# Mean and Expected Frequencies for Binomial Distribution

## Definition and Formula

• The mean (or expected value) of a binomial distribution is the average outcome of a large number of trials in an experiment.
• It is denoted by the Greek letter μ (mu) for the population mean, or ‘x̄’ (x-bar) for the sample mean.
• The mean of a binomial distribution is given by the formula μ = n*p, where ‘n’ is the number of trials and ‘p’ is the probability of success on each trial.

## Expected Frequencies

• The term expected frequency refers to the number of times a particular outcome is expected to occur in a set number of trials, based on the probability of that outcome.
• For a binomial distribution, the expected frequency of successes is given by n*p, which is also the mean of the distribution.
• The expected frequency of failures can be found by subtracting the expected frequency of successes from the total number of trials or n*(1-p).

## Significance and Applications

• Understanding the mean and expected frequencies helps to make predictions about the outcome of an experiment or investigation.
• For example, if you were tossing a fair coin 100 times, you could predict (using the mean of the binomial distribution) that it is most likely you would get heads approximately 50 times.
• Similarly, if a quality control inspector is examining items off a production line and each item has a 1% chance of being defective, the expected number of defective items in a sample of 500 can be found using the formula *np** = 500*0.01 = 5 items.
• It’s an important data parameter to handle large data sets in fields like business analytics, market research and many others.

## Binomial Distribution Characteristics and Mean

• A useful feature of a binomial distribution is that its mean is simply the most likely outcome of a large number of trials.
• It’s important to remember that the mean of a binomial distributed variable only provides the ‘centre’ of the distribution. It does not provide information about the ‘spread’ or ‘variance’ of the outcomes. For that, you would need to calculate the standard deviation or variance.
• Always recollect, the concept of mean and expected frequencies is tied to long-term averages. This means the more trials conducted, the more likely the average outcome will approach the mean.

## Limitations and Considerations

• It’s also important to remember that the formulary is based on the assumption that the probability of success stays consistent across trials. If this isn’t the case, then a different probability model might be more appropriate.
• Recognise that the idea of expected value is just a prediction based on probability, not a guarantee. The actual result of an experiment could be quite different from the expected value, especially if the number of trials is small.