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Correlation measures the relationship between two variables, including the strength and direction between them.

Directions

  • Either positive, negative, or flat (no relationship)

Mean

  • ๐‘ฅ โ†’ mean
  • ๐‘ฅ๐‘– โ†’ individual value

Standard Deviation

  • Measures the average deviation of the data from the mean
  • ๐œŽ=โˆšโˆ‘๐‘›๐‘–=1(๐‘ฅ๐‘–โˆ’๐‘ฅ)2๐‘›

Variance

  • Measures how spread out the data is form the mean
  • Units are squared units of the data. If ๐‘ฅ is in meters, variance is meters2
  • ๐‘ฃ=๐œŽ2=โˆ‘(๐‘ฅ๐‘–โˆ’๐‘ฅ2)๐‘›

Covariance

  • cov(๐‘ฅ,๐‘ฆ) measures how 2 variables vary together
  • Positive โ†’ both increase/decrease together
  • Negative โ†’ one increases and the other decreases
  • cov(๐‘ฅ,๐‘ฆ)=โˆ‘(๐‘ฅ๐‘–โˆ’๐‘ฅ)(๐‘ฆ๐‘–โˆ’๐‘ฆ)๐‘›โˆ’1
    • Similar to the variance formula, but we use ๐‘ฆ rather than squaring ๐‘ฅ

Correlation Coefficient

  • Referred to as ๐‘Ÿ, where ๐‘Ÿโˆˆ[โˆ’1,1]
  • ๐‘Ÿ=cov(๐‘ฅ,๐‘ฆ)๐œŽ๐‘ฅ๐œŽ๐‘ฆ,๐‘Ÿโˆˆ[โˆ’1,1]

Strength

  • The magnitude of ๐‘Ÿ represents the strength of the relationship
    • |๐‘Ÿ|=1 โ†’ perfect
    • |๐‘Ÿ|โˆˆ[0.8,0.9] โ†’ strong
    • |๐‘Ÿ|โˆˆ[0.7,0.5] โ†’ medium
    • |๐‘Ÿ|โˆˆ[0.1,0.4] โ†’ weak
    • |๐‘Ÿ|=0 โ†’ no relationship

Direction

  • Positive โ†’ variables increase together
  • Negative โ†’ variables move in opposite directions

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