Statistics Notes

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.
  

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.

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)

Quartiles:

• 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.

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.

• Rule of addition: P(A ∪ B) = P(A) + P(B) - P(A ∩ B)
• Rule of multiplication: P(A ∩ B) = P(A) P(B|A)
• Rule of subtraction: P(A') = 1 - P(A)
• P(A|B)=P(A n B)/P(B)
• Mutually Exclusive Events: P(A U B) = P(A)+P(B), where there are no outcomes in common or not overlap. P(A n B)=0
• When one event has no effect over another they are independant. To prove an independant event then P(A n B)=P(A) X P(B)