In This Guide
The Main Difference
Probability
Starts with known assumptions or a mathematical model and predicts the likelihood of future or uncertain outcomes.
Statistics
Starts with observed data and summarizes it, compares groups, estimates unknown quantities, or tests claims.
| Question | Probability | Statistics |
|---|---|---|
| Starting point | A model, assumptions, or known possible outcomes | Observed data from a sample, experiment, survey, or population |
| Main direction | Model to possible results | Data to conclusions about a model or population |
| Typical question | What is the chance of rolling a 6? | Does this die appear to be fair based on the rolls? |
| Typical output | Probability, expected value, distribution | Mean, median, standard deviation, estimate, interval, test result |
| Uncertainty | Describes randomness before or across repeated outcomes | Measures uncertainty in conclusions drawn from data |
A simple way to remember
Probability predicts data from a model. Statistics uses data to learn about the model or the population that produced it.
What Probability Studies
Probability measures how likely an event is. A probability is between 0 and 1, inclusive. A probability of 0 means an event is impossible under the model, while a probability of 1 means it is certain.
When all outcomes are equally likely, the probability of an event can be calculated as:
Example: Rolling a fair die
A fair six-sided die has six equally likely outcomes. The probability of rolling an even number is:
The favorable outcomes are 2, 4, and 6.
Complement rule
The probability that an event does not occur is:
Addition rule for mutually exclusive events
Multiplication rule for independent events
What Statistics Studies
Statistics is the study of collecting, organizing, analyzing, and interpreting data. It helps answer questions about patterns, variation, comparisons, relationships, and uncertainty.
Statistics may describe an entire population, but more often it uses a sample to learn about a larger population. That requires careful sampling and an understanding of how much sample results may vary.
Example: Survey data
Suppose 200 randomly selected students are asked how many hours they study each week. Statistics can be used to:
- calculate the sample mean and median;
- measure variability;
- compare subgroups;
- estimate the average for all students;
- evaluate whether the sample supports a claim.
Theoretical vs Experimental Probability
Theoretical probability
Theoretical probability is calculated from a mathematical model before collecting data. For a fair coin:
Experimental probability
Experimental probability is based on observed relative frequency:
Example: Coin tosses
Answer
The experimental probability is 0.47. This is close to, but not exactly equal to, the theoretical probability 0.50.
Random variation
Experimental results usually do not match theoretical probabilities exactly. With more independent trials, relative frequencies often become more stable.
Descriptive vs Inferential Statistics
Descriptive statistics
Descriptive statistics summarize the data that was actually observed. Common tools include:
- tables and graphs;
- mean, median, and mode;
- range, interquartile range, variance, and standard deviation;
- percentiles and proportions.
Inferential statistics
Inferential statistics uses sample data to make conclusions about a broader population. It includes estimation, confidence intervals, hypothesis tests, and regression inference.
| Task | Descriptive | Inferential |
|---|---|---|
| Summarize this sample | Yes | Not required |
| Estimate a population value | No | Yes |
| Display the distribution | Yes | May support inference |
| Quantify uncertainty | Usually not beyond the observed data | Yes |
| Test a claim | No | Yes |
Expected Value
Expected value is a probability concept that predicts the long-run average result of a random process. Multiply each possible value by its probability and add the products.
Example: Expected value of a game
Answer
The expected payout is $2 per play. This does not mean every play pays $2; it describes the long-run average.
Measures of Center and Spread
Statistics describes both the typical value and the variability of a data set. Two groups can have the same mean but very different spreads.
Mean
Median
The median is the middle value after ordering the data. When the number of values is even, the median is the mean of the two middle values.
Range
Population variance
Population standard deviation
Example: Mean, median, and range
Answer
The mean is 7, the median is 5, and the range is 10.
Samples, Populations, and Bias
A population is the full group of interest. A sample is the smaller group actually observed. A useful sample should represent the population and be selected in a way that reduces bias.
| Term | Meaning | Example |
|---|---|---|
| Population | The complete group being studied | All students at a university |
| Sample | The observed subset | 300 randomly selected students |
| Parameter | A numerical characteristic of a population | The true mean study time of all students |
| Statistic | A numerical characteristic calculated from a sample | The sample mean study time |
Common sources of bias
- Convenience sampling: using the easiest people to reach;
- voluntary response: people choose whether to participate;
- undercoverage: part of the population is difficult or impossible to select;
- response bias: wording or conditions influence answers;
- nonresponse: selected individuals do not participate.
A large sample is not automatically representative
A very large biased sample can still produce a misleading conclusion. Selection method is often more important than sample size alone.
How Probability and Statistics Work Together
Probability provides the mathematical foundation for statistical inference. Statistical methods use probability models to describe how sample results vary and how surprising an observed result would be under a particular assumption.
For example, suppose a company claims that 60% of customers prefer a new design. A statistician collects a random sample and observes the sample proportion. Probability is then used to evaluate how much that proportion could vary by chance if the claim were true.
Sample proportion
Here, x is the number of successes and n is the sample size.
Worked Examples
Example 1: Is this probability or statistics?
The possible outcomes and model are known before any card is drawn.
Answer
This is a probability question.
Example 2: Is this probability or statistics?
This question begins with observed data and asks what the data suggests about the process.
Answer
This is a statistics question that uses probability to judge whether the observed frequency is plausible.
Example 3: Experimental probability
Answer
The estimated probability is 0.35, or 35%.
Example 4: Comparing center and spread
Both data sets have mean 8:
Set A has no variability because every value equals the mean. Set B has a much larger spread.
Answer
The means are equal, but the distributions are not. Measures of spread are necessary to describe the difference.
Common Probability and Statistics Mistakes
Treating probability as a guarantee
A probability describes long-run behavior or uncertainty, not what must happen in the next trial.
Assuming experimental results must match theory exactly
Random samples vary. Exact equality is not expected in a limited number of trials.
Confusing independence with mutual exclusivity
Independent events can occur together. Mutually exclusive events cannot occur together.
Using the mean without checking outliers
A large or small outlier can pull the mean away from the center of most observations.
Ignoring variability
Two data sets can have the same mean but very different consistency and spread.
Generalizing from a biased sample
A sample must represent the population before its results can support a broader conclusion.
Claiming causation from association
A relationship between variables does not prove that one variable causes the other.
Mixing population and sample notation
Parameters describe populations; statistics describe samples. Their symbols and formulas may differ.
A Fast Classification Checklist
- Does the problem begin with known possible outcomes or a probability model?
- Does it ask what is likely to happen? That points toward probability.
- Does the problem begin with observed data?
- Does it ask for a summary, comparison, estimate, or conclusion? That points toward statistics.
- Is the probability theoretical or based on experimental frequency?
- Is the statistical task descriptive or inferential?
- Does the sample represent the target population?
- Are center, spread, shape, and outliers all considered?
- Does the conclusion acknowledge uncertainty?
Why the Difference Matters
Probability and statistics use many of the same ideas, but reversing their direction can lead to incorrect reasoning. A probability model may predict expected behavior, while observed data may suggest that the model is incomplete or inappropriate.
Understanding the difference also helps with exam questions. When a question gives a sample and asks about a population, it is usually statistical inference. When it gives possible outcomes and asks for a likelihood, it is usually probability.
Practice Probability and Statistics Questions
Practice probability, descriptive statistics, data interpretation, sampling, expected value, distributions, and mixed exam-style questions.