Exercises: Mean, median and mode

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

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

The ages of the participants in a robotics workshop are summarized 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, the median age and the mode.

In a quality control test, the time in seconds it takes for a computer 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 better represents the central tendency of this data? Justify your answer.

A 10th-grade student has taken five math exams throughout the term. His grades on the first four are: $6, 7, 5$ and $8$.

If he wants his exact average grade to be a $7$, what grade must he get on the fifth exam?

$20$ people were asked for their favorite color 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 which is the mode. Does it make sense to calculate the mean or the median in this case? Explain your answer.

Given the following two data sets referring to 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 sets and explain the difference in the calculation procedure when having an even or odd number of data points.

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

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

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 sneakers that increases their height by $4 \text{ cm}$, what will be the new mean height of the team? Will the mode change if there was a more frequent height?

In a study on water consumption in a community, the monthly usage in cubic feet ($m^3$) of several homes has been recorded, grouped into the following intervals:

• $[0, 10)$: $12$ homes

• $[10, 20)$: $25$ homes

• $[20, 30)$: $18$ homes

• $[30, 40)$: $5$ homes

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

The final grade for the Physics course is calculated using a weighted mean. The lab assignments are worth $20\%$, the research projects $30\%$ and the final exam $50\%$.

If a student has a $8$ on the labs, a $6$ on the projects and a $7,5$ on the final exam, calculate her final average grade.

In a shoe store, the number of pairs sold according to size during a weekend has been recorded. The mean size sold was $39,5$. Knowing 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 weight in pounds of a group of $40$ packages in a post office has been measured, obtaining the following distribution:

• $[0, 5)$: $6$ packages

• $[5, 10)$: $14$ packages

• $[10, 15)$: $12$ packages

• $[15, 20)$: $8$ packages

Find the interval where the median is located and estimate its exact value using the linear interpolation formula.

A logistics company has two industrial warehouses. In the Dallas warehouse, $45$ workers work with a mean salary of $1800 \text{ \$}$. In the Chicago warehouse, $30$ workers work with a mean salary of $2100 \text{ \$}$.

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 sports per day.

Calculate the mean time spent playing sports from the class midpoints extracted from the chart.

When calculating the mean of the grades of $25$ students, the teacher obtained a $6,4$. Later, reviewing the exams, he realized that he had mistakenly written down a $3$ for a student whose actual grade was a $8$.

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

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

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

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

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

• $[160, 180)$: $10$ 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, 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. The first month he buys $n$ shares at a mean price of $12 \text{ \$}$. The second month he buys twice as many shares as the first month at a mean price of $15 \text{ \$}$. The third month he buys $50$ shares at $10 \text{ \$}$.

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

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

• Mid-term exam (weighting $30\%$)

• Research project (weighting $20\%$)

• Final exam (weighting $50\%$)

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

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

It is known that they simultaneously satisfy the following conditions:

• 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 your answer step by step.

The following cumulative frequency polygon (ogive) shows the distribution of the distances obtained 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 in a frequency table with the intervals of the horizontal axis, what would be the modal class? Justify your answers based on the gradients of the graph.

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