Exercises: Covariance

In a bivariate study on $25$ individuals, the means $\bar{x} = 8$ and $\bar{y} = 15$ were obtained. Knowing that the sum of the products of the variables is $\sum x_i y_i = 3200$, calculate the covariance $s_{xy}$.

Given the following table with three pairs of values corresponding to the variables $X$ and $Y$:

Calculate the centre of gravity $(\bar{x}, \bar{y})$ and the covariance of the distribution.

In a study on sports performance, the number of weekly training hours ($X$) is related to the time in minutes that an athlete takes to complete a $10$ km race ($Y$). What sign do you expect the covariance $s_{xy}$ to have? Briefly give reasons for your answer.

For a set of $40$ bivariate data, it is known that $\bar{x} = 12$, $\bar{y} = 25$ and the covariance is $s_{xy} = -8,5$. Find the value of the sum of the products $\sum x_i y_i$.

In a thermodynamics experiment, the covariance between the pressure and volume of a gas under certain conditions is $s_{xy} = -15,4$. If, to simplify calculations, the researcher decides to add $2$ to all pressure values and subtract $5$ from all volume values, what will be the new value of the covariance?

Calculate the covariance of the following bivariate distribution given in a table of absolute frequencies $f_i$:

When studying the relationship between the amount of fertiliser applied to a plant (in grams) and its weekly growth (in millimetres), a covariance of $24,6$ is obtained. What does the sign of this statistical parameter indicate about the relationship between fertiliser and growth?

The centre of gravity of a bivariate distribution is $(5, 9)$. If the arithmetic mean of the products of the values of both variables is $\frac{\sum x_i y_i}{N} = 42$, determine the value of the covariance.

Based on records from the State Meteorological Agency (AEMET) in the province of Granada during a winter day, the altitude ($X$, in hundreds of metres) and the mean temperature ($Y$, in $^\circ\text{C}$) of four towns have been recorded: Motril $(0; 14)$, Lanjarón $(5; 10)$, Baza $(8; 7)$, and Trevélez $(15; 1)$.\na) Before performing any numerical calculation, reasonably estimate which sign you expect the covariance between altitude and temperature to have. Explain what you base your forecast on.\nb) With the given data, calculate the means $\bar{x}$ and $\bar{y}$, and find the exact value of the covariance $\sigma_{xy}$.\nc) Check whether the obtained sign matches your initial estimate and assess whether the result is consistent with the physical phenomenon observed in mountain areas.

In a school study on leisure habits, Marcos recorded the weekly hours spent studying ($X$) and the hours spent playing video games ($Y$) of $5$ classmates: $(1, 6)$, $(2, 5)$, $(3, 4)$, $(4, 2)$, and $(5, 3)$.\nTo calculate the covariance, Marcos writes in his notebook:\n"I calculate the means: $\bar{x} = 3$ and $\bar{y} = 4$.\nI multiply each pair and add them up: $\sum x_i y_i = 1\cdot 6 + 2\cdot 5 + 3\cdot 4 + 4\cdot 2 + 5\cdot 3 = 6 + 10 + 12 + 8 + 15 = 51$.\nI apply the formula: $\sigma_{xy} = 51 - 3 \cdot 4 = 51 - 12 = 39$.\nSince the covariance is positive ($\sigma_{xy} = 39 > 0$), I conclude that the more hours they play video games, the more hours they study."\na) Identify and explain in detail the mathematical error Marcos made in applying the formula.\nb) Explain why his qualitative conclusion is clearly contradictory when looking at the original data in the table.\nc) Correct the procedure and calculate the correct value of $\sigma_{xy}$, properly interpreting the obtained result.

The temperature $X$ (in $^\circ\text{C}$) and the strength $Y$ (in MPa) of a new plastic material have been measured in $5$ different tests, obtaining the following results:

Calculate the covariance of this bivariate distribution and interpret its sign in the context of the problem.

In an architectural study, the covariance between the length of a beam $X$ (measured in metres) and its maximum deflection $Y$ (measured in millimetres) is $s_{xy} = 3,5$. If it is decided to express the length in centimetres ($1\text{ m} = 100\text{ cm}$) and the deflection in micrometres ($1\text{ mm} = 1000\,\mu\text{m}$), calculate the new value of the covariance.

A survey was conducted among $20$ families on the number of vehicles they own ($X$) and the number of garage spaces they own ($Y$). The results are summarised in the following two-way table:

Calculate the centre of gravity and the covariance $s_{xy}$.

A set of $10$ bivariate data pairs has means $\bar{x} = 6$ and $\bar{y} = 8$, and its covariance is $s_{xy} = 4$. If it is discovered that an eleventh data pair is missing, which coincides exactly with the centre of gravity, that is, the pair $(6, 8)$, calculate the new covariance of the set of $11$ data values.

Look at the following scatter plot representing four observations obtained in a laboratory experiment:

From the integer coordinates of the points shown in the graph, determine the table of values and calculate the covariance of the distribution.

In a study on screen time, the age of the respondents ($X$) has been grouped into intervals, relating it to the number of smart devices they own ($Y$):

Using the class marks for the variable $X$, calculate the covariance of the distribution.

For a bivariate statistical variable $(X,Y)$ consisting of $8$ observations, the following data are known:

$\sum x_i = 40$, $\sum y_i = 64$, and $\sum x_i y_i = 280$.

Calculate the centre of gravity and the covariance.

In an economic study on the relationship between the inflation rate ($X$, in $\%$) and the unemployment rate ($Y$, in $\%$) of a country over $5$ consecutive years, the following pairs of values were recorded: $(2, 8)$, $(3, 6)$, $(4, 5)$, $(5, 4)$, $(6, 2)$. Calculate the covariance and indicate whether the results support a direct or inverse relationship between both rates.

Based on data from the National Statistics Institute (INE) on household equipment and consumption, the mean outdoor temperature during the month of July ($X$, in $^\circ\text{C}$) and the monthly electricity consumption per household ($Y$, in $\text{kWh}$) are analysed across five Spanish provincial capitals: Santander $(19, 170)$, A Coruña $(20, 180)$, Zaragoza $(26, 290)$, Madrid $(26, 300)$, and Sevilla $(29, 360)$.\na) Model the relationship between both variables by calculating the covariance $\sigma_{xy}$. What kind of joint trend does the obtained sign reflect in relation to the use of air conditioning systems?\nb) If an electricity distributor prefers to record consumption in megawatt-hours using the change of variable $Y' = 0,001 \cdot Y$, calculate the new covariance $\sigma_{xy'}$ applying the algebraic properties of covariance, without recalculating the products point by point.\nc) Explain why changing the unit of measurement alters the numerical value of the covariance but does not modify its qualitative interpretation at all.

In a vegetable canning factory in Navarra, the relationship between thermal sterilisation time ($X$, in minutes) and residual microbial count ($Y$, in colonies per gram) is monitored. In four analysed batches, the pairs $(15, 80)$, $(20, 50)$, $(25, 30)$, and $(30, k)$ were obtained, but the value $k$ of the fourth batch was accidentally erased from the control record.\nThe preserved technical quality report indicates that the sample covariance was exactly $\sigma_{xy} = -125$.\na) Express the marginal mean $\bar{y}$ and the average of the products $\dfrac{\sum x_i y_i}{4}$ in terms of the unknown $k$.\nb) Set up an equation using the covariance formula and solve for the exact value that $k$ must have.\nc) Explain what practical meaning it has for the factory's food safety that the covariance is negative, and what it would have indicated about the thermal process if a covariance close to $0$ had been obtained.

A new drug has been tested on a group of patients. The variable $X$ represents the administered dose (in mg) and the variable $Y$ the reduction in blood pressure (in mmHg). The results are shown in the following two-way table:

Calculate the covariance of this distribution and explain what the result suggests about the efficacy of the drug.

The trajectory of an object is analysed using $5$ points whose coordinates $(X,Y)$ are: $(-2, 4)$, $(-1, 1)$, $(0, 0)$, $(1, 1)$ and $(2, 4)$.

Calculate the covariance of this bivariate distribution. In view of the result, can it be stated that the variables $X$ and $Y$ are statistically independent? Justify your answer by linking it to the graph.

In a bivariate distribution of $3$ pairs of values, two of them are known: $(1, 2)$ and $(4, 6)$. The third pair is $(2, a)$, where $a$ is an unknown value. If it is known that the covariance of the distribution is $s_{xy} = 2$, set up an equation and find the value of $a$.

Given a bivariate variable $(X,Y)$, it is known that an exact linear functional relationship given by the equation $y_i = -3x_i + 5$ exists for all pairs of the distribution. Knowing that the variance of the variable $X$ is $s_x^2 = 4$, prove analytically and calculate the numerical value of the covariance $s_{xy}$.

In a company, the relationship between employees' length of service ($X$) and their monthly salary in hundreds of euros ($Y$) is studied. The following data are known: $\bar{x} = 10$, $\bar{y} = 20$, the variance of length of service is $s_x^2 = 16$ and the covariance is $s_{xy} = 24$. It is decided to create a "total valuation" index defined as $Z = X + Y$. Calculate the covariance between length of service ($X$) and this new index ($Z$).

A team of biologists measured the weight $X$ (in kg) and the length $Y$ (in cm) of a species of reptile, obtaining a covariance of $s_{xy} = 45$. For an international publication, they decide to transform the data: the weight will be expressed in pounds ($1\text{ kg} = 2,2\text{ lb}$) and the length in metres ($1\text{ m} = 100\text{ cm}$). Furthermore, due to an adjustment of the measuring equipment, they subtract $0,5\text{ m}$ from all lengths. What will be the value of the covariance in the new units?

A marketing department analyses advertising expenditure $X$ (in thousands of €) and generated sales $Y$ (in thousands of €) across $10$ different campaigns. The following sums are available:

$\sum x_i = 150$, $\sum y_i = 800$, $\sum x_i^2 = 2500$, $\sum y_i^2 = 70000$, $\sum x_i y_i = 12500$.

Calculate the original covariance. If for next year it is decided that all advertising expenditure will increase by $10\%$ and expected sales will increase by $20\%$, what will be the estimated covariance of the transformed data?

A set of bivariate data consists of the following $5$ tightly clustered points: $(1, 1)$, $(2, 2)$, $(2, 1)$, $(3, 2)$, $(3, 3)$; and an outlier: $(13, 15)$.

Calculate the covariance of the full set of $6$ points. Next, calculate the covariance excluding the outlier and compare the results to explain the sensitivity of covariance to extreme values.

A courier company audits four vans in its fleet by recording their registration year ($X$) and total accumulated kilometres ($Y$): Van 1 $(2018; 120\,000)$, Van 2 $(2019; 95\,000)$, Van 3 $(2021; 50\,000)$, and Van 4 $(2022; 15\,000)$.\nTwo analysts discuss which approach to follow to calculate the covariance:\n- Method A (Sergio): Directly apply the formula to the original data, calculating $\bar{x}$, $\bar{y}$, and products in the millions such as $2018 \cdot 120\,000$.\n- Method B (Marta): Define a change of origin and scale using $X' = X - 2020$ (years elapsed relative to 2020) and $Y' = \dfrac{Y}{1000}$ (kilometres in thousands), calculate $\sigma_{x'y'}$, and translate the result back to the original covariance.\na) Compare both methods and justify why Marta's proposal drastically minimises the risk of making calculation errors.\nb) Apply Method B: verify that $\bar{x}' = 0$, calculate $\sigma_{x'y'}$ in a simple way, and, applying the properties of linear transformations, deduce the exact value of the original covariance $\sigma_{xy}$.\nc) Based on this case, provide a reasoned justification for why adding a constant to a variable does not alter the covariance, whereas multiplying it by a constant does.

An environmental report analyses the relationship between the number of registered rural accommodations ($X$, in tens) and the solid waste collected in August ($Y$, in tonnes) across five districts of an autonomous community. In four districts with a fully rural profile, the data are: District A $(2, 10)$, District B $(3, 15)$, District C $(4, 20)$, and District D $(5, 25)$. However, District E presents an atypical profile due to the presence of a large industrial park and very little rural tourism, recording $(1, 80)$.\na) Model the relationship by calculating the covariance $\sigma_{xy}$ considering only the four typical rural districts (A, B, C, and D). Interpret the obtained sign in terms of tourist activity and waste generation.\nb) Include District E in the study ($N = 5$) and calculate the new covariance for all five districts combined.\nc) Compare both results and explain what phenomenon caused the radical change of sign in the covariance. In light of this, critically explain why a public manager would make a serious mistake if they used the overall covariance of the five districts to claim that "promoting rural tourism helps reduce the amount of waste generated".