What Is Expectation Value?

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  • Written By: Helen Akers
  • Edited By: Jessica Seminara
  • Last Modified Date: 15 October 2019
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Expectation value is the average value of an unknown variable obtained from a large amount of experiments. When an experiment or measurement is performed multiple times, the result of those experiments is expected to vary. An expectation value simply predicts the weighted average for all of those values. It is a concept that is commonly used in quantum physics as well as statistics.

The weighted average of all possible results is calculated by taking each potential variable and multiplying it by its probability. All of these values are added together and divided by the number of probabilities or possible outcomes. A simple illustration is the flipping of a coin. There is 50 percent probability that a flip will result in a heads and a 50 percent probability that it will result in a tails.

In this scenario there are two possible outcomes. Each potential result is multiplied by 0.50. Adding these two figures together gives a total of 1.00, which is then divided by two. The expected value for a coin toss is 0.50.

Most calculations involving the expectation value are more complex than a coin toss, however. They typically involve a variety of possibilities and weighted probabilities. An expectation value is simply a prediction. It cannot calculate an actual outcome, but rather just the mean or average of those outcomes.


Often, sample sizes are used to predict the outcomes of larger populations. When calculating the expectation value of a sample size, it is assumed that its mean is equal to the average of the larger population. Sample sizes are smaller representations of entire populations and are used since it is not feasible to test or measure every single variable that exists.

There is always the chance that the actual outcome of an experiment will deviate sharply from the average. Most of the time, about half of those values will be above the mean or expectation value. The other half will fall below the mean. With populations that are skewed towards the left of the mean, a higher proportion of actual experiment values will be below the mean. For populations that are skewed to the right, most of the actual values will be somewhere above the mean.

In terms of quantum physics, the expectation value is the average amount of energy that a large group of atom particles can be expected to have. The energy charge refers to the particle's orbit position. It dictates the number of degrees those particles are likely to move in response to external sources of energy or magnetic fields. The expectation value of energy would reflect an average amount of expected movement in a group of particles.


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