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 kilometers 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 of secondary 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 neighbors:

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

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.

Lucía'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 was carried out on the number of hours per week spent by 3rd year of secondary school students reading books for pleasure. The position measures 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 for 6 hours a week, in which quarter of the population are they located (between 0-25%, 25-50%, 50-75% or 75-100%)?

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.

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: $\text{RD\$}\ 66\,000, \text{RD\$}\ 72\,000, \text{RD\$}\ 75\,000, \text{RD\$}\ 78\,000$ and $\text{RD\$}\ 84\,000$. The manager has a salary of $\text{RD\$}\ 270\,000$.

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?