Scatter plots and correlation

Look at the following scatter diagrams shown below. Identify in each case whether the correlation is positive, negative or zero.

Match each of the following pairs of variables with the type of scatter diagram you would expect to obtain (increasing, decreasing or with no defined shape):

1. a) The number of kilometres driven by a car and the fuel remaining in its tank.
2. b) The height of a group of teenagers and their mark in an English exam.
3. c) An athlete's training hours and their personal best in a 100-metre sprint.
4. d) The outdoor temperature in summer and the energy consumption of air conditioning units.

A group of researchers wants to study whether there is a relationship between the time per week that teenagers spend on leisure screens (television, mobile phone or game console) and their academic performance. To do so, they have selected a sample of 8 students and have collected the following data:

1. a) Draw in your notebook the scatter plot associated with this two-variable distribution (screen hours on the horizontal axis and average grade on the vertical axis).

1. b) Looking at your graph, what type of correlation seems to exist between the two variables (direct or inverse)? Justify whether you think the relationship is strong or weak.

Look at the following scatter diagram representing the relationship between weekly training hours ($X$) and points scored in a season by a group of basketball players ($Y$).

Based on the shape of the scatter diagram, answer:

1. a) Is it a positive or negative relationship?
2. b) How would you describe the strength of the correlation: strong or weak?
3. c) What sign do you expect the covariance $\sigma_{xy}$ to have for these data?

We compare two scatter diagrams shown in the following graphs. Both show a positive relationship between the variables.

1. a) In which of the two graphs are the points closer to a possible regression line?
2. b) Which of the two will have a correlation coefficient $r$ closer to $1$?
3. c) If in graph A the value of $r$ is $1$, what reasonable value (between $0$ and $1$) could you assign to graph B?

The weight ($X$) and height ($Y$) of the players in a basketball team have been measured. The scatter diagram shows a very elongated and narrow shape following an upward trend.

What does a "narrow" diagram indicate regarding the strength of the correlation? What would we call the relationship between weight and height in this case (positive or negative)?

A music app has analysed data from 8 users to see the relationship between their age ($X$, in years) and the number of different songs they listen to per month ($Y$). The resulting scatter diagram is shown below:

Looking at the graph, can we say that the correlation is strong or weak? Justify whether the number of songs listened to tends to increase or decrease as age increases, according to this sample.

It is suspected that there is a relationship between the number of absences from class ($X$) and the final grade for the term ($Y$). The data for a class have been represented in the following diagram:

A red point appears on the graph that clearly deviates from the general trend of the group. What is this type of point called in statistics and what could it mean in this real-world context?

An influencer analyses the relationship between the number of hours spent editing a video ($X$) and the number of "likes" received in the first 24 hours ($Y$, in thousands). After uploading 10 videos, they obtain a very narrow and upward scatter diagram, except for one video which, after only 1 hour of editing, achieved a massive number of "likes" due to a viral challenge.

1. a) What is the technical term for a point that lies far from the general trend of the scatter diagram?
2. b) If the influencer decides to remove this special data point from the study, what would happen to the absolute value of the correlation coefficient $r$? Would it increase or decrease? Justify your answer.
3. c) Would the relationship between the variables become more or less linear after removing this point?

In statistics, it is essential to distinguish between a relationship that can be calculated with an exact formula and one that only shows a trend. State whether the relationship between the following pairs of variables is deterministic (functional) or statistical (correlation):

1. a) The number of hours a 1st Baccalaureate student spends playing video games and their mark in the Economics exam.
2. b) The length of the side of a square and its perimeter.
3. c) The litres of petrol put into a car and the total amount to pay at the petrol station.
4. d) The height of a teenager and their body weight.

A group of researchers analyses the relationship between the weekly time spent playing strategy video games (variable $X$, in hours) and the score obtained on a logical reasoning test (variable $Y$, from 0 to 100). The data from 10 teenagers show a very strong positive linear correlation, with the exception of one individual who spends 25 hours per week and scores only 10 points on the test.

Based on the graphical representation:

1. Explain, giving reasons, how point $P$ affects the value of Pearson's correlation coefficient $r$ and the gradient of the future regression line.

2. If we were to remove point $P$, could we state that a causal relationship exists between playing video games and logical reasoning? Justify your answer by analysing the nature of the variables.

The relationship between investment in social media advertising ($X$, in thousands of euros) and the number of new subscribers ($Y$, in hundreds) is studied for two different mobile apps, "App A" and "App B". The resulting scatter diagrams show that both have a mean investment of $\bar{x} = 5$ and a mean number of subscribers of $\bar{y} = 12$. However, App A presents a very scattered circular diagram, whereas App B presents a very narrow and elongated diagram with a negative gradient.

1. Sketch both scatter diagrams on the same set of axes, distinguishing them using colours or symbols.

2. Compare the correlation coefficients $r_A$ and $r_B$. In which case is it more reliable to make a prediction of the number of subscribers based on the investment?

3. What would it mean, in the context of marketing, for App B to have a negative gradient?

The direction of the relationship between two variables indicates how one varies when the other changes. Determine whether a positive correlation, negative correlation exists, or whether the variables are uncorrelated in the following cases:

1. a) The number of absences from class and the final mark for the course.
2. b) A person's shoe size and their average mark in Mathematics.
3. c) The outdoor temperature in summer and the energy consumption of air conditioning units.
4. d) The age in years of a mobile phone and its resale price on the second-hand market.

The covariance $\sigma_{xy}$ provides information about the direction of joint variation of two variables. Without carrying out complex calculations, look at the following data relating the temperature of a soft drink ($X$, in $^\circ C$) to the time it has been out of the fridge ($Y$, in minutes):

1. a) As time increases, does the temperature tend to increase or decrease?
2. b) Justify, based on your previous answer, whether the covariance will be positive, negative or zero.

Pearson's correlation coefficient, $r$, is a dimensionless measure that varies within a specific range. Of the following values, state which ones are not possible for a correlation coefficient and explain why:

1. a) $r = 0.85$
2. b) $r = -1.2$
3. c) $r = 2.5$
4. d) $r = -0.4$
5. e) $r = 0$

State whether the following statements about scatter diagrams are true or false:

1. a) If all the points in the scatter diagram lie exactly on an upward straight line, the correlation is perfect and positive ($r = 1$).
2. b) A circular scatter diagram indicates a very strong correlation.
3. c) In a scatter diagram, the independent variable is always represented on the vertical axis ($Y$).
4. d) A scatter diagram allows us to quickly visualize whether any relationship exists between two numerical variables.

Given the following four scatter diagrams associated with different social phenomena, match each one, giving reasons, to one of these correlation coefficients: $r_1 = 0.95$, $r_2 = -0.80$, $r_3 = 0.05$ and $r_4 = -0.10$.

Justify your choice based on the dispersion relative to a possible line and the direction of the slope of the points.

A sports psychologist studies the relationship between pre-match arousal level (anxiety, variable $X$) and free throw performance of basketball players (success percentage, variable $Y$). The literature suggests that an "optimal level" of arousal exists: if it is too low, the player is apathetic; if it is too high, the player freezes.

1. Draw a hypothetical scatter diagram reflecting this situation (known as the Yerkes-Dodson law).

2. If a data analyst were to calculate the covariance $s_{xy}$ of this scatter diagram and obtain a value close to zero, would it be correct to conclude that anxiety level does not affect performance? Justify your answer by analysing the geometry of the scatter diagram you have drawn.

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

The following table shows the distance covered ($X$, in km) and the time taken ($Y$, in min) by a cyclist on training rides:

1. a) Draw the scatter diagram.
2. b) Sketch freehand the line of best fit for the points (regression line).
3. c) Using the line you have drawn, estimate how long it would take the cyclist to cover 45 km. Do you think this estimation is reliable?

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

A study has been conducted on the relationship between the number of books read per year ($X$) and the score in a reading comprehension test ($Y$) for a group of students. The resulting scatter diagram is shown below:

1. a) State whether a deterministic (functional) relationship or a statistical relationship exists. Explain the difference between both concepts using this example.
2. b) Classify the correlation according to its sign and its strength (strong, moderate or weak).
3. c) Could we state that "reading more books necessarily causes getting a higher mark"? Explain the difference between correlation and causation.

A biologist measures the height of an exotic plant species ($Y$, in cm) as a function of the amount of nitrogenous fertiliser applied ($X$, in mg). She/He observes that, initially, the height increases rapidly with the fertiliser, but beyond a certain amount, the plant becomes saturated and the height stabilises or even decreases slightly due to toxicity.

1. Analyse the scatter diagram and determine whether a linear regression model would be suitable for describing this relationship across the entire domain of $X$.

2. If the biologist calculates Pearson's correlation coefficient for all the data and obtains a value close to $0$, does this mean that the variables are independent? Justify your answer based on the shape of the scatter diagram.

The following scatter diagram has been obtained, relating the monthly electricity consumption of a household ($X$, in kWh) to the mean outdoor temperature for that month ($Y$, in $^\circ C$).

1. Describe the relationship between the variables. Is it a linear relationship across its whole range?

2. Justify why the coefficient of determination $R^2$ for a linear fit would be very low, despite the evident relationship between the variables.

3. Propose a way to split the study of this scatter diagram in order to effectively apply linear regression techniques.

Explain, giving reasons, why the following data could not be correctly represented using a scatter diagram in the context of bivariate statistics:

Given the following pairs of variables, state for which of them it would make sense to construct a scatter diagram to analyse their relationship. Justify your decision in each case:

1. The names of the students in a class and their height.

2. The number of hours of study and the mark in an exam.

3. A person's eye colour and gender.

4. The price of a second-hand car and its distance driven.

In a study on urban demography, population density (inhabitants/$km^2$) is represented on the $X$ axis and the average rent price ($€/m^2$) on the $Y$ axis. The resulting scatter diagram displays a very peculiar structure: two clearly distinct groups of points ("clusters"). Group 1 corresponds to peripheral neighbourhoods and Group 2 to historic centre neighbourhoods.

1. If within each separate group the correlation is positive and strong, but when considering all points together the scatter diagram appears to have no clear trend, what problems would we encounter when attempting to calculate a single Pearson correlation coefficient for the entire city?

2. Explain how the existence of hidden qualitative variables (such as neighbourhood type) can "mislead" the interpretation of a bivariate scatter diagram.

Consider a scatter diagram with a very strong positive linear correlation ($r = 0.98$). Suppose that we transform all the values of variable $X$ using the operation $X' = X + 50$ and all the values of variable $Y$ using $Y' = 3 \times Y$.

1. Describe what changes the scatter diagram would undergo in the plane (translation, deformation, change of scale).

2. Reason, without performing calculations, whether the new correlation coefficient $r'$ will be greater than, less than, or equal to the original $r$.

3. Would the reliability of the predictions change if we used the new transformed scatter diagram? Explain why.