Correlation - Class 11 Economics - Chapter 6 - Notes, NCERT Solutions & Extra Questions
Renews every month. Cancel anytime
Your personal doubt-solving assistant
Chatterbot AI gives you 100% accurate answers to your questions in an instant.
Notes - Correlation | Class 11 Statistics For Economics | Economics
Comprehensive Correlation Class 11 Notes: Understanding Relationships Between Variables
Introduction to Correlation
Correlation is a statistical measure that describes the extent to which two variables are related. It provides insight into the direction and intensity of the relationship between the variables, facilitating better understanding and prediction of one variable based on the other.
Importance of Studying Correlation Studying correlation is essential because it helps identify and quantify relationships between different variables. For instance, understanding the relationship between temperature and ice cream sales can aid businesses in predicting peaks in demand.
Types of Correlation
Correlation is broadly categorized into three types:
Positive Correlation: When both variables move in the same direction. For example, as income increases, spending also increases.
Negative Correlation: When variables move in opposite directions. For example, as the price of apples decreases, the demand for apples increases.
No Correlation: When there is no discernible relationship between the variables. For example, the relationship between shoe size and intelligence.
Here's a visual representation of these types of correlation:
Tools for Measuring Correlation
Scatter Diagrams A scatter diagram is a graphical representation used to examine the form of the relationship between two variables. Data points are plotted on a graph to illustrate whether a relationship exists:
graph LR
A(X rises, Y rises)
B(X rises, Y falls)
C(No relationship between X and Y)
A -->|Positive Correlation| P(Upward Scatter)
B -->|Negative Correlation| N(Downward Scatter)
C -->|No Correlation| NC(Random Scatter)
Karl Pearson's Coefficient of Correlation This is a numerical measure of linear relationship between two variables $ X $ and $ Y $. It is calculated using the formula: $$ r = \frac{N \sum XY - (\sum X)(\sum Y)}{\sqrt{[N \sum X^2 - (\sum X)^2][N \sum Y^2 - (\sum Y)^2]}} $$
When to Use Pearson’s Correlation: Pearson’s Correlation is used when the relationship between the variables is linear and the data is normally distributed.
Spearman's Rank Correlation This method is used when the data does not meet the assumptions of Pearson’s correlation or when the relationships are non-linear. Spearman's rank correlation coefficient is calculated using the formula: $$ r_s = 1 - \frac{6 \sum D^2}{n(n^2 - 1)} $$ Where $ D $ is the difference between the ranks.
Handling Ties in Rankings: When there are ties, the average of the ranks is used. This ensures that the rank correlation still accurately reflects the relative ordering of the data.
When to Use Spearman’s Correlation: Spearman's correlation is preferred when dealing with ordinal data or when the assumptions of Pearson’s correlation are not met.
Properties of Correlation Coefficients
- Range: The value of the correlation coefficient lies between -1 and +1.
- No Unit of Measurement: It is a pure number.
- Sign Indication: Positive values indicate direct relationships, while negative values indicate inverse relationships.
- Independence from Origin and Scale Change: Changes in units of measurement or offsets do not affect the correlation coefficient.
Step-by-Step Calculation Examples
Example Using Karl Pearson's Coefficient
Given sample data on the number of years of schooling and annual yield per acre for farmers, the coefficient can be calculated as follows:
$$ \text{Given Data:} $$ $$ \begin{array}{c|c} \text{Years of Schooling} & \text{Annual Yield per Acre (in '000s)} \\ \hline 0 & 4 \\ 2 & 4 \\ 4 & 6 \\ 6 & 10 \\ 8 & 10 \\ 10 & 8 \\ 12 & 7 \\ \end{array} $$
Step-by-step, calculate the means, variances, and finally the correlation coefficient.
Example Using Spearman's Rank Correlation
Consider ranking five students in terms of height and weight:
$$ \text{Given Data:} $$ $$ \begin{array}{c|c|c} \text{Student} & \text{Height (Rank)} & \text{Weight (Rank)} \\ \hline A & 1 & 2 \\ B & 2 & 4 \\ C & 3 & 1 \\ D & 4 & 5 \\ E & 5 & 3 \\ \end{array} $$
Steps to compute include ranking the values, calculating the differences in ranks, squaring those differences, summing them up, and then applying the rank correlation formula.
Practical Applications of Correlation in Class 11
Classroom Activities and Data Analysis: Collect real data from students related to height, weight, and marks scored across subjects. Using scatter diagrams, Karl Pearson's coefficient, and Spearman's rank correlation can provide insightful correlation analyses.
Real-life Examples for Better Understanding:
- Studying the correlation between study hours and marks obtained.
- Analyzing the relationship between the number of hours exercised and weight loss.
Common Misinterpretations and Limitations
Correlation vs. Causation: It's crucial to understand that correlation does not imply causation. A high correlation simply indicates a relationship, not that one variable causes the other.
Non-linear Relationships: Not all relationships are linear, and correlation measures may not apply efficiently to non-linear relationships.
Conclusion
Understanding correlation is vital for gauging relationships between variables in statistics. Tools like scatter diagrams, Karl Pearson’s and Spearman’s correlation coefficients offer valuable insights, though they do not imply causation. Familiarity with these concepts helps in accurately interpreting data relationships.
Additional Resources
For further study, students can refer to recommended books and online resources that provide more practice problems and detailed explanations.
By mastering these concepts, students not only prepare well for their exams but also develop critical thinking skills essential for real-world data analysis.
🚀 Learn more about Notes with Chatterbot AI
Extra Questions - Correlation | Statistics For Economics | Economics | Class 11
The number of terms which are identical in the sequence 2, 5, 8, 11, ..., up to 60 terms and 3, 5, 7, 9, ..., up to 50 terms, is A 16 B 19 C 18 D 17
The number of terms which are identical in the sequence 2, 5, 8, 11, ..., up to 60 terms and 3, 5, 7, 9, ..., up to 50 terms, is
Considering the sequences provided:
First Sequence: 2, 5, 8, 11, ..., up to 60 terms
General formula: $a + (n-1)d$
Here, $a = 2$, $d = 3$
$T_{60} = 2 + (60-1) \cdot 3 = 2 + 177 = 179$
Hence, the last term of the first sequence is 179.
Second Sequence: 3, 5, 7, 9, ..., up to 50 terms
General formula: $a + (n-1)d$
Here, $a = 3$, $d = 2$
$T_{50} = 3 + (50-1) \cdot 2 = 3 + 98 = 101$
Hence, the last term of the second sequence is 101.
To Find Common Terms:
Common Difference:
The common difference of the common terms between the two sequences is the LCM of the differences.
$ \text{LCM}(3, 2) = 6 $
First Common Term: It is evident that 5 is the first common term.
Form the New Sequence with Common Difference:
Sequence: 5, 11, 17, ...,
To find the number of terms (n) such that the nth term is less than or equal to 101: $$ 5 + (n-1) \cdot 6 \leq 101$$ $$ 5 + 6n - 6 \leq 101 $$ $$ 6n - 1 \leq 101 $$ $$ 6n \leq 102$$ $$ n \leq 17$$
Therefore, the number of common terms in both the sequences is 17.
Answer: 17
💡 Have more questions?
Ask Chatterbot AINCERT Solutions - Correlation | Statistics For Economics | Economics | Class 11
The unit of correlation coefficient between height in feet and weight in kgs is.......
(i) $\mathrm{kg} /$ feet
(ii) percentage
(iii) non-existent
The correlation coefficient is a dimensionless measure that indicates the strength and direction of a linear relationship between two variables. It ranges from -1 to 1, where:
1 indicates a perfect positive relationship
-1 indicates a perfect negative relationship
0 indicates no linear relationship
Since it is dimensionless, the unit of the correlation coefficient between height in feet and weight in kilograms is non-existent.
So, the correct answer is:
(iii) non-existent
The range of simple correlation coefficient is.........
(i) 0 to infinity
(ii) minus one to plus one
(iii) minus infinity to infinity
The range of a simple correlation coefficient is from -1 to +1. So, the correct answer is:
(ii) minus one to plus one
If $r_{x y}$ is positive the relation between $X$ and $Y$ is of the type
(i) When $\mathrm{Y}$ increases $\mathrm{X}$ increases
(ii) When $\mathrm{Y}$ decreases $\mathrm{X}$ increases
(iii) When $\mathrm{Y}$ increases $\mathrm{X}$ does not change
If $r_{xy} $ is positive, it indicates a positive correlation between $ X $ and $ Y $. Therefore, the relation between $ X $ and $ Y $ is such that:
(i) When $ Y $ increases, $ X $ increases.
So, the correct relation is:
(i) When $ Y $ increases, $ X $ increases.
If $r_{x y}=0$ the variable $X$ and $Y$ are
(i) linearly related
(ii) not linearly related
(iii) independent
If $ r_{xy} = 0 $, it means that the variables $ X $ and $Y$ are not linearly related. They could still have a nonlinear relationship, or they might be independent, but linear correlation specifically measures linear relationships.
So the correct answer is: (ii) not linearly related
Of the following three measures which can measure any type of relationship
(i) Karl Pearson's coefficient of correlation
(ii) Spearman's rank correlation
(iii) Scatter diagram
The appropriate measure that can be used to measure any type of relationship among the given options is:
(iii) Scatter diagram
Explanation:
Karl Pearson's coefficient of correlation:
Measures the strength and direction of the linear relationship between two variables.
It is not suitable for non-linear relationships.
Spearman's rank correlation:
Measures the strength and direction of the monotonic relationship between two ranked variables.
Useful for ordinal data and non-parametric data but may not capture more complex relationships.
Scatter diagram:
A graphical representation that shows the relationship between two variables.
It can display any type of relationship (linear, non-linear, etc.) and is not restricted to specific types of data. It provides a visual understanding of the potential association, be it linear, quadratic, or more complex.
Thus, the scatter diagram is the most versatile among the three for measuring any type of relationship.
If precisely measured data are available the simple correlation coefficient is
(i) more accurate than rank correlation coefficient
(ii) less accurate than rank correlation coefficient
(iii) as accurate as the rank correlation coefficient
If precisely measured data are available, the simple correlation coefficient is:
(i) more accurate than rank correlation coefficient
The simple correlation coefficient (such as Pearson's correlation) uses the actual data values and can give a more precise measure of the linear relationship between the variables. In contrast, the rank correlation coefficient (such as Spearman's rank correlation) uses the ranks of the data and is generally used for ordinal data or when the data doesn't meet the assumptions required for Pearson's correlation. Hence, the precision of the simple correlation coefficient makes it more accurate when precise measurements are available.
Why is $\mathrm{r}$ preferred to covariance as a measure of association?
The correlation coefficient, denoted as $\mathrm{r}$, is preferred over covariance as a measure of association for several reasons:
Normalization: Correlation $\mathrm{r}$ is a standardized measure, which means it ranges between -1 and 1, making it easier to interpret. Covariance, on the other hand, is not standardized and its magnitude depends on the units of the variables, making it difficult to compare across different datasets.
Interpretability: The value of the correlation coefficient directly indicates the strength and direction of the linear relationship between two variables. An $\mathrm{r}$ value close to 1 indicates a strong positive linear relationship, while a value close to -1 indicates a strong negative linear relationship. A value of 0 indicates no linear relationship.
Unit Independence: Correlation is unit-free, meaning it is not affected by the scales of the variables. Covariance, however, is affected by the units of the variables, which can complicate interpretation when comparing covariances from different datasets.
Comparability: Because correlation is standardized, it allows for easy comparison of the strength of relationships across different pairs of variables, regardless of their original units.
For these reasons, $\mathrm{r}$ is generally more useful and meaningful as a measure of association compared to covariance.
Can $r$ lie outside the -1 and 1 range depending on the type of data?
No, the correlation coefficient $ r $ cannot lie outside the range of -1 to 1. This is because the correlation coefficient, which measures the strength and direction of the linear relationship between two variables, is mathematically constrained to fall within this range.
Here's a brief explanation:
$ r = 1 $ indicates a perfect positive linear relationship.
$ r = -1 $ indicates a perfect negative linear relationship.
$ r = 0 $ indicates no linear relationship.
This range is a fundamental property of the correlation coefficient due to its mathematical definition and how it is calculated from the covariance of the variables and the product of their standard deviations.
Does correlation imply causation?
No, correlation does not imply causation. Correlation simply means that there is a relationship between two variables, meaning that when one variable changes, the other tends to change in a predictable way. However, this relationship does not mean that one variable causes the change in the other. There could be other factors (confounding variables) that affect both variables, or it could be a coincidence.
For example, ice cream sales and drowning incidents tend to increase during the summer. While these two events are correlated, it does not mean that ice cream sales cause drowning incidents. Instead, both variables are affected by the common factor of warm weather.
When is rank correlation more precise than simple correlation coefficient?
Rank correlation is often more precise than the simple (or Pearson) correlation coefficient in the following scenarios:
Non-linear Relationships: If the relationship between the variables is non-linear, rank correlation (like Spearman's rho or Kendall's tau) can more accurately capture the monotonic relationship, whereas Pearson's correlation assumes linearity.
Ordinal Data: When the data is ordinal (rank-ordered), rank correlation is more appropriate. Pearson's correlation coefficient assumes interval or ratio scale data.
Outliers: Rank correlation is less sensitive to outliers than Pearson's correlation coefficient. Extreme values can disproportionately affect the Pearson correlation, but rank correlation, which uses ranks instead of actual data values, mitigates this issue.
Non-normal Distribution: If the data is not normally distributed, rank correlation can provide a more accurate measure of association. Pearson’s correlation assumes that the data is normally distributed.
Overall, rank correlation is more precise or appropriate in cases where data characteristics violate the assumptions underlying Pearson's correlation.
Does zero correlation mean independence?
Zero correlation does not necessarily imply independence. Correlation measures the linear relationship between two variables. A zero correlation means there is no linear relationship between the variables. However, they might still be dependent in a non-linear way.
For example, consider a case where one variable depends on the square of another variable. This non-linear relationship wouldn't be captured by correlation, which is a measure of linear dependence, thus resulting in a zero correlation despite the variables being dependent.
Can simple correlation coefficient measure any type of relationship?
No, the simple correlation coefficient, commonly known as Pearson's correlation coefficient, measures only the strength and direction of a linear relationship between two variables. It cannot accurately represent non-linear relationships or other types of dependencies between variables. For non-linear relationships, alternative measures such as Spearman's rank correlation or Kendall's tau coefficient are more appropriate.
Collect the price of five vegetables from your local market every day for a week. Calculate their correlation coefficients. Interpret the result.
To collect and calculate the correlation coefficients of the prices of five vegetables over a week, follow these steps:
Data Collection:
Record the prices of five vegetables (let’s say, Tomato, Potato, Onion, Carrot, and Cabbage) every day for a week. You should have 7 prices for each vegetable.
Arrange the data in a table, where rows represent days and columns represent the vegetables.
Data Entry:
Enter the daily price data for each vegetable in a spreadsheet or statistical software.
Calculating Correlation Coefficients:
Use a statistical software or a tool like MS Excel to compute the correlation coefficients between each pair of vegetables.
Here’s an example of how you can set this up in a spreadsheet:
Day | Tomato | Potato | Onion | Carrot | Cabbage |
---|---|---|---|---|---|
1 | 20 | 35 | 15 | 25 | 20 |
2 | 22 | 34 | 16 | 26 | 21 |
3 | 21 | 36 | 15 | 24 | 19 |
4 | 23 | 37 | 14 | 27 | 22 |
5 | 24 | 33 | 15 | 23 | 20 |
6 | 25 | 32 | 16 | 25 | 21 |
7 | 23 | 31 | 17 | 26 | 22 |
Interpreting the Results:
The correlation coefficient (r) ranges from -1 to +1.
+1: Perfect positive correlation. As one vegetable's price increases, the other’s price also increases.
0: No correlation. The price of one vegetable does not affect the price of the other.
-1: Perfect negative correlation. As one vegetable's price increases, the other’s price decreases.
You should interpret each pairwise correlation coefficient between the vegetables to understand the relationship between their price movements.
Let me further assist in calculating the correlation coefficients. Do you have specific price data you'd like to calculate this on, or should I use the hypothetical data above?
Measure the height of your classmates. Ask them the height of their benchmate. Calculate the correlation coefficient of these two variables. Interpret the result.
To calculate the correlation coefficient, you'll need to:
Collect Data: Measure the height of each classmate and ask them the height of their benchmate.
Organize Data: Create two lists or columns where one represents the heights of each classmate and the other represents the heights of their benchmates.
Calculate Correlation Coefficient: Use the formula for the Pearson correlation coefficient.
The formula for the Pearson correlation coefficient $ r $ is:
$$r = \frac{n(\sum xy) - (\sum x)(\sum y)}{\sqrt{[n \sum x^2 - (\sum x)^2][n \sum y^2 - (\sum y)^2]}} $$
Where:
$ n $ is the number of pairs,
$ x $ is the height of the classmate,
$ y $ is the height of the benchmate.
After calculating $ r $:
Interpret the result:
$r = 1 $: Perfect positive correlation.
$ r = -1 $: Perfect negative correlation.
$ r = 0 $: No correlation.
$ 0 < r < 1 $: Positive correlation.
$ -1 < r < 0 $: Negative correlation.
If you provide the data, I can compute the correlation coefficient for you!
List some variables where accurate measurement is difficult.
Accurate measurement can be challenging for several economic variables due to various complexities involved. Some of these variables include:
Shadow Economy (Informal/Underground Economy): Activities that are not recorded or reported to avoid taxes and regulations.
Consumer Confidence: Reflects people's expectations about the economic future, influenced by subjective perceptions.
Inflation Expectations: Public and market expectations of future inflation, which can be erratic and influenced by various factors.
Quality of Life: Involves subjective factors such as happiness, well-being, and life satisfaction.
Productivity in Services: Measuring output can be complex due to the intangible nature of services.
Unemployment: Underemployment and discouraged workers often make the true unemployment rate hard to gauge.
Environmental Degradation: Quantifying non-market damage to the environment and its economic impact.
Human Capital: Skills, knowledge, and experience possessed by an individual or population.
Technological Progress: Innovations and technological improvements and their comprehensive economic impacts.
Income Inequality: Discrepancies in income distribution within a population.
These variables often involve subjective judgments, non-market activities, or diffuse and indirect impacts, complicating their accurate measurement.
Interpret the values of $r$ as $1,-1$ and 0 .
In statistics, the correlation coefficient ( r ) measures the strength and direction of a linear relationship between two variables. Its values range from -1 to 1. Here is the interpretation for ( r ) when it equals 1, -1, and 0:
( r = 1 ) (Perfect Positive Correlation):
This means that there is a perfect positive linear relationship between the variables.
As one variable increases, the other variable also increases proportionally.
The data points lie exactly on a straight line with a positive slope.
( r = -1 ) (Perfect Negative Correlation):
This indicates a perfect negative linear relationship between the variables.
As one variable increases, the other variable decreases proportionally.
The data points lie exactly on a straight line with a negative slope.
( r = 0 ) (No Correlation):
This suggests that there is no linear relationship between the two variables.
Changes in one variable do not predict changes in the other variable in a linear sense.
The data points are scattered in such a way that no linear pattern can be identified.
Why does rank correlation coefficient differ from Pearsonian correlation coefficient?
The rank correlation coefficient (often Spearman's Rank Correlation Coefficient) and the Pearsonian (Pearson) correlation coefficient differ primarily in how they measure relationships between variables and what kinds of relationships they capture. Here are the main differences:
Rank Correlation Coefficient (Spearman's Rank)
Nature of Data:
Ordinal Data: Spearman's rank correlation can be used with ordinal data (data that can be ranked but may not have a meaningful numerical difference between ranks).
Non-linear relationships: It measures the strength and direction of the association between two ranked variables, capturing even monotonic (not strictly linear) relationships.
Calculation Method:
Rank Order: Based on the ranks of the data rather than the raw data values.
Uses the formula: $$ r_s = 1 - \frac{6 \sum d_i^2}{n(n^2 - 1)}$$ where $d_i$ is the difference between the ranks of corresponding variables and (n) is the number of observations.
Robustness:
Outliers: Less sensitive to outliers since it uses ranks.
Pearsonian Correlation Coefficient
Nature of Data:
Interval/Ratio Data: Pearson's correlation is suitable for continuous, interval, or ratio data.
Linear relationships: It measures the strength and direction of the linear relationship between two continuous variables.
Calculation Method:
Raw Data: Based on the actual data values.
Uses the formula: $$ r = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum (x_i - \bar{x})^2 \sum (y_i - \bar{y})^2}} $$ where $x_i$ and $y_i$ are the data points, and $\bar{x}) and (\bar{y}$ are the means of $x$ and $y$, respectively.
Robustness:
Outliers: More sensitive to outliers which can significantly affect the correlation coefficient.
Summary
Rank Correlation (Spearman): Used for ordinal data or non-linear relationships; based on rank order.
Pearson Correlation: Used for continuous data measuring linear relationships; based on actual data values.
Because they are designed to capture different aspects of relationships between variables and are based on different mathematical principles, they can yield different values for the same dataset.
Calculate the correlation coefficient between the heights of fathers in inches (X) and their sons (Y)
X | 65 | 66 | 57 | 67 | 68 | 69 | 70 | 72 |
Y | 67 | 56 | 65 | 68 | 72 | 72 | 69 | 71 |
The Pearson correlation coefficient between the heights of fathers (X) and their sons (Y) is approximately 0.438. This indicates a moderate positive correlation between the two sets of heights.
Here is the plot illustrating the relationship:
Calculate the correlation coefficient between $\mathrm{X}$ and $\mathrm{Y}$ and comment on their relationship:
X | -3 | -2 | 1 | 1 | 2 | 3 |
Y | 9 | 4 | 1 | 1 | 4 | 9 |
Calculation of Correlation Coefficient
The Pearson correlation coefficient for the provided datasets is calculated as: $$ r \approx -0.174 $$
Comment on the Relationship
Correlation Value: The correlation coefficient $ r \approx -0.174 $ indicates a weak negative linear relationship between $X$ and $Y$.
Direction of Relationship: Since $ r $ is negative, as $ X $ increases, $ Y $ tends to decrease, but the relationship is very weak.
Plot:
The plot visually confirms the weak negative correlation.
Conclusion: The data shows a weak negative correlation, suggesting only a slight inverse relationship between the variables $X$ and $Y$. This implies that as one variable increases, the other tends to decrease, but not strongly.
Calculate the correlation coefficient between $\mathrm{X}$ and $\mathrm{Y}$ and comment on their relationship
X | 1 | 3 | 4 | 5 | 7 | 8 |
Y | 2 | 6 | 8 | 10 | 14 | 16 |
It seems there was an issue with understanding the correlation computation format. Let's manually verify the calculation using Karl Pearson's formula for correlation coefficient ( r ).
Given data:
$X = {1, 3, 4, 5, 7, 8} $
$ Y = {2, 6, 8, 10, 14, 16}$
The formula for the Pearson correlation coefficient is:
$$ r = \frac{N \sum XY - \sum X \sum Y}{\sqrt{N \sum X^2 - (\sum X)^2} \cdot \sqrt{N \sum Y^2 - (\sum Y)^2}} $$
Let's calculate based on this formula.
Step-by-Step Calculation:
Sum of (X) and (Y):$$ \sum X = 1 + 3 + 4 + 5 + 7 + 8 = 28 $$ $$ \sum Y = 2 + 6 + 8 + 10 + 14 + 16 = 56 $$
Sum of (X^2) and (Y^2):$$ \sum X^2 = 1^2 + 3^2 + 4^2 + 5^2 + 7^2 + 8^2 = 1 + 9 + 16 + 25 + 49 + 64 = 164 $$ $$ \sum Y^2 = 2^2 + 6^2 + 8^2 + 10^2 + 14^2 + 16^2 = 4 + 36 + 64 + 100 + 196 + 256 = 656 $$
Sum of (XY):$$ \sum XY = (1 \cdot 2) + (3 \cdot 6) + (4 \cdot 8) + (5 \cdot 10) + (7 \cdot 14) + (8 \cdot 16) = 2 + 18 + 32 + 50 + 98 + 128 = 328 $$
Number of data points (N):$$ N = 6 $$
Calculate (r):$$ r = \frac{6 \cdot 328 - 28 \cdot 56}{\sqrt{6 \cdot 164 - 28^2} \cdot \sqrt{6 \cdot 656 - 56^2}} $$
Breaking it down: $$ r = \frac{1968 - 1568}{\sqrt{984 - 784} \cdot \sqrt{3936 - 3136}} $$
Simplifying further: $$ r = \frac{400}{\sqrt{200} \cdot \sqrt{800}} $$ $$ r = \frac{400}{\sqrt{160000}} $$ $$ r = \frac{400}{400} = 1 $$
Conclusion
The Pearson correlation coefficient $ r $ is 1, indicating a perfect positive linear relationship between $ X $ and $ Y $. This means that as $ X $ increases, $ Y $ increases proportionally.
💡 Have more questions?
Ask Chatterbot AI