Reading Direction and Strength from r

An energy company finds $r = -0.86$ between the outdoor temperature and the amount of gas homes use for heating. Describe the correlation.

Look at the sign for the direction: $r = -0.86$ is negative = Negative: warmer days go with less gas

Take the absolute value for the strength: $\lvert -0.86 \rvert = 0.86$ = $\lvert r \rvert = 0.86$

Compare with the rule of thumb: $0.86 \ge 0.7$ = Strong

Put direction and strength together: negative + strong = Strong negative linear correlation

Answer: There is a strong negative linear correlation ($r = -0.86$): on warmer days, homes tend to use less gas for heating.

Reading Scatter Plots A to D

Look at the scatter plots A, B, C and D in this lesson. Describe each one, and say whether $r$ tells the whole story.

Plot A: temperature and cold drinks: The points rise from left to right and lie very close to a straight line, $r \approx 0.99$ = Strong positive

Plot B: screen time and sleep: The points drift slightly downward but are widely scattered, $r \approx -0.36$ = Weak negative

Plot C: shoe size and test score: No upward or downward trend, $r \approx 0.02$ = Almost no linear relationship

Plot D: fertilizer and tomatoes: The points form a clear hill-shaped curve, but $r = 0$ = Strong curved relationship that $r$ does not see

Answer: A is strong positive, B is weak negative, C shows almost no linear relationship. D has $r = 0$ even though the pattern is very clear: $r$ only measures straight-line relationships, so you must always look at the plot.

Writing an Interpretation Sentence

For 40 students, the correlation between minutes spent reading per day and the score on a vocabulary test is $r = 0.58$. Write an interpretation sentence.

Direction from the sign: $r = 0.58 > 0$ = Positive

Strength from $\lvert r \rvert$: $0.4 \le 0.58 < 0.7$ = Moderate

Name both variables in context: minutes of reading per day and vocabulary score = Context included

Describe the tendency with "tend to": More reading goes with higher scores on average, not for every single student = Use "tend to"

Answer: There is a moderate positive linear correlation ($r = 0.58$) between minutes spent reading per day and vocabulary test score: students who read more tend to score higher.

Changing Units Does Not Change r

For plot A, $r \approx 0.99$ with temperatures in °C. A colleague converts every temperature to °F with $F = 1.8C + 32$. What is $r$ now?

Convert one value to see what happens: $25\,°C$: $1.8 \times 25 + 32 = 77\,°F$ = $77\,°F$

Describe the conversion: Every $x$ is multiplied by the same positive number $1.8$ and then $32$ is added = The points are stretched and shifted, not reshaped

Check the pattern: The order of the points and how close they are to a line stay the same = Same pattern

State $r$: $r$ has no units, so it does not depend on the units of $x$ or $y$ = $r \approx 0.99$

Answer: $r$ stays about $0.99$. Changing units (multiplying by a positive number and adding a constant) never changes $r$.

The Effect of One Outlier

Six students practiced 1 to 6 hours and scored 52, 58, 65, 70, 78 and 83 on a quiz: $r \approx 0.999$. A seventh student practiced 7 hours but was ill on the quiz day and scored 40 (plot E). With this point, $r \approx 0.14$. Interpret both values.

Interpret $r$ without the outlier: $r \approx 0.999$, so $\lvert r \rvert \ge 0.7$ = Strong positive, almost a perfect line

Interpret $r$ with the outlier: $r \approx 0.14$, so $\lvert r \rvert < 0.4$ = Weak positive

Find what changed: Only one point was added: $(7, 40)$, far below the pattern = One outlier changed the strength from strong to weak

Report honestly: Give $r$ with and without the outlier and explain the unusual point = Mention the outlier

Answer: Without the outlier the correlation is strong and positive ($r \approx 0.999$); with it, $r \approx 0.14$ looks weak. One unusual point hides a clear pattern, so report both values and explain the outlier.

Mistake: Thinking $r = -0.8$ is weaker than $r = 0.5$ because $-0.8$ is a smaller number

Why: The sign only tells the direction. The strength is the distance from $0$, which is $\lvert r \rvert$.

Correct: Compare $\lvert -0.8 \rvert = 0.8$ with $\lvert 0.5 \rvert = 0.5$: $r = -0.8$ is the stronger correlation.

Mistake: Concluding "no relationship" whenever $r$ is close to $0$

Why: $r$ only measures straight-line relationships. A hill or U-shaped curve can have $r = 0$ (plot D).

Correct: Say "no **linear** relationship" and always look at the scatter plot for curves.

Mistake: Believing $r$ changes when you switch units, for example from cm to inches

Why: Changing units multiplies every value by the same positive number (and may add a constant). The shape of the scatter plot stays the same.

Correct: $r$ has no units: the same data in cm, m or inches give the same $r$.

Mistake: Reading $r$ without checking for outliers

Why: A single unusual point can make a strong correlation look weak (plot E) or a weak one look strong.

Correct: Plot the data first. If there is an outlier, report $r$ with and without it.

Planning Energy Use

Energy companies use the correlation between temperature and gas use to plan how much gas they need in winter.

With $r = -0.86$, colder days strongly tend to go with higher gas use, so a cold forecast is a good warning sign.

Fuel Use and Driving Speed

Car engineers know that fuel use is high at very low and very high speeds and lowest in between. This curved pattern fools $r$.

Speeds of 30, 50, 70, 90, 110, 130 and 150 km/h give fuel use of 8.0, 6.5, 5.5, 5.2, 5.5, 6.5 and 8.0 liters per 100 km: a U-shaped curve.

Checking Data for Outliers

Data analysts always check scatter plots for outliers before they report a correlation, because one wrong or unusual value can change $r$ a lot.

In plot E, six students show $r \approx 0.999$, but one ill student with 7 hours of practice and a score of 40 brings $r$ down to about $0.14$.

Q: Is $r = -0.9$ a weak correlation because it is negative?

A: No. The minus sign only means the direction is negative. $\lvert -0.9 \rvert = 0.9$, so it is a strong negative correlation, stronger than $r = 0.5$.

Q: Why are there different strength tables in different books?

A: Where "strong" begins is a convention, not a mathematical fact. This lesson uses $0.7$ and $0.4$ as a rule of thumb. Other tables, like the one in the Correlation Coefficient lesson, use more bands, for example "very strong" for $0.8$ and above: in this lesson, "very strong" is simply part of strong. In every table, a larger $\lvert r \rvert$ means a stronger linear relationship.

Q: If I measure height in meters instead of centimeters, does $r$ change?

A: No. Dividing every height by $100$ does not change the shape of the scatter plot, only the numbers on the axis. $r$ has no units, so it stays exactly the same.

Q: Can an outlier make a correlation look stronger?

A: Yes. A cloud of points with no pattern ($r$ near $0$) plus one point far away in the top right corner can give $r$ above $0.9$. That is why you always check the plot.