Correlation & Line of Best Fit
Read a scatter plot, describe how two measured variables move together (direction and strength), and use a line of best fit to predict without confusing correlation with cause.
Science almost never studies one number at a time. It asks whether two things move together: does more exercise go with lower resting heart rate, does a higher dose go with a bigger effect, does more screen time go with less sleep. A scatter plot puts the pairs on a graph, the correlation coefficient r turns the pattern into one number from -1 to +1 (direction and strength), and a line of best fit lets you predict a value you did not measure. Epidemiologists use correlation to spot links between exposures and disease, then design studies to test them. Biostatisticians fit regression lines to clinical data to estimate how much an outcome changes per unit of treatment. Lab scientists build standard curves (a line of best fit) to turn a machine reading into a concentration. Doctors reading a study need to know the one rule that keeps all of this honest: correlation does not prove causation, because a hidden third variable can make two unrelated things rise together.
- Common Core · HSS-ID.B.6Represent two quantitative variables on a scatter plot and describe how they are related, including fitting a function such as a line to the data.
- Common Core · HSS-ID.C.8Compute (using technology) and interpret the correlation coefficient of a linear fit as a measure of the direction and strength of the relationship.
- Common Core · HSS-IC.B.6Evaluate reports based on data, including whether a claimed relationship is causal or only a correlation.
- Ohio · Ohio HS S.ID.6Fit a linear model to bivariate data shown on a scatter plot and use it to solve problems and make predictions.
- NGSS · SEP-4Analyzing and Interpreting Data: identify relationships in data, including trends and correlations, and distinguish correlation from causation.
- Plot ordered pairs on a coordinate grid: A scatter plot is just many (x, y) points, so students must place a point from its two coordinates before they can read a cloud of them.
- Read slope and use y = mx + b: A line of best fit is a linear equation; predicting from it means substituting an x value and computing y.
- Read values off a graph's axes: Direction, strength, and predictions are all read from the axes, so students must trace a point to its x and y values.
Prerequisites are inferred: pending teacher review.
Re-learn the skill with worked practice and clear examples.
Describe a scatter plot in three moves: name the direction (do points rise, fall, or drift with no pattern), judge the strength with r (how tightly the points hug a line, from -1 to +1), and, when there is a clear trend, draw a line of best fit and use it to predict. A rising cloud is positive, a falling cloud is negative, and a shapeless cloud is near zero.
Using the scatter plot shown, how would you describe the correlation between the two variables?
Reviewed- A.Strong positive correlation
- B.Strong negative correlation
- C.No correlation
- D.The point at the top is an error
Show the worked solution ▾
Answer: A. Strong positive correlation
- Step 1: Scan left to right: As x increases, the points move steadily upward, so the direction is positive.
- Step 2: Judge the tightness: The points hug a rising line closely with no scatter against it, so the relationship is strong.
Why it's right: The points climb tightly from lower left to upper right, which is a strong positive correlation.
- B: Negative means the points fall as x rises; here they rise.
- C: No correlation would be a shapeless cloud, not a clear rising line.
- D: The top point continues the same rising pattern, so it is not an error.
Aligned to Common Core HSS-ID.C.8: interpret direction and strength · reading level ~grade 9
A study reports a correlation coefficient of r = -0.85 between hours of screen time and hours of sleep. What does this tell you?
Reviewed- A.A weak link, because the number is negative
- B.A strong link where more screen time goes with less sleep
- C.A strong link where more screen time goes with more sleep
- D.No relationship, because r is below zero
Show the worked solution ▾
Answer: B. A strong link where more screen time goes with less sleep
- Step 1: Read the sign: The sign is negative, so as one variable goes up the other goes down: more screen time goes with less sleep.
- Step 2: Read the size: The size is 0.85, which is close to 1, so the relationship is strong.
Why it's right: r = -0.85 has a negative sign (more screen time, less sleep) and a large size (0.85 is close to 1), so it is a strong negative relationship.
- A: A negative sign does not mean weak; strength comes from how close the size is to 1.
- C: The negative sign means the variables move in opposite directions, not the same direction.
- D: A negative r is a real relationship; only r near 0 means little to no relationship.
Aligned to Common Core HSS-ID.C.8: interpret the correlation coefficient · reading level ~grade 9
The line of best fit for a data set is y = 2x + 3. Use the figure to predict y when x = 6.
Reviewed- A.9
- B.11
- C.15
- D.18
Show the worked solution ▾
Answer: C. 15
- Step 1: Substitute x: Put x = 6 into the line: y = 2(6) + 3.
- Step 2: Compute: 2 times 6 is 12, and 12 + 3 = 15.
Why it's right: Substituting x = 6 into y = 2x + 3 gives y = 2(6) + 3 = 12 + 3 = 15.
- A: This is 2(3) + 3; it uses x = 3, not x = 6.
- B: This is 2(4) + 3; it uses x = 4, not x = 6.
- D: This adds 6 to 2(6) instead of adding the constant 3.
Aligned to Ohio HS S.ID.6: use a linear model to predict · reading level ~grade 9
- A nurse-researcher describes a scatter plot of dose vs. blood level as a strong positive correlation before trusting a prediction from it.
- A student uses the line of best fit y = 2x + 3 to predict a value at x = 6 that was never measured directly.
- A lab reads r = -0.85 for screen time vs. sleep and reports it as a strong negative link, not proof that screens cause sleep loss.
Fill these in as you work through the lesson.
- Scatter plot (one point per subject, from two measurements):
- Correlation coefficient (r) (a number from -1 to +1: direction and strength):
- Line of best fit (the line closest to all the points; used to predict):
- Correlation is not causation (a hidden third variable can link two things):
The correlation coefficient r runs from to ; its sign gives the and its size gives the , and a strong correlation still does not prove .
- How do you tell a positive correlation from a negative one just by looking at the cloud of points?
- Which is stronger, r = 0.5 or r = -0.9, and why does the sign not decide strength?
- Give an example of two variables that are correlated because of a hidden third variable, not because one causes the other.
The line of best fit is y = 2x + 3. To predict y at x = 6, substitute: y = 2(6) + 3 = 12 + 3 = ____.
The vocabulary of this topic, shown in the way you will meet it.