Covariance
In the scatter diagram of height and weight, the points rise from left to right: the taller, the heavier. In one of hours of screen time and hours of sleep, they fall: the more screen time, the less sleep. Covariance is the number that sums up, in a single figure, which way that cloud of points leans.
What covariance measures
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…
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 $Y$: | $X$ | 2 | 4 | 6 | |:---:|:---:|:---:|:---:| | $Y$ | 3 | 5 | 10 |…
Given the following table with three pairs of values corresponding to the variables $X$ and $Y$:
| $X$ | 2 | 4 | 6 |
|---|---|---|---|
| $Y$ | 3 | 5 | 10 |
Calculate the centre of gravity $(\bar{x}, \bar{y})$ and the covariance of the distribution.
The sign of the covariance
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.…
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?
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…
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.
Calculation with a frequency table
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…
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:
| Temperature ($X$) | 10 | 20 | 30 | 40 | 50 |
|---|---|---|---|---|---|
| Strength ($Y$) | 85 | 78 | 65 | 50 | 32 |
Calculate the covariance of this bivariate distribution and interpret its sign in the context of the problem.
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…
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:
| $X \backslash Y$ | 0 | 1 | 2 |
|---|---|---|---|
| 1 | 4 | 6 | 0 |
| 2 | 1 | 5 | 4 |
Calculate the centre of gravity and the covariance $s_{xy}$.
Properties and a warning about outliers
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…
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 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…
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.