Exercises: Statistical parameters

In a study on the use of social media, a group of teenagers were 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 is as follows:

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

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

During a week of training, an athlete has recorded the miles she has run each day to prepare for a race:

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

Find the median ($Me$) of the distance run. What is the meaning of this value in relation to the total number of days trained?

The following graph shows the maximum temperatures recorded in a mountain lodge during the first 5 days of January.

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

A teacher has organized the number of spelling mistakes their students made in the last dictation into 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 e-sports teams have 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 8th grade students has been measured and the data has been ordered from lowest to highest. 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 measure $172 \text{ cm}$ or less?

Look at the following representation of a data set on a number line. The dots represent the number of pets that several neighbors have:

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

A group of 20 teenagers has recorded how many times they have been to the movies in the last six months. The data obtained is as follows:

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

Organize this 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 applications, "App A" and "App B", are competing to be the most stable. The response time (in milliseconds) has been measured in 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 operation for a user.

An e-sports club has recorded the time (in minutes) its members take 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 midpoints.

Sarah's grades in her first four math exams were $6,5$, $4$, $8$ and $7,5$. If she still has a fifth exam to take, what grade must she get for her final mean 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 8th grade students spend reading books for pleasure. The measures of position 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%)?

In a Physical Education class, the recovery time of two groups after a race has been 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? Justify your answer based on the dispersion.

A youth clothing store wants to analyze 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 (who 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 sets of data representing the points scored by two basketball players in their last 5 matches:

• Player A: $18, 20, 22, 19, 21$

• Player B: $10, 30, 5, 25, 30$

Without performing 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 scores?

3. If you had to choose one 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, their 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 the heights to carry out the calculation.

Two eSports players, Alex and Bea, have 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 for one player to have a much higher standard deviation than the other if their means were similar.

A courier company analyzes the weight of the packages it transports. The data has been grouped in the following interval frequency table:

Weight in lbs $[L_i, L_s)$	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 ($x_i$) for each interval.

Look at the following box plot representing the weekly hours of use of a mobile app in a sample of teenagers:

Based on the graph, answer:

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 if it shows skewness. In which half of the data is there greater spread?

A teacher has graded the exams of a class of $25$ students. The mean was $6,4$ and the standard deviation was $1,2$. After reviewing one of the exercises, they decide to add $0,5$ points to all the students in the class.

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

It is known that in a very large data distribution that follows a "bell curve", $68\%$ of the data is found in the interval $(\bar{x} - \sigma, \bar{x} + \sigma)$.

In a light bulb factory, the mean lifespan of a model is $1200$ hours with a standard deviation of $150$ hours. If $5000$ light bulbs are produced:

1. a) How many light bulbs are expected to last between $1050$ and $1350$ hours?
2. b) If a light bulb lasts more than $1500$ hours, how many standard deviations from the mean is it? Do you consider it to be a common value or an outlier?

Make up a series of $6$ statistical data points (positive integers) that simultaneously meet 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 your series meets all the requirements.

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

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 in the company? Explain the effect the manager's salary (extreme value) has on the mean and why the median is a more robust parameter in this case.
3. c) If they want to give an image of prosperity in the company's advertising, which parameter would be used? Would it be misleading?