Key takeaway: A random sample is a group selected purely by chance from a larger population, so that nobody’s choice shapes who is included and the group tends to mirror the whole.
Grade 77.SP.A.1Statistics and Data
Definition of Random Sample
A random sample is a small group selected from a larger population by chance alone. Names drawn from a hat, numbers produced by a random generator, or every tenth item picked after a random start all qualify, because nobody’s choice influences who gets in.
Why bother? Asking everyone is often too slow or too costly. A sample chosen by chance tends to look like the whole group, with similar proportions of ages, opinions and habits. A group picked for convenience, such as friends at your lunch table, tends to lean one way, and conclusions drawn from it can be wrong for the population.
Formula and notation
Sample size is written \(n\) and population size \(N\). To predict from a sample, scale up its proportion:
\[\text{prediction} = \frac{\text{count in sample}}{n} \times N\]
The larger the random sample, the closer the prediction usually lands to the truth.
Key facts
- Fair selection means each person in the group could just as likely be the one drawn.
- More chosen members generally give a steadier prediction than only a few.
- Convenience picks, like the first 20 people in line, are usually biased.
- Two separate draws from one group will give slightly different results.
Random Sample example
A middle school has 840 students. A computer picks 40 student ID numbers at random and each of those students is asked whether they prefer pizza for the spring party. In the sample, 14 choose pizza. Predict how many students in the whole school prefer pizza.
-
Proportion in the sample: \(\frac{14}{40} = 0.35\).
-
Apply it to the school: \(0.35 \times 840 = 294\).
About 294 students would be expected to prefer pizza.
Common mistake
Watch out
To learn the school’s favorite sport, Tavon surveys only the 25 players on the basketball team and reports that 80% of students love basketball.
Why it is wrong
Team members are not chosen by chance, and they are far more likely than other students to pick basketball. This convenience group is biased. A fair estimate needs a random sample, such as numbers drawn from the full student roster.
Related terms and lessons
Browse the glossary: A–Z list · Grade 7 vocabulary · Statistics and Data