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Statistics - Scatter Diagrams and Correlation

Grade 9IGCSE

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

🔑Concepts

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Scatter Diagram: A graph used to display the relationship (correlation) between two sets of numerical data (bivariate data).

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Positive Correlation: As one variable increases, the other variable also increases. The points generally follow an upward trend from left to right.

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Negative Correlation: As one variable increases, the other variable decreases. The points generally follow a downward trend from left to right.

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Zero (No) Correlation: There is no apparent relationship between the two variables; points are scattered randomly.

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Strength of Correlation: Described as 'Strong' if the points lie very close to a straight line, or 'Weak' if they are more spread out but still show a trend.

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Line of Best Fit: A straight line drawn through the data points that best represents the trend. It should pass through the mean point and have an equal distribution of points above and below it.

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Interpolation: Estimating a value within the range of the given data points using the line of best fit (usually reliable).

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Extrapolation: Estimating a value outside the range of the given data points (less reliable as the trend may not continue).

📐Formulae

Mean of x values: xˉ=∑xn\text{Mean of } x \text{ values: } \bar{x} = \frac{\sum x}{n}

Mean of y values: yˉ=∑yn\text{Mean of } y \text{ values: } \bar{y} = \frac{\sum y}{n}

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

💡Examples

Problem 1:

A researcher collects data on the age of a car (years) and its current market value ($). What type of correlation would you expect to see on a scatter diagram?

Solution:

Strong Negative Correlation

Explanation:

As the age of a car increases, its market value typically decreases. Because this relationship is usually very consistent, it is considered a strong negative correlation.

Problem 2:

The mean of the x-coordinates (hours studied) is 5, and the mean of the y-coordinates (test scores) is 65. If a student draws a line of best fit, which specific coordinate must the line pass through?

Solution:

(5,65)(5, 65)

Explanation:

In statistics, the line of best fit for a scatter diagram is mathematically required to pass through the mean point, represented by (xˉ,yˉ)(\bar{x}, \bar{y}).

Problem 3:

Using a line of best fit y=2x+10y = 2x + 10, where xx is the number of sunny hours and yy is the number of visitors to a park, estimate the number of visitors if there are 12 sunny hours.

Solution:

34 visitors

Explanation:

Substitute x=12x = 12 into the linear equation: y=2(12)+10⇒y=24+10=34y = 2(12) + 10 \Rightarrow y = 24 + 10 = 34. This process is called interpolation if 12 hours is within the range of original data.