2.9 Mean and Spread of a Discrete Distribution
A Parameter Belongs to the Model
A rental service uses a stated daily-refund model: refunds total $0 with probability 0.50, $4 with probability 0.30, and $10 with probability 0.20. Let X be the dollar total for a day under this model. The probabilities describe the defined population of model-days; they are supplied model values, not estimates inferred from an unexplained sample.
A parameter numerically describes a population or probability distribution. It has one fixed value while the model remains fixed. Individual days vary, and a sample average may vary, but the model’s mean and standard deviation do not change merely because the latest day has a different result.
Worked Example: Weight Each Value by Its Chance
The expected value, also called the mean, is written or . For countable or finite discrete values,
Here runs over the possible values, and is each value’s probability. This is a weighted average, not the ordinary average of the three dollar amounts. More probable values contribute more often in the long run.
The expected refund is $3.20 per model-day. No single day has to produce $3.20—in fact, this model never does. Over many days the average refund total approaches $3.20 per day. “Expected” names that long-run average, not a guaranteed or most likely daily result.
Measure Deviations from the Mean
The standard deviation, written or , describes typical distance from the mean. First calculate the probability-weighted squared deviations:
| x: dollars | P(X=x) | x−3.20 | (x−3.20)²P(X=x) |
|---|---|---|---|
| 0 | 0.50 | −3.20 | 5.120 |
| 4 | 0.30 | 0.80 | 0.192 |
| 10 | 0.20 | 6.80 | 9.248 |
| Total | 1.00 | 14.560 |
Their sum is the variance, square dollars. Taking the square root restores dollars:
Squaring makes negative and positive deviations contribute positively. The probabilities weight them by how often they occur. The mean inside every deviation is the same distribution mean, $3.20. Variance and standard deviation are both fixed model parameters, but their units differ: square dollars versus dollars.
Say What the Numbers Describe
Under this refund model, daily totals typically deviate about $3.82 from the mean daily total of $3.20. This is a description of variation among daily totals in the population. It is not a promise that every total is within one SD, and it does not describe the change between today’s total and tomorrow’s total.
The same arithmetic works for counts, lengths or other discrete outcomes. Keep the unit and population in the interpretation. A mean without its time period or observational unit can be mistaken for a total; a standard deviation without its reference mean can be mistaken for a difference between consecutive observations. The complete interpretation identifies the random variable, the population/model, the long-run average and the typical deviation about that average.
Try It Yourself
The following original practice applies the ideas above to a fresh setting. Show the reasoning requested in each part. Answers and task-specific rubrics are on the separate solutions page.
Original generated practice for the Fall 2026 course.
Glossary and deeper reading · 9 refreshers
These editorial refreshers supplement the reading. Definitions and examples are added guidance; original source wording and recorded answers remain in their original context.
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Probability
Probability quantifies how likely an event is under a specified chance model. It lies between 0 and 1, with larger values representing greater likelihood. For example, drawing a red card from a well-shuffled standard deck has probability one half.
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Population
A population is the full set of individuals or measurements a study aims to describe. It must be defined clearly before deciding whether a sample supports generalization. For example, all students enrolled in a particular school this year form a population.
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Population parameter
A population parameter is a numerical feature of the population or probability model being studied. It is fixed for that defined population, although its value may be unknown. For example, the true proportion of all enrolled students who walk to school is a parameter.
Read more: Population parameter in the captured Statistics reading →
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Probability distribution
A probability distribution describes a random variable’s possible values and their probabilities. A discrete distribution lists probability masses, while a continuous distribution describes probabilities through areas under a density. For example, a distribution can list the probabilities of 0, 1, 2, or 3 red cards in three draws.
Read more: Probability distribution in the captured Statistics reading →
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Standard deviation
Standard deviation measures spread around the mean using squared deviations and a square root. A sample standard deviation uses the sum of squared deviations divided by one less than the sample size before taking the square root. For example, a larger standard deviation of travel times indicates greater variability, in the original time units.
Read more: Standard deviation in the captured Statistics reading →
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Mean
The mean is a measure of center often called an average. For a finite data set, the arithmetic mean is the sum of the values divided by their number; for a random variable, the mean is its probability-weighted expected value when that value exists. Means can be pulled toward extreme values and are not generally resistant. For example, the arithmetic mean of 2, 4, and 9 is 5.
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Variance
Variance is the square of the standard deviation. It measures spread in squared units, with the precise averaging rule depending on whether it describes a sample or a probability distribution. For example, a standard deviation of 3 centimeters corresponds to a variance of 9 square centimeters.
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Observation
An observation is a recorded value or set of values for an individual or occasion. The individual may be a person, object, or event rather than a human subject. For example, one plant’s measured height is an observation in a plant-growth study.
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Random variable
A random variable assigns a numerical value to each outcome of a chance process. Its value is uncertain before the outcome is observed, and its distribution gives the probabilities of possible values. For example, the number of red cards in three draws is a random variable.
Read more: Random variable in the captured Statistics reading →