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Stats week 3 Formula
| Question | Answer |
|---|---|
| What is the formula for the Mean | x̄ = Σxᵢ / n, ȳ = Σyᵢ / n |
| What is the formula for Sample Standard Deviation | sₓ = √[ Σ(xᵢ − x̄)² / (n − 1) ], sᵧ = √[ Σ(yᵢ − ȳ)² / (n − 1) ] |
| What is the formula for Covariance | cov(x, y) = Σ(xᵢ − x̄)(yᵢ − ȳ) / (n − 1) |
| What is the formula for the Correlation Coefficient? | r = cov(x, y) / (sₓ × sᵧ) |
| What is the formula for the Regression Line | ŷ = a + b·x, where b = r × (sᵧ / sₓ), and a = ȳ − b·x̄ |
| x̄ | Mean (average) of x values |
| ȳ | Mean (average) of y values |
| Σ | Summation symbol (add all values) |
| xᵢ | Individual x value (the i-th data point) |
| yᵢ | Individual y value (the i-th data point) |
| n | Sample size (number of data points) |
| sₓ | Sample standard deviation of x |
| sᵧ | Sample standard deviation of y |
| cov(x, y) | Covariance between x and y |
| r | Correlation coefficient (ranges from -1 to +1) |
| ŷ | Predicted y value (from regression line) |
| b | Slope of regression line |
| a | Y-intercept of regression line |
| What does (x̄)represent? | The center or average value of a dataset |
| What does standard deviation measure? | The spread or variability of data around the mean |
| What does covariance tell us? | Whether two variables tend to increase together (positive) or inversely (negative) |
| What's the difference between covariance and correlation? | Correlation is standardized (between -1 and +1), covariance is not standardized |
| Why do we divide by (n-1) instead of n? | It's Bessel's correction - provides an unbiased estimate for sample statistics |
| What does r = 1 mean? | Perfect positive linear relationship |
| What does r = -1 mean? | Perfect negative linear relationship |
| What does r = 0 mean? | No linear relationship between variables |
| To calculate correlation (r), what must you calculate first? | Standard deviations (sₓ and sᵧ) and covariance |
| To find the regression line, what do you need? | The means (x̄, ȳ), standard deviations (sₓ, sᵧ), and correlation (r) |
| What's the first step in calculating standard deviation? | Calculate the mean (x̄ or ȳ) |
| How are the slope (b) and correlation (r) related? | b = r × (sᵧ / sₓ) - slope depends on correlation and standard deviations |
| If r = 0, what is the slope (b) of the regression line? | b = 0 (horizontal line) |
| The regression line always passes through which point? | (x̄, ȳ) - the point of the means |
| If sₓ is large, what does that tell you about the x data? | The x values are widely spread out from their mean |
| If cov(x,y) is negative, what happens as x increases? | y tends to decrease |
| What does ŷ represent in a regression? | The predicted value of y for a given x |
| What's the difference between ŷ and y? | ŷ is the predicted value from the regression line; y is the actual value |
| What's the difference between xᵢ and x̄? | xᵢ is an individual data point; x̄ is the average of all x values |
| Can correlation (r) be greater than 1? | No, r is always between -1 and +1 |
Created by:
Producer456