Exercises: Statistical measures

In a study on social media use, a group of teenagers was asked how many hours a day they spend watching videos on TikTok. The results obtained are:

$2, 3, 1, 4, 2, 3, 2, 5, 2, 1$

Identify the population of the study and calculate the mode of the hours of use.

Seven friends compare the number of unread messages they have on their messaging app. The data are as follows:

$12, 15, 8, 20, 15, 14, 11$

Calculate the arithmetic mean ($\bar{x}$) of the number of unread messages for this group.

During a training week, an athlete recorded the kilometres she ran each day to prepare for a race:

$5, 8, 10, 12, 15, 18, 21$

Find the median ($Me$) of the distance covered. What does this value mean in relation to the total number of days trained?

The graph below shows the maximum temperatures recorded at a mountain shelter during the first 5 days of January.

Knowing that the temperatures were $4^\circ C, 6^\circ C, 2^\circ C, 8^\circ C$ and $5^\circ C$, calculate the range of the temperature over those days.

A teacher organised the number of spelling mistakes made by their students in the last dictation in a frequency table:

Mistakes ($x_i$)	No. of students ($f_i$)
$0$	$8$
$1$	$12$
$2$	$6$
$3$	$4$

Calculate the mean number of mistakes in the class using the data from the table.

Two esports teams recorded their scores in the last 5 matches.

• Team A: $85, 88, 90, 92, 95$

• Team B: $60, 75, 90, 105, 120$

Both teams have the same mean, $\bar{x} = 90$. Without performing complex standard deviation calculations, look at the data and explain which of the two teams is more consistent (has less dispersion) and why.

The height of a group of 3rd Year students was measured and the data were ordered from smallest to largest. The values of the quartiles obtained are:

$Q_1 = 158 \text{ cm}, Q_2 = 165 \text{ cm}, Q_3 = 172 \text{ cm}$

1. a) What is the value of the median?
2. b) What percentage of students is $172 \text{ cm}$ tall or less?

Observe the following representation of a data set on a number line. The dots represent the number of pets owned by several neighbours:

From the graph, write down the list of data and calculate the mode ($Mo$) and the range.

Cian wants to calculate the median of the marks he has obtained in seven Mathematics tests throughout the school year: $4$, $8$, $3$, $9$, $5$, $7$, and $2$. On his exam, he writes the following justification:

"Since there are $7$ marks, the median is the value that occupies the central position, that is, the fourth element in the list. I count from left to right and the fourth number is $9$, so the median is $Me = 9$".

1. a) Identify and explain with reasoning what conceptual error Cian has made.
2. b) Without needing to calculate it yet, why is it impossible for the median of that set of marks to be $9$?
3. c) Correct the procedure and calculate the real value of the median and the mode of his marks.

In a school athletics competition, a shot put athlete has recorded the following distances (in metres) across her five attempts: $8.1$; $8.3$; $8.0$; $8.4$, and $12.2$.

1. a) Estimate: without performing operations on paper, estimate between which two whole numbers the arithmetic mean will lie and reason whether you think the median will be greater than, less than, or equal to said estimated mean. Justify your answer by observing how the values are distributed.
2. b) Check: calculate exactly the mean ($\bar{x}$) and the median ($Me$) of the throws.
3. c) Evaluate: compare the results with your initial estimate. Explain in your own words how the last throw ($12.2$ m) has influenced the mean and why it has barely affected the median.

A group of 20 teenagers recorded how many times they went to the cinema in the last six months. The data obtained are as follows:

$2, 3, 0, 1, 5, 3, 2, 1, 0, 0, 2, 4, 3, 2, 1, 1, 6, 2, 0, 3$

Organise these data into a frequency table and calculate the measures of central tendency: the arithmetic mean ($\bar{x}$), the median ($Me$) and the mode ($Mo$). Which of them do you consider best represents the group?

Two messaging apps, "App A" and "App B", are competing to be the most stable. The response time (in milliseconds) was measured over 10 different attempts, obtaining the following results:

• App A: $\bar{x} = 120$ ms; $\sigma = 15$ ms

• App B: $\bar{x} = 125$ ms; $\sigma = 5$ ms

Calculate the coefficient of variation ($CV$) for both applications and explain with reasons which of the two is more "reliable" or consistent in its performance for a user.

An e-sports club recorded the time (in minutes) taken by its members to complete a difficult level of a video game. The results are shown in this grouped data table:

Time (min)	[10, 20)	[20, 30)	[30, 40)	[40, 50)
No. of players	4	12	8	6

Calculate the arithmetic mean and the standard deviation of the distribution using the class marks.

Aoife's marks in her first four Mathematics exams were $6.5$, $4$, $8$ and $7.5$. If she still has a fifth exam to take, what mark must she get for her final mean mark to be exactly $7$? Justify your answer by setting up an equation.

A survey has been carried out on the number of hours per week that 3rd Year students spend reading books for pleasure. The position parameters obtained are:

$Mín = 0$; $Q_1 = 2$; $Me = 4$; $Q_3 = 7$; $Máx = 12$

Draw the corresponding box-and-whisker plot (Box-plot). If a student reads 6 hours a week, in which quarter of the population are they (between 0-25%, 25-50%, 50-75% or 75-100%)?

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

In a PE class, the recovery time of two groups after a run was measured. Both groups have the same mean heart rate, $\bar{x} = 85$ bpm. However, the variance of Group 1 is $S^2 = 16$ and that of Group 2 is $S^2 = 100$.

1. a) Calculate the standard deviation of each group.
2. b) In which of the two groups do the students have a more similar fitness level to each other? Reason your answer based on the dispersion.

A youth clothing shop wants to analyse the age of its customers. To do this, it takes a sample of 10 people on a Saturday morning:

$14, 15, 16, 14, 22, 15, 17, 14, 35, 16$

1. a) Calculate the range and the mean age.
2. b) The median is $15.5$ years. If we remove the 35-year-old customer (which is considered an outlier), calculate the new mean and the new median. Which parameter has changed the most? Why do you think this happens?

Look at the following two data sets, which represent the points scored by two basketball players in their last 5 games:

• Player A: $18, 20, 22, 19, 21$

• Player B: $10, 30, 5, 25, 30$

In the figure you have the same data represented on a number line (each dot is a game).

Without doing complex variance calculations, look at the data and answer:

1. Do they have the same arithmetic mean?

2. Which of the two players is more "consistent" in their scoring?

3. If you had to choose one of them to play the last minute of a final, which one would you choose and why? Use the concepts of mean and dispersion in your answer.

A sociological research team collects a set of $5$ positive integers corresponding to the number of books read per year by a small group of students. The original data have been misplaced, but the final report keeps the following parameters intact:

• The mode is unique and equal to $4$.

• The median is $5$.

• The range is $8$.

• The arithmetic mean is $7$.

Deduce in an orderly and reasoned manner what the $5$ original numbers are. Justify each step by explaining which statistical property allows you to determine or rule out each of the values.

A tech design startup in Galway has a staff of $8$ people. Their net monthly salaries in euros are: $1\,200$, $1\,250$, $1\,300$, $1\,300$, $1\,400$, $1\,450$, $1\,500$, and $6\,600$ (the latter corresponds to the founding partner).

1. a) Calculate the arithmetic mean, the median, and the mode of the salaries.
2. b) In a job offer, the company states: "The average salary of our team is $2\,000$ € per month". For their part, the workers' representatives argue that the most representative salary is around $1\,300$ €–$1\,400$ €. Compare both measures of central tendency (mean versus median): which one describes the typical compensation of the staff more accurately? Justify your answer by analysing the impact of the outlier.
3. c) To study the dispersion of these salaries, two alternatives are proposed: calculating the total range or calculating the interquartile range ($IQR = Q_3 - Q_1$). Justify which of the two measures is preferable in this case.

In a group of $10$ friends, the mean height is $172$ cm. If a person who is $194$ cm tall joins the group, what will the new mean be? If another person who was $161$ cm tall then leaves, what will the final mean of the group be? Justify step by step how each change affects the total sum of heights to carry out the calculation.

Two eSports players, Alex and Bea, recorded their scores in the last $8$ matches of a tournament.

• Alex: $85, 90, 88, 92, 87, 89, 91, 88$

• Bea: $70, 98, 65, 100, 75, 95, 80, 100$

1. a) Calculate the mean and standard deviation for both players.
2. b) Use the coefficient of variation ($CV$) to determine which of the two is a more consistent and reliable player for the team. Explain what it means in this context if one player has a much higher standard deviation than the other when their means are similar.

A courier company analyses the weight of the packages it transports. The data have been grouped into the following frequency table by intervals:

Weight in kg $[L_i, L_s)$	No. of packages ($f_i$)
$[0, 2)$	$12$
$[2, 4)$	$25$
$[4, 6)$	$38$
$[6, 8)$	$15$
$[8, 10)$	$10$

Calculate the arithmetic mean, the variance and the standard deviation of the distribution. To do this, first identify the class marks ($x_i$) of each interval.

Look at the following box plot (box-and-whisker plot) representing the weekly hours of use of a mobile app in a sample of teenagers:

Based on the graph, answer the following:

1. a) What is the value of the median, the lower quartile ($Q_1$) and the upper quartile ($Q_3$)?
2. b) Calculate the interquartile range ($RIC$).
3. c) Reason whether the distribution is symmetrical or skewed. In which half of the data is there greater spread?

A teacher marked the exams of a class of $25$ students. The mean turned out to be $6.4$ and the standard deviation $1.2$. After reviewing one of the questions, the teacher decides to increase everyone's mark by $0.5$ points.

1. a) What will the new mean and the new standard deviation be?
2. b) If instead of adding a fixed amount, the teacher decided to increase all marks by $10\%$, how would both measures be affected? Justify your answer based on the properties of statistical measures.

Create a set of $6$ statistical data points (positive integers) that simultaneously satisfy the following conditions:

1. The mode is $4$.

2. The median is $5$.

3. The range is $10$.

4. The mean is greater than the median.

Show the calculations that verify that your set meets all the requirements.

In a small company, the salaries of the $5$ employees are: $\text{€}\ 1100, \text{€}\ 1200, \text{€}\ 1250, \text{€}\ 1300$ and $\text{€}\ 1400$. The manager has a salary of $\text{€}\ 4500$.

1. a) Calculate the mean and the median of the salaries, including the manager.
2. b) Which of the two measures better represents the "typical" salary of the company? Explain the effect that the manager's salary (an extreme value) has on the mean and why the median is a more robust measure in this case.
3. c) If the company wants to project an image of prosperity in its advertising, which measure would it use? Would it be misleading?

Based on historical series from Met Éireann, the annual precipitation (in litres per square metre, $\text{l/m}^2$) of two Irish cities has been recorded over five representative years:

• Galway: $1\,200$, $1\,300$, $1\,100$, $1\,400$, $1\,000$

• Dublin: $200$, $150$, $300$, $100$, $250$

1. a) Calculate the mean ($\bar{x}$), the variance ($s^2$), and the standard deviation ($s$) of the precipitation for each city.
2. b) A person states: "The volume of rain in Galway is much more irregular than in Dublin because its standard deviation is practically double". Use the coefficient of variation ($CV = s / \bar{x}$) expressed as a percentage to refute or support this statement with mathematical rigor.
3. c) Justify why in this study it is methodologically correct to use the coefficient of variation with precipitation in $\text{l/m}^2$, whereas it would make no mathematical sense to apply it directly to average annual temperatures measured in degrees Celsius ($^\circ\text{C}$).

A food cooperative in Wexford packages bags of rice whose nominal weight is set at $1\,000$ g. To check the calibration of two packaging lines, quality control takes a sample of $5$ packages from each machine:

• Machine A: $995$, $1\,000$, $1\,005$, $998$, $1\,002$ (in grams).

• Machine B: four packages have been recorded with weights of $980$, $1\,020$, $990$, $1\,010$ (in grams), but the weight of the fifth package was not recorded due to a computer error.

1. a) Deduce what the weight of the fifth package of Machine B must be so that both lines have exactly the same arithmetic mean of $1\,000$ g.
2. b) With the deduced value, calculate the standard deviation ($s$) and the percentage coefficient of variation ($CV_{\%}$) of each of the two machines.
3. c) The factory's quality protocol requires that the coefficient of variation of a packaging machine does not exceed $1\%$. Give a reasoned verdict on whether both machines comply with the standard and explain what economic or commercial damage the cooperative would suffer if it used the more dispersed machine, even though its mean is exactly $1\,000$ g.