7.4 Errors in Sampling
7.4: Errors in Sampling
7.4.1: Expected Value and Standard Error
Expected value and standard error can provide useful information about the data recorded in an experiment.
Learning Objective
Solve for the standard error of a sum and the expected value of a random variable
Key Takeaways
Key Points
- The expected value (or expectation, mathematical expectation, EV, mean, or first moment) of a random variable is the weighted average of all possible values that this random variable can take on.
- The expected value may be intuitively understood by the law of large numbers: the expected value, when it exists, is almost surely the limit of the sample mean as sample size grows to infinity.
- The standard error is the standard deviation of the sampling distribution of a statistic.
- The standard error of the sum represents how much one can expect the actual value of a repeated experiment to vary from the expected value of that experiment.
Key Terms
- standard deviation
- a measure of how spread out data values are around the mean, defined as the square root of the variance
- continuous random variable
- obtained from data that can take infinitely many values
- discrete random variable
- obtained by counting values for which there are no in-between values, such as the integers 0, 1, 2, ….
Expected Value
In probability theory, the expected value (or expectation, mathematical expectation, EV, mean, or first moment) of a random variable is the weighted average of all possible values that this random variable can take on. The weights used in computing this average are probabilities in the case of a discrete random variable, or values of a probability density function in the case of a continuous random variable.
The expected value may be intuitively understood by the law of large numbers: the expected value, when it exists, is almost surely the limit of the sample mean as sample size grows to infinity. More informally, it can be interpreted as the long-run average of the results of many independent repetitions of an experiment (e.g. a dice roll). The value may not be expected in the ordinary sense—the “expected value” itself may be unlikely or even impossible (such as having 2.5 children), as is also the case with the sample mean.
The expected value of a random variable can be calculated by summing together all the possible values with their weights (probabilities):
where represents a possible value and represents the probability of that possible value.
Standard Error
The standard error is the standard deviation of the sampling distribution of a statistic. For example, the sample mean s the usual estimator of a population mean. However, different samples drawn from that same population would in general have different values of the sample mean. The standard error of the mean (i.e., of using the sample mean as a method of estimating the population mean) is the standard deviation of those sample means over all possible samples of a given si