus 2 end of fraction equals numerator 60 space plus space 60 space plus space 70 space plus space 80 over denominator 10 end of fraction equals 270 over 10 equals 27

Thus, the average price of trips for that day was R$27.00.

Exercise 6

(Enem 2015) A contest consists of five stages. Each stage is worth 100 points. Each candidate's final score is the average of their grades over the five steps. The classification follows the descending order of the final scores. The tiebreaker is based on the highest score in the fifth stage.

Image for question resolution.

The final ranking order for this contest is

a) A, B, C, E, D.
b) B, A, C, E, D.
c) C, B, E, A, D.
d) C, B, E, D, A.
e) E, C, D, B, A.

Correct answer: b) B, A, C, E, D.

We need to determine the average of the five candidates.

We write e1 + e2 + e3 + e4 as the sum of the candidates' first four grades.

Candidate for

numerator 1 space plus space 2 space plus space 3 space plus space 4 over denominator 4 end of fraction equal to 90

Thus,

and 1 space plus space and 2 space plus space and 3 space plus space and 4 space equals space 90 space. space 4 and 1 space plus space and 2 space plus space and 3 space plus space and 4 space equal to 360

Candidate A's five-step average

numerator 1 space plus space 2 space plus space 3 space plus space 4 space plus space 5 over denominator 5 end of fraction equal to

We've already determined the sum of the first four steps, which equals 360. From the table, we take the score of the fifth stage, 60.

Calculating the average, we have:

numerator and 1 space more space and 2 space more space and 3 space more space and 4 space more space and 5 over denominator 5 end of fraction equal to numerator 360 space plus space 60 over denominator 5 end of fraction equal to 420 over 5 equal to 84

Candidate A's average scores in the first five stages were 84 points.

Repeating the reasoning for the other candidates, we have:

Candidate B:
In the first four stages,

numerator 1 space plus space 2 space plus space 3 space plus space 4 over denominator 4 end of fraction equals 85 and 1 space plus space 2 space plus space 3 space plus space 4 space equals space 85 space. space 4 space equals space 340

In the five steps,

numerator 1 space more space 2 space more space 3 space more space 4 space more space 5 over denominator 5 end of fraction equals numerator 340 space plus space 85 over denominator 5 end of fraction equals 85

Candidate C:
In the first four stages,

numerator 1 space plus space 2 space plus space 3 space plus space 4 over denominator 4 end of fraction equals 80 and 1 space plus space 2 space plus space 3 space plus space 4 space equals space 80 space. space 4 space equals space 320

In the five steps,

numerator 1 space more space 2 space more space 3 space more space 4 space more space 5 over denominator 5 end of fraction equal to numerator 320 space plus 95 over denominator 5 end of equal fraction to 83

Candidate D:
In the first four stages,

numerator 1 space plus space 2 space plus space 3 space plus space 4 over denominator 4 end of fraction equals 60 and 1 space plus space 2 space plus space 3 space plus space 4 space equals space 60 space. space 4 space equals space 240

In the five steps,

numerator 1 space more space 2 space more space 3 space more space 4 space more space 5 over denominator 5 end of fraction equal to numerator 240 space plus 90 over denominator 5 end of equal fraction to 66

Candidate E:

In the first four stages,

numerator 1 space plus space 2 space plus space 3 space plus space 4 over denominator 4 end of fraction equals 60 and 1 space plus space 2 space plus space 3 space plus space 4 space equals space 60 space. space 4 space equals space 240

In the five steps,

numerator 1 space more space 2 space more space 3 space more space 4 space more space 5 over denominator 5 end of fraction equal to numerator 240 space plus 100 over denominator 5 end of equal fraction to 68

In descending order of scores, we have:

B 85
THE 84
Ç 83
AND 68
D 66

Exercise 7

(UFT 2013) The average height of the 35 adult Indians in a village is 1.65 m. Analyzing only the heights of the 20 men, the average is equal to 1.70 m. What is the average, in meters, of heights if we consider only women?

a) 1.46
b) 1.55
c) 1.58
d) 1.60
e) 1.65

Correct answer: c) 1.58

There are 35 people in the village, 20 of whom are men, 15 are women.

35 = 20 + 15

Average height of women.

Calling Sm the sum of women's heights, we have:

straight S with straight m subscript over 15 equals straight x

Soon, straight S with subscript straight m equal to 15 spaces. straight space x

Where x is the mean of women's heights.

Average height of men.

S with h subscript over 20 equals 1 comma 70
S with h subscript equal to 20 space. space 1 comma 70 space equals space 34

Where Sh is the sum of men's heights.

Average of all people in the village

Calling S, the sum of the heights of all the people in the village, this is the sum of the heights of men plus women.

Averaging the entire village, we have:

S over 35 equals numerator S m space plus space S h over denominator 35 end of fraction equals 1 comma 65

Substituting the values ​​of Sh and Sm, we have:

numerator 15 x space plus space 34 over denominator 35 end of fraction equals 1 comma 65

Solving the equation for x,

numerator 15 x space plus space 34 over denominator 35 end of fraction equals 1 comma 65 15 x space plus space 34 space equals space 1 comma 65 space. space 35 15 x space plus space 34 space equals space 57 comma 75 15 x space equals space 57 comma 75 space minus space 34 15 x space equal to space 23 comma 75 x space equal to space numerator 23 comma 75 over denominator 15 end of fraction equal to 1 comma 58

if we consider only women, 1.58 m is the average height.

Exercises 8

(EsSA 2012) The arithmetic mean of all candidates in a competition was 9.0, of the selected candidates it was 9.8 and the eliminated ones was 7.8. What percentage of candidates are selected?

a) 20%
b) 25%
c) 30%
d) 50%
e) 60%

Correct answer: e) 60%

1st step: determine the percentage ratio of the selected

We must determine the ratio of those selected to the total number of candidates.

S on T

Where S is the number of selected candidates and T is the total number of candidates.

However, the number T of the total number of candidates is equal to the sum of those selected plus those eliminated.

T = S + E

Where E is the total eliminated.

Thus, the reason we must determine is:

numerator S over denominator S plus E end of fraction

2nd step: determine a relationship between S and E

We have that the total average was 9. In this way,

numerator n T over denominator T end of fraction equal to space 9

Where nT is the sum of all grades. This sum is the addition of the grades of the selected nS, plus the grades of the eliminated, nE.

nT = nS + nE

Then,

numerator n T over denominator T end of fraction equals numerator n S space plus space n E space over denominator S space plus space E end of fraction space equals space 9 (equation I)

Also, we have to:

numerator n S over denominator S end of fraction equal to 9 comma 8 therefore, n S space equal to 9 comma 8 space. S space

and

numerator n E over denominator E end of fraction equal to 7 comma 8 therefore, n E space equal to space 7 comma 8. AND

Substituting in equation I, we have:

numerator 9 comma 8 S space plus space 7 comma 8 E over denominator S space plus space E end of fraction equal to 9

Writing S in function of E:

9 comma 8 S space plus space 7 comma 8 E space equals 9 space. left parenthesis S space plus space E right parenthesis 9 comma 8 S space plus space 7 comma 8 E space equals space 9 S space plus space 9 E 9 comma 8 S space minus space 9 S space equals space 9 E space minus space 7 comma 8 E 0 comma 8 S space equal to space 1 comma 2 E S equal to numerator 1 comma 2 over denominator 0 comma 8 end of fraction E S space equal to 1 comma 5. AND

3rd step: replace in the reason

the reason is

numerator S over denominator S plus E end of fraction

Replacing S,

numerator 1 comma 5 And over denominator 1 comma 5 And space plus space And end of fraction equals numerator 1 comma 5 And over denominator 2 comma 5 And end of fraction equals 0 comma 6

4th step: transform into percentage

To turn it into a percentage, we multiply by 100

0.6 x 100 = 60%

Therefore, 60% is the percentage of selected candidates.

Fashion

Exercise 9

In a movie theater, popcorn is sold in packs of three sizes. After entering a session, the management carried out a survey to find out which of the packages was the most sold.

In order of sales, these were the values ​​noted by the popcorn cashier.

20,30
17,50
17,50
17,50
20,30
20,30
11,40
11,40
17,50
17,50
11,40
20,30

Based on the fashion of the values, determine which size of popcorn was the best seller.

Right answer:

Fashion is the most repeated element. Each element repeated itself:

11.40 three times

17.50 x five times

20.30 x four times

Thus, the average popcorn was the most sold, as 17.50 is the most repeated value.

Exercise 10

(Navy 2014) Review the chart below.

Image for question resolution.

Check the option that shows the data mode in the table above.

a) 9
b) 21
c) 30
d) 30.5
e) 31

Correct answer: b) 21

Fashion is the most repeated element. Element 21 repeats 4 times.

Exercise 11

(Enem 2016) When starting its activities, an elevator operator records both the number of people that enter as the number of people leaving the elevator on each floor of the building where it works. The painting shows the records of the elevator operator during the first climb from the ground floor, where he and three other people depart, to the fifth floor of the building.

Table associated with the resolution of the issue.

Based on the chart, what is the fashion for the number of people in the elevator going up from the ground floor to the fifth floor?

a) 2
b) 3
c) 4
d) 5
e) 6

Correct answer: d) 5.

We must consider the number of people entering, the number leaving and the number of people remaining.

entered went out remain for walking
5th floor 7 already had + 2 6 7 + 2 - 6 = 3
4th floor 5 already had + 2 0 5 + 2 = 7
3rd floor 5 already had + 2 2 5 + 2 - 2 = 5
2nd floor 5 already had + 1 1 5 + 1 - 1 = 5
1 ° floor 4 already had + 4 3 4 + 4 - 3 = 5
Ground floor 4 0

4 - 0 = 4

Thus, the fashion is 5, as it is the number of people that repeats the most.

Exercise 12

(UPE 2021) In the summer of 2018, a large appliance store recorded the number of fan units sold for 10 consecutive days, as shown in the table below. With this, it was possible to verify the sales volume per day and the variation in the number of sales from one day to the next.

Image for question resolution.

What is the mode of variations in the number of daily sales in the period considered?

a) 53
b) 15
c) 7
d) 4
e) 2

Correct answer: d) 4.

The variation in the number of sales is the difference between one day and the previous one.

Day 2 - Day 1 53 - 46 7
Day 3 - Day 2 38 - 53 - 15
Day 4 - Day 3 45 - 38 7
Day 5 - Day 4 49 - 45 4
Day 6 - Day 5 53 - 49 4
Day 7 - Day 6 47 - 53 -6
Day 8 - Day 7 47 - 47 0
Day 9 - Day 8 51 - 47 4
Day 10 - Day 9 53 - 51 2

With 4 being the most repeated difference, 4 is fashion.

learn more about Average, Fashion and Median.

You may be interested in:

  • Arithmetic Average Exercises
  • Arithmetic average
  • Weighted Arithmetic Average
  • Statistics - Exercises
  • Statistic
  • Geometric Mean
  • Relative Frequency
  • Standard deviation
  • Dispersion Measures
  • Variance and standard deviation
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