Bivariate Data

Grade 11 · statistics · 101 practice problems · read aloud

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Bivariate Data Analysis

What is Bivariate Data? 🎯

Bivariate data involves two variables for each individual or item. We analyze them to see if a relationship exists. This is crucial for answering questions like: "Does more study time lead to higher test scores?" or "Is there a link between height and shoe size?"

Step-by-Step Guide

  1. Collect Data: Gather paired values (X, Y).
  2. Create a Scatter Plot: Graph the points to visualize the relationship.
  3. Analyze the Form: Is it linear, curved, or no pattern?
  4. Determine Direction: Positive (upward slope) or Negative (downward slope)?
  5. Assess Strength: How closely do the points follow a trend?
  6. Calculate Correlation (r): A number from -1 to 1 quantifying the strength and direction.
  7. Find Line of Best Fit: Use technology or calculation (y = mx + b).

Worked Examples

Example 1: Study Time vs. Test Score

Data: (1, 60), (2, 70), (3, 75), (4, 85)

Step 1: Plot points. Step 2: Form is roughly linear. Step 3: Direction is positive. Step 4: Strength is strong (points are close to a line). Step 5: Correlation (r) ≈ 0.98. Step 6: Line of best fit: y ≈ 8.5x + 55.

Interpretation: There's a strong positive relationship. For each additional hour studied, the score increases by about 8.5 points.

Example 2: Hours on Phone vs. Sleep

Data: (2, 8), (4, 7), (6, 6), (8, 5)

Step 1: Plot points. Step 2: Form is linear. Step 3: Direction is negative. Step 4: Strength is perfect. Step 5: Correlation (r) = -1. Step 6: Line: y = -0.5x + 9.

Interpretation: A perfect negative correlation. Each extra hour on the phone is associated with 0.5 hours less sleep.

Common Mistakes ⚠️

Correlation vs. Causation: Just because two variables are related does NOT mean one causes the other. (e.g., Ice cream sales and drowning incidents both increase in summer, but one doesn't cause the other).

Extrapolation: Don't use your line of best fit to make predictions far outside your data range. It's unreliable.

Forgetting Units: Always include units in your slope interpretation (e.g., "points per hour").

Tips & Tricks

Memory Aid: Correlation 'r' values: 0 = none, ±0.5 = weak, ±0.8 = strong, ±1 = perfect.

Slope Interpretation: "For every 1 unit increase in X, Y changes by [slope] units."

Tech is Your Friend: Use graphing calculators or software (like Desmos) to calculate 'r' and the line of best fit accurately.

Practice Suggestions

  • Collect your own data! Measure hand span vs. height in your class.
  • Find real-world scatter plots in news articles and interpret them.
  • Practice calculating correlation and the line of best fit from small datasets.
  • Always state the direction, form, and strength when describing a relationship.

Practice problems

6 of the 101, worked through step by step — try them before opening the answer.

1 Create a scatter plot for the data: (9,14), (11,18), (13,22), (15,26), (17,30), (19,34). Calculate the correlation coefficient r.

Hint: Consider how the relationship between the variables affects the correlation coefficient. For example, if points lie perfectly on a straight line with positive slope, what would r be?

Show the answer

Answer: 1.0

  1. Calculate the means: x̄ = (9+11+13+15+17+19)/6 = 84/6 = 14, ȳ = (14+18+22+26+30+34)/6 = 144/6 = 24
  2. Calculate the sums needed for r: Σ(x-x̄)(y-ȳ) = (9-14)(14-24)+(11-14)(18-24)+(13-14)(22-24)+(15-14)(26-24)+(17-14)(30-24)+(19-14)(34-24) = (-5)(-10)+(-3)(-6)+(-1)(-2)+(1)(2)+(3)(6)+(5)(10) = 50+18+2+2+18+50 = 140
  3. Calculate Σ(x-x̄)² = (-5)²+(-3)²+(-1)²+(1)²+(3)²+(5)² = 25+9+1+1+9+25 = 70
  4. Calculate Σ(y-ȳ)² = (-10)²+(-6)²+(-2)²+(2)²+(6)²+(10)² = 100+36+4+4+36+100 = 280
  5. Calculate r = 140/√(70×280) = 140/√19600 = 140/140 = 1.0

The answer is 1.0.

2 Create a scatter plot for the data: (7, 18), (9, 22), (11, 26), (13, 30), (15, 34), (17, 38). Describe the correlation and estimate the line of best fit.

Hint: Plot the points on a coordinate plane and observe the pattern they form. Consider how the y-values change as x increases.

Show the answer

Answer: Strong positive correlation; y = 2x + 4

  1. Plot the points (7,18), (9,22), (11,26), (13,30), (15,34), (17,38) on a coordinate plane.
  2. Observe the pattern: As x increases by 2, y increases by 4. This indicates a strong positive linear correlation.
  3. Calculate the slope: change in y / change in x = 4/2 = 2.
  4. Use point-slope form with (7,18): y - 18 = 2(x - 7)
  5. Simplify: y - 18 = 2x - 14 → y = 2x + 4
  6. Verify with another point: For x=9, y=2(9)+4=22, which matches the data. The line of best fit is y = 2x + 4 with strong positive correlation.

3 Create a scatter plot for the data: (3,7), (5,11), (7,15), (9,19), (11,23), (13,27), (15,31), (17,35), (19,39), (21,43) and determine the correlation coefficient.

Hint: Calculate the covariance and standard deviations of both variables to find the correlation coefficient.

Show the answer

Answer: 1.0

  1. Calculate the means of x and y. x̄ = (3+5+7+9+11+13+15+17+19+21)/10 = 120/10 = 12 ȳ = (7+11+15+19+23+27+31+35+39+43)/10 = 250/10 = 25
  2. Calculate the covariance. Cov(x,y) = Σ[(xᵢ - x̄)(yᵢ - ȳ)] / (n-1) = [(3-12)(7-25) + (5-12)(11-25) + (7-12)(15-25) + (9-12)(19-25) + (11-12)(23-25) + (13-12)(27-25) + (15-12)(31-25) + (17-12)(35-25) + (19-12)(39-25) + (21-12)(43-25)] / 9 = [(-9)(-18) + (-7)(-14) + (-5)(-10) + (-3)(-6) + (-1)(-2) + (1)(2) + (3)(6) + (5)(10) + (7)(14) + (9)(18)] / 9 = [162 + 98 + 50 + 18 + 2 + 2 + 18 + 50 + 98 + 162] / 9 = 660 / 9 = 73.333...
  3. Calculate the standard deviations. s_x = √[Σ(xᵢ - x̄)² / (n-1)] = √[(81+49+25+9+1+1+9+25+49+81)/9] = √[330/9] = √36.666... = 6.055... s_y = √[Σ(yᵢ - ȳ)² / (n-1)] = √[(324+196+100+36+4+4+36+100+196+324)/9] = √[1320/9] = √146.666... = 12.110...
  4. Calculate the correlation coefficient. r = Cov(x,y) / (s_x × s_y) = 73.333... / (6.055... × 12.110...) = 73.333... / 73.333... = 1.0 The correlation coefficient is 1.0, indicating a perfect positive linear relationship.

4 Create a scatter plot for Aroha's data: (3,7), (5,11), (7,15), (9,19), (11,23), (13,27), (15,31), (17,35), (19,39), (21,43). Calculate the correlation coefficient r.

Hint: Look for a consistent pattern in how the y-values change as the x-values increase. Consider using the formula for Pearson's correlation coefficient with sums of products and squares.

Show the answer

Answer: 1

  1. Calculate the mean of x-values: (3+5+7+9+11+13+15+17+19+21)/10 = 120/10 = 12
  2. Calculate the mean of y-values: (7+11+15+19+23+27+31+35+39+43)/10 = 250/10 = 25
  3. Calculate Σ(x-x̄)(y-ȳ): (3-12)(7-25)+(5-12)(11-25)+(7-12)(15-25)+(9-12)(19-25)+(11-12)(23-25)+(13-12)(27-25)+(15-12)(31-25)+(17-12)(35-25)+(19-12)(39-25)+(21-12)(43-25) = (-9)(-18)+(-7)(-14)+(-5)(-10)+(-3)(-6)+(-1)(-2)+(1)(2)+(3)(6)+(5)(10)+(7)(14)+(9)(18) = 162+98+50+18+2+2+18+50+98+162 = 660
  4. Calculate Σ(x-x̄)²: (-9)²+(-7)²+(-5)²+(-3)²+(-1)²+(1)²+(3)²+(5)²+(7)²+(9)² = 81+49+25+9+1+1+9+25+49+81 = 330
  5. Calculate Σ(y-ȳ)²: (-18)²+(-14)²+(-10)²+(-6)²+(-2)²+(2)²+(6)²+(10)²+(14)²+(18)² = 324+196+100+36+4+4+36+100+196+324 = 1320
  6. Calculate r = Σ(x-x̄)(y-ȳ)/√[Σ(x-x̄)²Σ(y-ȳ)²] = 660/√(330×1320) = 660/√435600 = 660/660 = 1 The correlation coefficient r = 1, indicating a perfect positive linear relationship.

5 Create a scatter plot for Liam's data: (1,5), (5,13), (9,21), (13,29), (17,37), (21,45), (25,53), (29,61), (33,69), (37,77). Calculate the correlation coefficient r.

Hint: Examine the pattern of change between x and y values. When the relationship is perfectly linear, the correlation coefficient will be either 1 or -1. Look at whether y increases or decreases as x increases.

Show the answer

Answer: 1.0

  1. Calculate the mean of x-values: (1+5+9+13+17+21+25+29+33+37)/10 = 190/10 = 19
  2. Calculate the mean of y-values: (5+13+21+29+37+45+53+61+69+77)/10 = 410/10 = 41
  3. Calculate the sum of (x-mean_x)(y-mean_y): (1-19)(5-41) + (5-19)(13-41) + (9-19)(21-41) + (13-19)(29-41) + (17-19)(37-41) + (21-19)(45-41) + (25-19)(53-41) + (29-19)(61-41) + (33-19)(69-41) + (37-19)(77-41) = (-18)(-36) + (-14)(-28) + (-10)(-20) + (-6)(-12) + (-2)(-4) + (2)(4) + (6)(12) + (10)(20) + (14)(28) + (18)(36) = 648 + 392 + 200 + 72 + 8 + 8 + 72 + 200 + 392 + 648 = 2640
  4. Calculate the sum of (x-mean_x)^2: (-18)^2 + (-14)^2 + (-10)^2 + (-6)^2 + (-2)^2 + (2)^2 + (6)^2 + (10)^2 + (14)^2 + (18)^2 = 324 + 196 + 100 + 36 + 4 + 4 + 36 + 100 + 196 + 324 = 1320
  5. Calculate the sum of (y-mean_y)^2: (-36)^2 + (-28)^2 + (-20)^2 + (-12)^2 + (-4)^2 + (4)^2 + (12)^2 + (20)^2 + (28)^2 + (36)^2 = 1296 + 784 + 400 + 144 + 16 + 16 + 144 + 400 + 784 + 1296 = 5280
  6. Calculate r = sum[(x-mean_x)(y-mean_y)] / sqrt[sum(x-mean_x)^2 * sum(y-mean_y)^2] = 2640 / sqrt(1320 * 5280) = 2640 / sqrt(6969600) = 2640 / 2640 = 1.0 The correlation coefficient is 1.0, indicating a perfect positive linear relationship.

6 Create a scatter plot for the data: (9, 14), (11, 18), (13, 22), (15, 26), (17, 30), (19, 34). Describe the correlation and estimate the slope of the line of best fit.

Hint: Examine how the y-values change as the x-values increase. A consistent pattern in the change can help identify both the correlation type and the slope.

Show the answer

Answer: positive, 2

  1. Plot the points (9, 14), (11, 18), (13, 22), (15, 26), (17, 30), (19, 34) on a coordinate plane.
  2. Analyze the distribution. As the x-value increases, the y-value also increases consistently. This indicates a positive linear correlation.
  3. To estimate the slope, calculate the change in y over the change in x for several points. For example, from (9, 14) to (11, 18): change in y = 18 - 14 = 4, change in x = 11 - 9 = 2, slope = 4 / 2 = 2. From (13, 22) to (15, 26): change in y = 26 - 22 = 4, change in x = 15 - 13 = 2, slope = 4 / 2 = 2. The slope is consistently 2.
  4. The correlation is positive, and the estimated slope of the line of best fit is 2.
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