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Statistics and Probability - Scatter Diagrams and Correlation

Grade 8IGCSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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Bivariate Data: Data that involves two different variables to see if there is a relationship between them.

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Scatter Diagram: A graphical representation where individual data points are plotted on an x-y axis to identify trends.

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Positive Correlation: Both variables increase together (the points generally move from bottom-left to top-right).

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Negative Correlation: As one variable increases, the other decreases (the points generally move from top-left to bottom-right).

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No Correlation: There is no apparent relationship or pattern between the variables.

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Strength of Correlation: Refers to how closely the points follow a straight line (classified as Strong or Weak).

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Line of Best Fit: A straight line drawn through the center of the data points that best represents the trend. It should have roughly an equal number of points above and below it.

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Interpolation: Predicting a value within the range of the plotted data (usually reliable).

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Extrapolation: Predicting a value outside the range of the plotted data (usually unreliable/risky).

📐Formulae

General Equation of Line of Best Fit: y=mx+c\text{General Equation of Line of Best Fit: } y = mx + c

Mean of x (Mean Point Coordinate 1): xˉ=∑xn\text{Mean of } x \text{ (Mean Point Coordinate 1): } \bar{x} = \frac{\sum x}{n}

Mean of y (Mean Point Coordinate 2): yˉ=∑yn\text{Mean of } y \text{ (Mean Point Coordinate 2): } \bar{y} = \frac{\sum y}{n}

Note: The Line of Best Fit should ideally pass through the mean point (xˉ,yˉ)\text{Note: The Line of Best Fit should ideally pass through the mean point } (\bar{x}, \bar{y})

💡Examples

Problem 1:

A student records the number of hours spent studying (xx) and the test score achieved (yy) for 5 students: (2, 40), (4, 60), (5, 70), (7, 85), (8, 90). Describe the correlation.

Solution:

Strong Positive Correlation

Explanation:

As the number of hours spent studying increases, the test scores also increase consistently. The points lie very close to a potential straight line, indicating a strong relationship.

Problem 2:

Given a scatter plot where the line of best fit is drawn, a student uses the line to estimate the height of a plant at 10 weeks, even though the data collected only goes up to 6 weeks. What is this technique called and is it reliable?

Solution:

Extrapolation; Not reliable.

Explanation:

Extrapolation is the process of estimating values outside the known data range. It is considered unreliable because the trend may change (e.g., the plant might stop growing or die) beyond the observed timeframe.

Problem 3:

Calculate the mean point (xˉ,yˉ)(\bar{x}, \bar{y}) for the following data: x=[2,4,6]x = [2, 4, 6], y=[10,20,30]y = [10, 20, 30].

Solution:

(4,20)(4, 20)

Explanation:

To find xˉ\bar{x}, calculate (2+4+6)/3=4(2+4+6)/3 = 4. To find yˉ\bar{y}, calculate (10+20+30)/3=20(10+20+30)/3 = 20. The line of best fit for this data should pass through the point (4,20)(4, 20).