Back to Exercise: Compute and interpret the correlation coefficient

Exercises: Compute and Interpret the Correlation Coefficient

Work through each section in order. The correlation coefficient r measures
the STRENGTH and DIRECTION of a LINEAR association on a scale from -1 to +1.
Read its SIGN for direction and its MAGNITUDE (distance from 0) for strength.
Where a problem gives a calculator or spreadsheet value, use it as your
computed r. Remember: r sees only straight-line patterns, an outlier can
swing it, and a high r never proves causation.

Grade 10·21 problems·~30 min·Common Core Math - HS Statistics and Probability·group·hss-id-c-8

Printable layout

Warm-Up: Reading the Correlation Scale

These problems review how to read a given r value.

The correlation coefficient $r$ measures two things about a linear association. Which pair does it report?

The direction (from its sign) and the strength (from its magnitude) of a linear association.

The mean and the standard deviation of the data.

The slope and the y-intercept of the regression line.

The number of data points and the units of measurement.

A data set has correlation coefficient $r = -0.95$ . Which statement best describes this association?

A very strong negative linear association.

A weak association, because $r$ is negative.

A weak positive association.

No linear association at all.

A regression line has slope $12$ (dollars per hour) and the fit has correlation coefficient $r = 0.6$ . A student says "the correlation is 12." What is the correct value and meaning of $r$ here?

The correlation is $r = 0.6$ , a unitless measure of fit strength on $-1$ to $+1$ ; the slope $12$ is a separate rate with units.

The correlation is $12$ , the same as the slope.

The correlation is $12$ dollars per hour.

The correlation is $0.6$ dollars per hour.

On the correlation scale, match the endpoints and middle. Which row is correct?

$r = +1$ : perfect positive (points on an upward line); $r = -1$ : perfect negative (points on a downward line); $r = 0$ : no linear association.

$r = +1$ : no association; $r = 0$ : perfect fit; $r = -1$ : weak fit.

$r = +1$ : strongest; $r = -1$ : weakest; $r = 0$ : moderate.

$r = 0$ : perfect downward line; $r = +1$ : scatter; $r = -1$ : scatter.

Fluency Practice

Estimate r from each plot, or read and interpret a given r value.

$A scatter plot whose points form a tight, nearly straight upward band rising from lower left to upper right.$

The scatter plot below shows points that lie in a tight upward band, hugging a rising line closely. Which value of $r$ best matches this plot?

$r \approx 0.95$

$r \approx 0.5$

$r \approx 0$

$r \approx -0.9$

Two relationships are summarized: relationship P has $r = 0.5$ and relationship Q has $r = 0.9$ . Which relationship has the STRONGER linear association, and how do their plots compare?

Q is stronger; its points hug the line tightly, while P is a noticeably looser cloud.

P is stronger, because $0.5$ means "half" the points fit perfectly.

They are equally strong, since both $r$ values are positive.

P is stronger, because a smaller $r$ means less scatter.

$A scatter plot whose points drift upward overall but spread out loosely around the trend.$

The scatter plot below shows points that drift upward overall but spread out in a loose cloud around the trend. Which value of $r$ best matches this plot, and is the association strong or moderate?

$r \approx 0.5$ — a moderate (loose) positive association.

$r \approx 0.95$ — a strong (tight) positive association.

$r \approx 0.5$ — a strong positive association, since $0.5$ is large.

$r \approx -0.5$ — a moderate negative association.

A student enters bivariate data into a graphing calculator and runs LinReg with diagnostics on. The output reports $r = 0.82$ and slope $b = 3.4$ . To one decimal place, what is the value of the correlation coefficient $r$ ?

In a spreadsheet, a student selects two data columns and enters $=$ CORREL(A2:A20, B2:B20). The cell displays $-0.67$ . To two decimal places, what is the correlation coefficient $r$ for this data?

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