How two variables are related: correlation, direct and inverse variation
TEAS 7 outline: Explain the relationship between two variables
- The independent variable is the input (x); the dependent variable responds (y).
- Rising dots mean positive correlation; falling dots mean negative correlation.
- Direct variation, y = kx: y ÷ x stays the same.
- Inverse variation, y = k/x: x × y stays the same.
- Correlation does not imply causation.
Key facts
- The independent variable is the input (x, horizontal axis); the dependent variable is the output that responds to it (y, vertical axis).
- Positive correlation: as x increases, y tends to increase. Negative correlation: as x increases, y tends to decrease.
- Points with no pattern, or a flat horizontal pattern, show no relationship: knowing x does not help predict y.
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- The correlation coefficient r is always between −1 and 1; values closer to −1 or 1 mean a stronger straight-line relationship.
- Direct variation is y = kx: the ratio y ÷ x is the same constant k for every pair, and doubling x doubles y.
- Inverse variation is y = k/x: the product x × y is the same constant k for every pair, and doubling x halves y.
- Correlation does not imply causation: two variables can move together without one causing the other.
Quick check1 question · 2 more at the end
A home health agency pays its aides $22 for each home visit. An aide's weekly pay, P, is found with the equation P = 22v, where v is the number of visits made that week. Which statement correctly describes the variables?
Correct answer: B.
The independent variable is the input, and the dependent variable is the result that responds to it. Pay depends on how many visits the aide makes, so the number of visits, v, is independent and the pay, P, is dependent. In an equation written as y = kx, x is the independent variable and y is the dependent one; the fixed number 22 is a constant, not a variable.
A This reverses the roles; the number of visits drives the pay, not the other way around.
B Correct: pay depends on the number of visits, so v is independent and P is dependent.
C 22 is a fixed rate (the constant of variation), so it is not a variable at all.
D P changes in response to v, so P depends on v and cannot be independent.
Full lesson
Each part starts with its main idea. Open a part when you want the details.
The short answer: The independent variable (x) is the input you choose; the dependent variable (y) responds to it. On a scatter plot, rising dots show a positive correlation and falling dots a negative one. In direct variation, y = kx, so doubling x doubles y. In inverse variation, y = k/x, so as x increases, y decreases.
1 Which variable is independent and which is dependent?The independent variable is the input, and the dependent variable is the result that responds to it. 1 min
The independent variable is the input, and the dependent variable is the result that responds to it. In an equation like y = 4x + 10, x is the independent variable and y is the dependent variable.
Quick test: say "___ depends on ___." Total cost depends on the number of items bought, so cost is dependent and number of items is independent.
On a graph, the independent variable goes on the horizontal x-axis and the dependent variable on the vertical y-axis.
2 How do I read a scatter plot?A scatter plot shows the relationship by whether the dots rise or fall from left to right. 1 min
A scatter plot shows the relationship by whether the dots rise or fall from left to right.
| What the dots do | Name | What it means |
|---|---|---|
| Rise left to right | Positive correlation | As x increases, y tends to increase |
| Fall left to right | Negative correlation | As x increases, y tends to decrease |
| No pattern, or flat | No correlation | Knowing x does not help you predict y |
The closer the dots sit to a straight line, the stronger the relationship. Statisticians measure this with the correlation coefficient r, which runs from −1 to 1.
Worked example 1. A clinic resells used wheelchairs. Age in years and price: (1, $600), (2, $540), (3, $470), (4, $400), (5, $330). Describe the relationship.
- Name the variables: age is independent (x), price is dependent (y).
- Read left to right: as age goes up, price goes down every time.
- Name the direction: falling dots mean a negative correlation.
- Judge strength: the drops are steady, so the points sit close to a line and the relationship is strong.
- Say it in words: "As age increases, price tends to decrease."
Classic wrong turn: saying the table proves that age causes the price drop. Data can show that two things move together, but correlation does not imply causation.
3 What is direct variation?In direct variation, y = kx, so doubling x always doubles y. 1 min
In direct variation, y = kx, so doubling x always doubles y. The number k is the constant of variation, and y ÷ x equals k for every pair. There is no added starting amount.
Worked example 2. A pump delivers fluid at a steady rate and gives 150 mL in 2 hours. How much does it give in 5 hours?
- Write the formula: V = kt.
- Substitute the known pair: 150 = k × 2.
- Solve: k = 150 ÷ 2 = 75 (mL per hour).
- Write the equation: V = 75t.
- Answer: V = 75 × 5 = 375 mL.
Classic wrong turn: calling every straight line "direct variation." B = 35w + 150 has a starting amount, so doubling w does not double B. Test it: y ÷ x must stay the same.
4 What is inverse variation?In inverse variation, y = k/x, so when one variable goes up the other goes down. 1 min
In inverse variation, y = k/x, so when one variable goes up the other goes down. Here the product x × y equals k for every pair.
Worked example 3. Two pharmacy technicians take 6 hours to restock a storeroom. Working at the same rate, how long would 3 technicians take?
- Write the formula: t = k/n.
- Substitute: 6 = k/2.
- Solve: k = 6 × 2 = 12 (technician-hours of work).
- Answer: t = 12 ÷ 3 = 4 hours.
Classic wrong turn: setting up 2/6 = 3/t and getting 9 hours. Sense check: more helpers should mean less time.
With positive numbers, every inverse variation falls from left to right, but not every falling pattern is inverse variation. Check whether x × y stays the same.
Common mistakes
The traps students fall into most. Read these before you practice.
Watch out: A negative correlation is still a relationship; no correlation means no pattern, not a falling pattern.
Watch out: y = 35x + 150 is a straight line but not direct variation, because of the added starting amount; check that y ÷ x stays the same.
Watch out: Inverse variation needs x × y to stay constant, not just y going down; a steadily falling straight line is not inverse variation.
Watch out: For an inverse problem, do not set up a regular proportion: more workers should mean fewer hours, not more.
Watch out: A strong correlation in a table does not prove that one variable causes the other.
How to remember it
One rule separates direct from inverse variation in every problem you meet.
Rule to remember: direct, divide (y ÷ x stays the same); inverse, multiply (x × y stays the same).
Memory hook: "The Dependent Depends." If you can say "y depends on x," then y is dependent.
Self-check:
- Name the independent variable in "pay depends on hours worked."
- If y varies directly with x and x triples, what happens to y? What if y varies inversely?
- Sketch a scatter plot with a negative correlation and one with no correlation.
Show answers
- Hours worked.
- Use the test: "pay depends on hours worked."
- Pay is dependent, so hours worked is independent.
- Direct: y triples. Inverse: y becomes one-third as big.
- Direct, y = kx: multiply x by 3 and y is multiplied by 3.
- Inverse, y = k/x: x × y must stay the same. So if x is multiplied by 3, y is divided by 3.
- What your sketches should show:
- Negative correlation: dots fall from the upper left to the lower right.
- No correlation: dots scattered with no pattern, or in a flat line.