Exercises: Mean, median and mode

In a botany experiment, the number of leaves on $8$ plants of the same species was measured after two weeks of growth. The results obtained are: $12, 15, 14, 12, 16, 12, 18, 13$.

Calculate the arithmetic mean, median and mode of these data.

The ages of the participants in a robotics workshop are summarised in the following frequency table:

• $14$ years old: $5$ people

• $15$ years old: $8$ people

• $16$ years old: $4$ people

• $17$ years old: $3$ people

Determine the mean age, median age and mode.

In a quality control test, the time in seconds that a computer takes to boot up is measured. The following times are obtained: $12, 11, 13, 12, 14, 85$.

Calculate the mean and the median. Which of the two measures best represents the central tendency of these data? Justify your answer.

A 4th ESO student has taken five mathematics exams throughout the term. Their marks in the first four are: $6, 7, 5$ and $8$.

If they want their exact mean mark to be $7$, what mark must they get in the fifth exam?

$20$ people were asked about their favourite colour for the interface design of a new mobile app. The results are:

• Blue: $8$ people

• Green: $5$ people

• Red: $4$ people

• Yellow: $3$ people

State the mode. Does it make sense to calculate the mean or the median in this case? Give reasons for your answer.

Given the following two data sets concerning the number of books read in a year by two groups of young people:

• Set A: $2, 4, 5, 7, 9$

• Set B: $1, 3, 4, 6, 8, 10$

Find the median for each of the data sets and explain the difference in the calculation procedure when having an even or odd number of data values.

The following bar chart shows the number of siblings that students in a class have.

From the chart, construct the frequency table and calculate the mode and the arithmetic mean.

You can do it on paper and send a photo with your answer.

The mean height of a basketball team of $10$ players is $192 \text{ cm}$.

If all the players decide to wear a special model of shoes that increases their height by $4 \text{ cm}$, what will the new mean height of the team be? Will the mode change if there was a most frequent height?

In the subject of Applied Mathematics in 4th year of ESO, the grading criteria for a term set the following weights: daily work and homework count for $20\,\%$, written tests for $50\,\%$, and the term project for $30\,\%$. Lucas has obtained a mark of $8$ on homework, a $4$ on written tests, and a $6$ on the project. To calculate his final mark, Lucas performs the following calculation:

$$\bar{x} = \frac{8 + 4 + 6}{3} = \frac{18}{3} = 6$$

And states with conviction: "I've passed the term with an average mark of 6!".

1. a) Explain with reasons what conceptual error Lucas makes by applying the simple arithmetic mean instead of the weighted mean.
2. b) Carry out the correct calculation of Lucas's final mark and state whether he actually passes the term (remember that the minimum passing mark is $5$).

The State Meteorological Agency (AEMET) recorded the following daily maximum temperatures ($^\circ\text{C}$) during a week in July at the Córdoba airport station:

$$39,\quad 41,\quad 42,\quad 40,\quad 44,\quad 41,\quad 41$$

1. a) Before carrying out any written calculations, estimate at a glance between which two integer values the weekly mean temperature will lie and what you guess the median value will be. Write down your estimation and justify the visual intuition or quick elimination you used.
2. b) Calculate accurately the arithmetic mean, the median, and the mode of this set of records.
3. c) Check: did your initial estimation match the calculated results? If during the weekend the hottest record ($44\,^\circ\text{C}$) had reached a record value of $47\,^\circ\text{C}$, would the median have changed? And the mean? Justify your answer without recalculating the entire sum.

In a study on water consumption in a community, the monthly usage in cubic metres ($m^3$) of several households was recorded, grouped into the following intervals:

• $[0, 10)$: $12$ households

• $[10, 20)$: $25$ households

• $[20, 30)$: $18$ households

• $[30, 40)$: $5$ households

Calculate the mean water consumption and state which is the modal class.

The final mark for Physics is calculated using a weighted mean. Laboratory practicals are worth $20\%$, research projects $30\%$ and the final exam $50\%$.

If a student scores $8$ in the practicals, $6$ in the projects and $7,5$ in the final exam, calculate her final mean mark.

In a shoe shop, the number of pairs sold according to size over a weekend was recorded. The mean size sold was $39,5$. Given the following data, determine the number of pairs sold of size $40$ (represented by $f$).

• Size $38$: $4$ pairs

• Size $39$: $8$ pairs

• Size $40$: $f$ pairs

• Size $41$: $5$ pairs

The mass in kilograms of a group of $40$ parcels at a post office was measured, obtaining the following distribution:

• $[0, 5)$: $6$ parcels

• $[5, 10)$: $14$ parcels

• $[10, 15)$: $12$ parcels

• $[15, 20)$: $8$ parcels

Find the interval containing the median and estimate its exact value using linear interpolation.

A logistics company has two warehouses. At the Madrid warehouse, $45$ workers work with a mean salary of $1800 \text{ \euro}$. At the Barcelona warehouse, $30$ workers work with a mean salary of $2100 \text{ \euro}$.

Calculate the mean salary of all the workers in the company as a whole.

The following histogram represents the time (in minutes) that a group of people spend playing sport per day.

Calculate the mean time spent playing sport from the class marks extracted from the graph.

You can do it on paper and send a photo with your answer.

When calculating the mean mark of $25$ students, the teacher obtained $6,4$. Later, when checking the exams, he realized that he had mistakenly recorded a $3$ for a student whose actual mark was $8$.

Calculate the new correct mean without needing to know all individual marks.

In a small tech company, the monthly salaries of its $7$ employees are:

$1200 \text{ \euro}, 1250 \text{ \euro}, 1300 \text{ \euro}, 1200 \text{ \euro}, 1400 \text{ \euro}, 1350 \text{ \euro}, 8500 \text{ \euro}$.

Calculate the mean and the median. Write a short text explaining to future employees which of the two measures more realistically represents the typical salary they will earn and why.

In a small footwear cooperative in Elche, $5$ people work. The following statistical information is available regarding their gross monthly salaries:

• The median of the salaries is exactly $1\,300$ €.

• The mean salary of the five workers is $1\,450$ €.

• The distribution is unimodal (there is a single mode) and its value is $1\,150$ €.

• The most experienced craftsperson, who earns the most among the staff, receives $2\,100$ €.

1. a) Deduce with reasons, step by step, the exact monthly salary of each of the $5$ employees of the cooperative.
2. b) If the following month a sixth person joins on an internship training contract with a salary of $950$ €, what will the new median salary of the cooperative be? Will the mode be affected? Justify your answer without performing unnecessary calculations.

Based on data from the Household Budget Survey of the National Statistics Institute (INE), the monthly expenditure on public transport (travel passes, metro, and buses) of a sample of $100$ young people aged $18$ to $24$ in a Spanish metropolitan area has been analysed:

• Interval $[0, 10)\text{ €}$: $15$ young people.

• Interval $[10, 20)\text{ €}$: $25$ young people.

• Interval $[20, 30)\text{ €}$: $40$ young people.

• Interval $[30, 50)\text{ €}$: $20$ young people.

1. a) Identify the class mark of each interval and calculate the estimated mean monthly expenditure per young person.
2. b) Determine with reasons which interval is the modal class, taking into account that not all intervals have the same width. Why, in distributions with intervals of different widths, is it not enough to look only at which interval has the highest absolute frequency?
3. c) If the transport authority introduces a general $50\,\%$ discount on all fares starting next month, what will the new estimated mean monthly expenditure of this group of young people be? Justify your answer by applying the properties of the mean, without reconstructing the frequency table.

In a medical centre, the cholesterol level of $50$ patients was recorded. The arithmetic mean of the distribution is $194 \text{ mg/dL}$. Two frequencies, $f_1$ and $f_2$, have accidentally been erased from the table.

• $[160, 180)$: $10$ patients

• $[180, 200)$: $f_1$ patients

• $[200, 220)$: $f_2$ patients

• $[220, 240)$: $8$ patients

Set up and solve a system of equations to find the absolute frequencies $f_1$ and $f_2$.

Let $X$ be a statistical variable with mean $\bar{x} = 15$. A new variable $Y$ is defined by the linear transformation $y_i = 3x_i - 4$.

Prove algebraically, by applying the properties of summation and the arithmetic mean, that the mean of the new variable is $\bar{y} = 3\bar{x} - 4$, and calculate its numerical value.

An investor buys shares in a company for three consecutive months. In the first month, they buy $n$ shares at a mean price of $12 \text{ \euro}$. In the second month, they buy double the number of shares bought in the first month at a mean price of $15 \text{ \euro}$. In the third month, they buy $50$ shares at $10 \text{ \euro}$.

If the overall mean price of all the shares bought across the three months is $13 \text{ \euro}$, how many shares did they buy in the first month?

In the assessment of a subject, the final mark is calculated using three components:

• Midterm exam (weighting $30\%$)

• Research project (weighting $20\%$)

• Final exam (weighting $50\%$)

A student has $4,5$ in the midterm exam and $6$ in the project. The final exam mark must be an integer or have a single decimal place. What is the minimum mark they must achieve in the final exam for their weighted mean to be at least $5,0$? If the maximum mark is $10$, is it possible for them to achieve a mean of $8,0$?

Consider a set of five positive integers ordered from smallest to largest: $a, b, c, d, e$.

They are known to satisfy the following conditions simultaneously:

• The mode is $4$.

• The median is $5$.

• The arithmetic mean is $6$.

• The range (difference between the largest and the smallest) is $7$.

Determine the exact value of the five numbers, reasoning step by step.

The following cumulative frequency polygon (ogive) shows the distribution of distances achieved by a group of $100$ athletes in the long jump event.

From the graph, estimate the value of the median. If the data were grouped into a frequency table with the intervals on the horizontal axis, what would the modal class be? Justify your answers based on the slopes of the graph.

The mean of a data set of $n$ values is $\bar{x}$. If we add a new value of $D$, the new mean increases by $2$ units. If, instead of $D$, we add a value of $D - 15$, the new mean decreases by $1$ unit relative to the original mean $\bar{x}$.

Set up the corresponding equations and calculate the original number of data values $n$.

A metal alloy is composed of two metals $A$ and $B$. The density of metal $A$ is $3 \text{ g/cm}^3$ and that of metal $B$ is $8 \text{ g/cm}^3$.

If the mean density of the resulting alloy is $6 \text{ g/cm}^3$, what percentage of the total volume of the alloy corresponds to metal $B$? (Note that the mean density of an alloy behaves like the weighted mean of the densities of its components with respect to their volume fractions).

In a parcel logistics company, $25$ employees work. For the company collective agreement negotiations, management and the union representation analyse the list of gross monthly salaries:

• $18$ warehouse workers earn $1\,200$ € each.

• $4$ administrative staff earn $1\,600$ € each.

• $2$ shift managers earn $2\,800$ € each.

• $1$ operations director earns $9\,500$ €.

1. a) Calculate the arithmetic mean, the median, and the mode of the company's salary distribution.
2. b) In the company newsletter, management states: "The average salary of our workforce exceeds $1\,700$ €, so salaries are well above the sector average". In contrast, the union declares: "That figure distorts reality; the salary that best represents the typical worker in the company is $1\,200$ €". Analyse which measure of central tendency supports each of the two positions. Justify critically which of the two measures (mean or median) is more suitable for describing the representative worker's salary, explaining the impact of the director's salary as an outlier.
3. c) If the director raises her salary to $12\,000$ € and the other salaries remain frozen, state with reasons how the mean and the median will change. What fundamental property of the median is demonstrated by this fact?

A civic platform studying access to rented housing collects data from $40$ young people in a city who rent. The data are initially grouped into the following monthly expenditure brackets:

• Bracket A $[400, 600)\text{ €}$: $12$ people.

• Bracket B $[600, 800)\text{ €}$: $16$ people.

• Bracket C $[800, 1\,000)\text{ €}$: $8$ people.

• Bracket D $[1\,000, 1\,400)\text{ €}$: $4$ people.

1. a) Calculate the estimated mean monthly rent for this sample using class marks and determine which is the modal class by analysing the frequency density of the brackets.
2. b) When reviewing the original contracts of the $4$ people in bracket D, it is verified that their exact actual rents are: $1\,050$ €, $1\,100$ €, $1\,150$ €, and $1\,300$ €. Calculate the exact actual mean of these four rents and compare it with the class mark you assigned to that interval. Was the mean expenditure overestimated or underestimated by using the class mark? Explain what conceptual limitation class marks have when estimating parameters in grouped distributions.
3. c) The city council grants a direct subsidy of $100$ € per month to all people in the study who pay $700$ € or more in rent. It is known that exactly half of the people in bracket B ($8$ young people) and all of brackets C and D ($12$ young people) receive the subsidy. What will be the new net mean monthly rent paid by the sample of $40$ young people? Deduce and calculate the result by reasoning about the total subsidised amount and the concept of the mean as an equitable share, without redoing the calculations interval by interval.