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Statistics and Probability - Correlation and Lines of Best Fit

Grade 8IB

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

🔑Concepts

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Bivariate data involves the relationship between two different variables, usually denoted as xx (independent variable) and yy (dependent variable).

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A scatter plot is a mathematical diagram using Cartesian coordinates to display values for two variables for a set of data.

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Correlation describes the direction and strength of the relationship between the variables: Positive correlation means yy increases as xx increases; Negative correlation means yy decreases as xx increases.

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The strength of correlation is 'Strong' if the points are clustered closely around a line, and 'Weak' if they are widely dispersed.

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The Line of Best Fit (Trend Line) is a straight line that best represents the data on a scatter plot. It is used to make predictions.

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The mean point (xˉ,yˉ)(\bar{x}, \bar{y}) is the point calculated from the average of all xx-values and all yy-values. The line of best fit must pass through this point.

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Interpolation is the process of predicting a value inside the range of the given data points, which is generally more reliable than extrapolation.

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Extrapolation is the process of predicting a value outside the range of the given data points; it assumes the trend will continue and may be less accurate.

📐Formulae

xˉ=∑xn\bar{x} = \frac{\sum x}{n}

yˉ=∑yn\bar{y} = \frac{\sum y}{n}

m=y2−y1x2−x1m = \frac{y_2 - y_1}{x_2 - x_1}

y=mx+cy = mx + c

💡Examples

Problem 1:

A student records the number of hours spent studying (xx) and the test scores (yy) for five students: (2,40),(4,60),(5,70),(7,85),(2,45)(2, 40), (4, 60), (5, 70), (7, 85), (2, 45). Calculate the mean point (xˉ,yˉ)(\bar{x}, \bar{y}).

Solution:

First, find xˉ\bar{x}: xˉ=2+4+5+7+25=205=4\bar{x} = \frac{2 + 4 + 5 + 7 + 2}{5} = \frac{20}{5} = 4 Next, find yˉ\bar{y}: yˉ=40+60+70+85+455=3005=60\bar{y} = \frac{40 + 60 + 70 + 85 + 45}{5} = \frac{300}{5} = 60

Explanation:

The mean point is calculated by finding the arithmetic mean of all xx-coordinates and all yy-coordinates separately. The mean point is (4,60)(4, 60).

Problem 2:

A line of best fit passes through the mean point (5,12)(5, 12) and has a gradient of m=1.5m = 1.5. Find the equation of the line in the form y=mx+cy = mx + c.

Solution:

Substitute m=1.5m = 1.5, x=5x = 5, and y=12y = 12 into the equation y=mx+cy = mx + c: 12=(1.5×5)+c12 = (1.5 \times 5) + c 12=7.5+c12 = 7.5 + c c=12−7.5c = 12 - 7.5 c=4.5c = 4.5 Therefore, the equation is y=1.5x+4.5y = 1.5x + 4.5.

Explanation:

To find the equation of the line of best fit, we use the gradient and the coordinates of the mean point to solve for the yy-intercept (cc).

Problem 3:

Given the equation of the line of best fit y=2x+10y = 2x + 10, where xx is the temperature in degrees Celsius and yy is the number of ice creams sold. Predict the number of ice creams sold when the temperature is 25∘C25^{\circ}C.

Solution:

Substitute x=25x = 25 into the equation: y=2(25)+10y = 2(25) + 10 y=50+10y = 50 + 10 y=60y = 60

Explanation:

By substituting the independent variable (xx) into the equation of the line of best fit, we can predict the value of the dependent variable (yy).