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Teacher Guide: Correlation Word Problems

Solve real correlation problems: compute r step by step from a small data set, interpret it in context, judge causal claims and see how an outlier changes r.

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Class quiz

10 questions on Correlation. Students join with a name, you see everyone's score.

For Teachers

Learning Objectives
  • Compute the correlation coefficient rr by hand for a data set of 4 to 6 points
  • Interpret the direction and strength of rr in the context of a word problem
  • Decide whether a causal claim is justified and name a possible confounding variable
  • Describe how an outlier changes rr and decide when it may be removed
  • Check calculations with the zero-sum of deviations and the range −1≤r≤1-1 \le r \le 1
Prerequisites
  • • Introduction to Correlation: positive, negative and no correlation in scatter plots
  • • Correlation Coefficient: the formula for rr and the strength scale
  • • Computing the mean of a small data set
  • • Squares and square roots with a calculator
Discussion Starters
  • 1. The correlation between umbrella sales and accidents is r=0.95r = 0.95. How could you check whether the rain, and not the umbrellas, explains it?
  • 2. If you measured study time in minutes instead of hours, would rr change? Why or why not?
  • 3. When is it honest to remove an outlier, and when is it cheating?
  • 4. Why could a correlation be very strong in one town and weak in another for the same two variables?
Common Misconceptions

A bigger SxyS_{xy} always means a stronger correlation

Remediation: Show two data sets with the same rr but different units: SxyS_{xy} grows with the units, rr does not. Only the ratio SxySxxSyy\frac{S_{xy}}{\sqrt{S_{xx} S_{yy}}} measures strength.

If rr is close to 11, the claim of cause and effect must be true

Remediation: Collect classic examples (umbrellas and traffic accidents, TVs and life expectancy, shoe size and reading) and let students find the confounding variable in each.

One point cannot matter much in a data set

Remediation: Use the fertilizer example: compute rr for four points (r=1r = 1), then add the fifth point and recompute (r=0r = 0).

Differentiation Ideas

For Struggling Students:

  • • Give a deviation table with the columns already labeled: xi−xˉx_i - \bar{x}, yi−yˉy_i - \bar{y}, product, squares
  • • Use data sets whose means are whole numbers
  • • Let students check each step with the calculator before moving on

For On-Level Students:

  • • Compute rr for 4 to 6 points and round to 2 decimal places
  • • Write a full context sentence for each rr (direction, strength, variables)
  • • Judge causal claims and name a confounding variable

For Advanced Students:

  • • Find a fifth point that turns a positive rr negative
  • • Prove that adding a constant to every yy value leaves rr unchanged
  • • Design a controlled experiment that could test a causal claim from the news
Standards Alignment
  • HSS.ID.C.8 (CCSS.MATH.CONTENT.HSS.ID.C.8)

    Compute (using technology) and interpret the correlation coefficient of a linear fit

  • HSS.ID.C.9 (CCSS.MATH.CONTENT.HSS.ID.C.9)

    Distinguish between correlation and causation

  • HSS.ID.B.6 (CCSS.MATH.CONTENT.HSS.ID.B.6)

    Represent data on two quantitative variables on a scatter plot, and describe how the variables are related

Lesson Resources
  • visualScatter Plots with Trend Line

    The study-hours data and the outlier data set, with the trend line drawn

  • activityDeviation Table Race

    Pairs fill in a table of deviations, products and squares for a 5-point data set and compare their rr

  • worksheetHeadline Detective

    Students judge six news headlines based on correlations and name a confounding variable for each

Lesson Content

Everything students see: definition, examples, common mistakes, applications. Tap to open.

Definition

A correlation word problem gives you paired data from a real situation and asks you to measure, interpret and judge the relationship. You already know the formula for the correlation coefficient rr from the lesson *Correlation Coefficient*. Here you use it as a full method.
The five-step method for a small data set:
  1. 1.Find the means xˉ\bar{x} and yˉ\bar{y}.
  2. 2.Find each deviation xi−xˉx_i - \bar{x} and yi−yˉy_i - \bar{y}.
  3. 3.Add the products: Sxy=∑(xi−xˉ)(yi−yˉ)S_{xy} = \sum(x_i - \bar{x})(y_i - \bar{y}).
  4. 4.Add the squares: Sxx=∑(xi−xˉ)2S_{xx} = \sum(x_i - \bar{x})^2 and Syy=∑(yi−yˉ)2S_{yy} = \sum(y_i - \bar{y})^2.
  5. 5.Divide:
    r=SxySxx⋅Syyr = \frac{S_{xy}}{\sqrt{S_{xx} \cdot S_{yy}}}
Then answer the question in context:
  • Direction: the sign of rr (positive: both rise together; negative: one rises while the other falls).
  • Strength: the size of ∣r∣|r| (scale below).
  • Causation: a strong rr alone never proves that one variable causes the other. Look for a confounding variable (also called a lurking variable) and ask whether the data come from a controlled experiment.
  • Outliers: one unusual point can make rr much stronger, much weaker or even change its sign.
rStrength
0.000.00 to 0.190.19very weak
0.200.20 to 0.390.39weak
0.400.40 to 0.590.59moderate
0.600.60 to 0.790.79strong
0.800.80 to 1.001.00very strong

Worked Examples

Five students recorded their study hours and quiz scores: (1,64)(1, 64), (2,63)(2, 63), (3,77)(3, 77), (4,71)(4, 71), (5,75)(5, 75). Compute rr and interpret it.
1

Find the means

xˉ=1+2+3+4+55=3\bar{x} = \frac{1+2+3+4+5}{5} = 3, yˉ=64+63+77+71+755=3505=70\bar{y} = \frac{64+63+77+71+75}{5} = \frac{350}{5} = 70 → xˉ=3\bar{x} = 3, yˉ=70\bar{y} = 70
2

Find the deviations

xi−xˉx_i - \bar{x}: −2,−1,0,1,2-2, -1, 0, 1, 2 yi−yˉy_i - \bar{y}: −6,−7,7,1,5-6, -7, 7, 1, 5 → Each deviation sum is 00 (a quick check)
3

Multiply the deviations and add

(−2)(−6)+(−1)(−7)+(0)(7)+(1)(1)+(2)(5)=12+7+0+1+10(-2)(-6) + (-1)(-7) + (0)(7) + (1)(1) + (2)(5) = 12 + 7 + 0 + 1 + 10 → Sxy=30S_{xy} = 30
4

Square the deviations and add

Sxx=4+1+0+1+4=10S_{xx} = 4 + 1 + 0 + 1 + 4 = 10 Syy=36+49+49+1+25=160S_{yy} = 36 + 49 + 49 + 1 + 25 = 160 → Sxx=10S_{xx} = 10, Syy=160S_{yy} = 160
5

Apply the formula

r=3010⋅160=301600=3040r = \frac{30}{\sqrt{10 \cdot 160}} = \frac{30}{\sqrt{1600}} = \frac{30}{40} → r=0.75r = 0.75
6

Interpret in context

Positive sign, ∣r∣=0.75|r| = 0.75 is in the band 0.600.60 to 0.790.79 → Strong positive correlation

Common Mistakes

Multiplying the raw values xiyix_i y_i instead of the deviations

Why it's wrong: The formula r=SxySxxSyyr = \frac{S_{xy}}{\sqrt{S_{xx} S_{yy}}} uses the distances from the means. Raw products are always large and positive for positive data and give a wrong rr.

Correct: Subtract the means first. A quick check: the deviations of each variable add up to 00.

Forgetting the square root or rounding too early

Why it's wrong: 3010⋅160≈0.019\frac{30}{10 \cdot 160} \approx 0.019 is not rr. Rounding 520\sqrt{520} to 2323 gives 0.9570.957 instead of 0.9650.965.

Correct: Take the square root of Sxx⋅SyyS_{xx} \cdot S_{yy} and round only the final answer to 2 decimal places.

Treating a strong rr as proof that one variable causes the other

Why it's wrong: Observational data can show a strong correlation because a confounding variable drives both, as rainy weather drives umbrella sales and traffic accidents.

Correct: Say "is associated with". Only a randomized controlled experiment can justify a causal claim.

Ignoring an outlier or deleting it without a reason

Why it's wrong: One point can change rr from 11 to 00. Ignoring it hides the problem; deleting it just because it is inconvenient is dishonest.

Correct: Plot the data, compute rr with and without the outlier and remove it only with a clear reason (a measuring error, a snail), stating that you did.

Why It Matters

Correlation numbers appear in news articles, school reports and science projects every day, and many of them are misread:
  • Health news: "People who drink coffee live longer" is usually a correlation from a survey, not proof.
  • Sports: coaches check whether training hours really go with better race times before changing a plan.
  • Business and energy: a shop compares temperature with sales, a family compares outdoor temperature with heating use.
  • Science projects: one broken measurement (an outlier) can ruin a result if nobody checks the scatter plot.
If you can compute rr yourself, describe it in plain words and spot a false causal claim or a misleading outlier, you can judge statistics instead of just believing them.

Real World Applications

Sports Science

Coaches compare weekly training hours with race times to check whether more training goes with faster runs.

Example:

For a running club the sums are Sxy=−45S_{xy} = -45, Sxx=25S_{xx} = 25 and Syy=100S_{yy} = 100. A negative rr means: more training, lower (faster) times.

1Try It Yourself

A running club records training hours per week and 5 km times (minutes) for its members. The sums are Sxy=−45S_{xy} = -45, Sxx=25S_{xx} = 25 and Syy=100S_{yy} = 100.

What is the correlation coefficient rr?

Step 1: Write the mathematical expression

Compute r=SxySxx⋅Syyr = \frac{S_{xy}}{\sqrt{S_{xx} \cdot S_{yy}}}:

Reading the News Critically

Journalists and readers must decide whether a reported correlation supports a claim about cause and effect.

Example:

A report finds r=0.83r = 0.83 between the number of TVs per 100 people in a country and life expectancy, and a blog says TVs make people live longer.

2Try It Yourself

Across many countries, the number of TVs per 100 people and life expectancy have r=0.83r = 0.83. A blog concludes that buying more TVs makes people live longer.

Is the causal claim justified?

Step 1: Choose the right answer

Choose your verdict:

Quality Control of Sensors

Engineers check measurement data for faulty readings, because one bad value can hide a perfect relationship.

Example:

A temperature sensor is tested at five settings: (2,10)(2, 10), (4,14)(4, 14), (6,18)(6, 18), (8,22)(8, 22) and a faulty reading (10,6)(10, 6).

3Try It Yourself

A sensor gives the readings (2,10)(2, 10), (4,14)(4, 14), (6,18)(6, 18), (8,22)(8, 22), (10,6)(10, 6). The last reading is a known error. The means are xˉ=6\bar{x} = 6 and yˉ=14\bar{y} = 14.

What is rr once the faulty reading is removed?

Step 1: Write the mathematical expression

Compare rr with and without the error:

Key Takeaways

  • 1To compute rr by hand: means, deviations, SxyS_{xy}, SxxS_{xx}, SyyS_{yy}, then r=SxySxxSyyr = \frac{S_{xy}}{\sqrt{S_{xx} S_{yy}}}
  • 2Check your work: the deviations of each variable add up to 00, and rr must lie between −1-1 and 11
  • 3Round only the final value of rr, usually to 2 decimal places
  • 4Interpret rr in context: the sign gives the direction, ∣r∣|r| gives the strength
  • 5A strong correlation from observational data does not justify a causal claim; look for confounding variables
  • 6One outlier can strengthen, weaken or even reverse rr: plot the data and compare rr with and without it

Frequently Asked Questions

Why do the deviations always add up to zero?

The mean is the balance point of the data: the values above it and below it cancel out exactly. If your deviations do not add up to 00, you have a mistake in the mean or in a subtraction.

Does adding the same number to every value change rr?

No. Adding a constant moves all points together, so the deviations stay the same. Multiplying by a positive number (for example hours to minutes) does not change rr either. Multiplying one variable by a negative number flips the sign of rr.

When is it fine to remove an outlier?

When you have a clear reason that is not about the result: a measuring error, a typing mistake, a plant eaten by a snail. Always say that you removed it and, if possible, report rr both with and without the point.

How can we ever show causation?

With a randomized controlled experiment: assign the treatment (for example fertilizer) at random to some subjects and not to others, keep everything else the same, and compare. Random assignment spreads confounding variables evenly over both groups.

Glossary

Deviation
The distance of a value from the mean, xi−xˉx_i - \bar{x}; it can be positive or negative
SxyS_{xy}
The sum of the products of deviations, ∑(xi−xˉ)(yi−yˉ)\sum(x_i - \bar{x})(y_i - \bar{y}); its sign is the sign of rr
SxxS_{xx} and SyyS_{yy}
The sums of squared deviations of xx and of yy; they are never negative
Outlier
A data point far from the pattern of the others; it can change rr a lot
Confounding variable
A variable not included in the study that affects both variables and can create a misleading correlation
Controlled experiment
A study in which the researcher assigns the treatment at random; the only kind of study that can justify a causal claim

Formula Card

Correlation coefficient

r=SxySxx⋅Syyr = \frac{S_{xy}}{\sqrt{S_{xx} \cdot S_{yy}}}

$S_{xy} = \sum(x_i - \bar{x})(y_i - \bar{y})$, $S_{xx} = \sum(x_i - \bar{x})^2$, $S_{yy} = \sum(y_i - \bar{y})^2$

Checks

∑(xi−xˉ)=0,−1≤r≤1\sum(x_i - \bar{x}) = 0, \quad -1 \le r \le 1

If a deviation sum is not $0$ or $|r| > 1$, there is a calculation error

Strength scale

∣r∣: 0.2 weak, 0.4 moderate, 0.6 strong, 0.8 very strong|r|: \ 0.2 \ \text{weak}, \ 0.4 \ \text{moderate}, \ 0.6 \ \text{strong}, \ 0.8 \ \text{very strong}

Each word starts at its value; below $0.20$ the correlation is very weak

More in This Topic

Introduction to Correlation

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Correlation Coefficient

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Interpreting Correlation

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Correlation vs Causation

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