Introduction

• A hat (^ or - ) above a greek letter refers to the fact that we are working with a sample rather than a population.

​​ Definitions

• Population

  • A population refers to a group of animals that are part of the overall breeding structure
    • e.g. All dairy cows in New Zealand

• Sample

  • A subset of animals from a population
    • e.g. 30 dairy cows from New Zealand

• Sample Mean

  

  Where $y_i$ is an observed trait value on an animal in the sample

  • An average of all observed traits

• Standard Deviation

  • A measure of how spread out the data is

    • Note - To give an idea of its measure, In a normal distribution 99.7% of the data will be within 3 standard deviations

      

• Variance

  • An indication of the range of possible values that $y_i$ could be. For example a min of 1 and a max of 3 will have a smaller variance than a min of 0 and a max of 4.

  • $StandardDeviation^2$

    

• Coefficient of Variation

  • Represents the degree of variation relative to the size of the mean

    

• Covariance

  • Measure how two traits vary together. A trait such as milk solids against a trait such as avg days on penicillin:

    

  • Can be positive or negative

    • Positive means when one grows so does the other
    • Negative means when one grows the other shrinks

• Correlation Coefficient

  • Describes the same thing as Covariance, but is a bit easier to calculate and interpret

    

  • Ranges between -1 and +1

​​ Normal Distribution

• Described using the mean and variance - N(mean, variance)
• A standard normal distribution (SND) = N(0,1)

​​ Commonly used values for an SND

• z-value
  • Used to express standard deviations from the mean
    • 2.5 is 2.5 standard deviation from the mean (positively)
  • $z_i = y_i-μ/σ$
• Percentage point (p)
  • The portion of population above the z-value
• Confidence Interval
  • Gives the portion of the population within a z-value and it’s inverse

• Selection Intensity
  • $i$
  • the average value of the portion $p$ of the population that is above the z-value

​​ The Empirical Rule

For any bell-shaped curve - normal distribution

• What % of values for within 1 standard deviation of the mean in either direction
  • 68%
• What % of values fall within 2 standard deviations of the mean in either direction
  • 95%
• What % of values fall with 3 standard deviations of the mean in either direction
  • 99.7%