Statistics Notes



1.1 Mathematical Models In Porbability And Statist

What is a mathematical model? A simplification of a real world situation. These are useful because:

  • They are quick and easy to produce.
  • They can simplify a more complex situation.
  • Improve our understanding of the real world as certain variable can be changed.
  • Allows predictions to be made.
  • They can help provide control.

However, they should be treated with caution because:

  • It is a simplification of the real problem so does not include all aspects of the problem.
  • The model may only work in certain situations.
  • Only work for a restricted range of values.
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1.2 Designing a mathematical model

You can design a mathematical model by:

  • 1. A real world problem is observed.
  • 2. A mathematical model is devised.
  • 3. The mathematical model is used to make predictions about the expected behaviour of the real world problem.
  • 4. Experimental data is collected from the real world.
  • 5. Compare predicted and observed outcomes.
  • 6. Statistical tests are used to assess how well the model describes the real world.
  • 7. The mathematical model is refined, if necessary, to improve the match of predicted outcomes with observed experimental data.
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2.1 Representation and summary of data

Quantitative data: Numerical observations.

Qualitative data: Non-numerical observations.

Continuous Variable: Can take any value in a given range.

Discrete Data: Can only take specific values in a given range.

Oulier: An extreme value that lies outside the overall pattern of data. This is any value that is:

  • Greater than upper quartile + (1.5 X IQR)
  • Less than lower quartile - (1.5 X IQR)


  • Q2-Q1 = Q3 -Q2 then the distribution is symmetrical.
  • Q2-Q1 < Q3-Q2 then the distribution is positively skewed.
  • Q2-Q1 > Q3-Q2 then the distribution is negatively skewed.
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Additional Info

Can also use measures of location for skewness:

  • mode=median=mean describes a distribution which is symmetrical.
  • mode<median<mean describes a distribution with positive skewness.
  • mode>median>mean describes a distribution with negative skewness.

You can calculate 3(mean - median) divided by standard deviation to find skewness.

  • The larger the number the greater the amount of skewness.
  • The closer to zero the more symmetrical the skewness is.
  • Positive number means positive skewness, and negative number means negative skewness.
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