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Maths Advanced

Expected Value, Variance, and Standard Deviation

What is expected value, variance and standard deviation?

In this post, we use discrete random variables and associated probabilities to solve practical problems, as a part of the Prelim Maths Advanced course under the topic Statistical Analysis and sub-part Discrete Probability Distributions, specifically focusing on expected value, variance and standard deviation. We:

  • Use relative frequencies obtained from data to obtain point estimates of probabilities associated with a discrete random variable
  • Recognise uniform discrete random variables and use them to model random phenomena with equally likely outcomes
  • Examine simple examples of non-uniform discrete random variables, and recognise that for any random variable, X , the sum of the probabilities is 1
  • Recognise the mean or expected value, E(X) = \mu , of a discrete random variable X as a measure of centre, and evaluate it in simple cases
  • Recognise the variance, Var(), and standard deviation ( \sigma ) of a discrete random variable as measures of spread, and evaluate them in simple cases
  • Use Var(X) = E((X - \mu)^{2}) = E(X^{2})  -  \mu^{2} for a random variable and Var(x) = \sigma^{2} for a dataset.

 

What is expected value?

The following two videos will introduce you to expected values E(X) , and how to approach and set-out the calculations.

Part 1

 

Part 2

 

What is variance?

The following two videos will introduce you to variance, and how to set-out the required calculations.

Part 1

 

Part 2

 

What is standard deviation?

The following two videos will introduce you to standard deviation, and how to go about visualising and calculating it.

Part 1

 

Part 2