A sample in statistics is a part of a population that is actually observed or measured, such as 60 students surveyed out of a school of 900. You use the sample to estimate facts about the whole population. A random sample, where every member has the same chance to be chosen, tends to represent the population fairly.
Random samples do not match the population exactly. Their estimates scatter around the true value, and bigger samples scatter less.
Formal definition and notation
A sample is a subset of a population selected for study. In a simple random sample of size n, every group of n individuals is equally likely to be chosen. Numbers computed from the sample, called statistics (such as x̄ or p̂), estimate the matching population parameters. Different random samples of the same size give somewhat different statistics; this sampling variability shrinks as n grows.
Symbol
Read it as
Meaning
n
n
the sample size
x̄
x bar
the sample mean
p̂
p hat
the sample proportion
Key facts
A sample is useful only if it is representative: it looks like the population in the ways that matter.
Random selection (drawing names, random numbers) is the standard way to get a representative sample.
Convenient samples, like friends or volunteers, are often biased; see sampling bias.
Two random samples from the same population usually give different results. Larger samples vary less.
To estimate a population count: (sample proportion) × (population size).
The sample mean x̄ estimates the population mean μ; it is rarely exactly equal.
Sample examples, step by step
Example 1: Sample proportion
Warm-up
In a random sample of 45 students, 18 said pizza is their favorite lunch. What percent of the sample chose pizza?
1845 = 0.4.
0.4 = 40%.
Answer: 40% of the sample chose pizza.
Example 2: Predict for the population
Core
A school has 900 students. In a random sample of 60, 21 ride a bike to school. Predict how many students in the school ride a bike.
Sample proportion: 2160 = 0.35.
Apply it to the population: 0.35 × 900 = 315.
Answer: About 315 students ride a bike to school.
Example 3: Estimate a mean from a sample
Test-style
To estimate the mean word length in a novel, Kai picks 6 words at random and counts letters: 4, 7, 3, 5, 9, 2. Estimate the mean word length, and say why a second sample might differ.
Sample mean: 4 + 7 + 3 + 5 + 9 + 26 = 306 = 5.
Use 5 letters as the estimate for the whole book.
Another random 6 words would likely give a different mean; a larger sample would give a steadier estimate.
Answer: The estimated mean word length is 5 letters.
Common mistakes
✗ MistakeThinking “random” means “whoever is handy”: surveying the first 30 students in the lunch line.
✓ InsteadGive every student an equal chance, for example by drawing 30 names from the full school list.
A handy group may differ from the population, for example by grade or schedule.
✗ MistakeExpecting a sample to match the population exactly: the sample says 35%, so exactly 315 of 900 students bike.
✓ InsteadSay “about 315.” Another sample might give 30% or 40%.
Sample results vary by chance; they estimate the population value.
✗ MistakeThinking a huge sample always fixes a badly chosen sample: 5,000 online volunteers must be accurate.
✓ InsteadA smaller random sample usually beats a large biased one.
Size reduces random variation but not bias in how people were chosen.
A sample space lists every possible outcome of a chance experiment, like {H, T}. It is a probability idea, not a group of people surveyed.
Quick check: 3 questions
Question 1 of 3
Which is a random sample of students at a school?
Show answer and explanation
Answer: B. Drawing from the full list gives every student the same chance. Early arrivals, volunteers and one class all leave some kinds of students out.
Question 2 of 3
In a random sample of 80 students, 12 play chess. About how many of the 1,000 students in the school play chess?
Show answer and explanation
Answer: B.1280 = 0.15, and 0.15 × 1000 = 150. 12 is the sample count, 120 uses 12%, and 80 is the sample size.
Question 3 of 3
Two random samples from the same population give means of 14.2 and 15.1. What is the best explanation?
Show answer and explanation
Answer: C. Different random samples naturally give different statistics; that is sampling variability. Neither sample is necessarily wrong, and averaging them gives an estimate, not an exact value.
Sample: frequently asked questions
What makes a good sample?
A good sample is representative: it reflects the population in the ways that matter for the question. The best way to get one is random selection, where every member has an equal chance to be picked. A larger random sample gives a more precise estimate. Volunteers, friends and whoever is nearby usually make poor samples.
How big should a sample be?
Big enough that the estimate is precise enough for the purpose. Larger random samples vary less, so their results are more reliable, but the gains slow down as the size grows. A well-chosen random sample of a few hundred people can describe a large population surprisingly well, while a biased sample of thousands can be far off.
PopulationA population in statistics is the whole group of people, animals or objects you want to learn about, such as every student in a school or every bulb a factory makes. Measuring all of them is often too costly, so you study a sample, a part of the population, and use it to draw conclusions about the whole group.Statistics · Grade 7, Statistics, Algebra 1SubsetA subset is a set whose elements all belong to another set. If every element of A is also in B, then A is a subset of B, written A ⊆ B. Every set is a subset of itself, the empty set is a subset of every set, and a set with n elements has 2n subsets.Probability · Algebra 1, StatisticsNumberA number is an amount or value used to count, measure, order and compare. We write numbers with digits (7), words (seven) or other symbols, but the number itself is the amount: 7 dots, the word seven and the numeral 7 all show the same number. Numbers include whole numbers, fractions, decimals and negatives.Number System · Kindergarten, Grade 1, Grade 2Sampling BiasSampling bias happens when the way a sample is chosen makes some members of the population more likely to be picked than others, so the sample does not represent the population. Its results lean the same wrong way every time. Surveying only gym-goers about exercise, or counting only volunteers who reply, are classic examples.Statistics · Grade 7, Algebra 1, StatisticsLengthLength is how long something is: the distance from one end of an object to the other. You measure it by counting equal units laid end to end with no gaps or overlaps, such as inches, feet, centimeters or meters. A new pencil is about 7 inches long.Measurement · Grade 1, Grade 2, Grade 3Sample SpaceThe sample space of a chance experiment is the set of all its possible outcomes, usually written S. For one coin flip, S = {H, T}; for one number cube, S = {1, 2, 3, 4, 5, 6}. Organized lists, tables and tree diagrams help you write it out, and the probabilities of all its outcomes add to 1.Probability · Grade 7, Grade 8, StatisticsOutcomeAn outcome is one possible result of a chance experiment, such as rolling a 4 on a number cube or getting tails on a coin flip. The set of all outcomes is the sample space, and a group of outcomes is an event. When outcomes are equally likely, each one has probability 1 divided by the number of outcomes.Probability · Grade 7, Grade 8, StatisticsDataData are facts or values collected to answer a question, such as survey answers, measurements or counts. Numerical data are numbers you measure or count, like heights; categorical data are labels or groups, like favorite colors. Data are organized in tables and shown in graphs so patterns are easy to see.Statistics · Grade 2, Grade 3, Grade 6StatisticsStatistics is the branch of math that collects, organizes, analyzes and interprets data to answer questions. Descriptive statistics summarize data with graphs and numbers such as the mean and median. Inferential statistics use a sample to draw conclusions about a whole population, with a stated amount of uncertainty.Statistics · Grade 6, Grade 7, Grade 8ProportionA proportion is an equation that says two ratios are equal, such as 12 = 36. It is read 1 is to 2 as 3 is to 6. In a true proportion the cross products are equal, which lets you find a missing value. Proportions are often used to solve scale, percent, and rate problems.Ratios and Proportions · Grade 6, Grade 7, Grade 8MeanThe mean is the sum of the values divided by the number of values. It is the arithmetic average: add every value and divide by how many there are. For 3, 7, 5 and 9, the mean is 24 ÷ 4 = 6. The mean is the balance point of the data, so one outlier can pull it far.Statistics · Grade 5, Grade 6, Grade 7Relative FrequencyRelative frequency is how often a value or event occurs compared with the total number of observations: frequency ÷ total. It can be a fraction, a decimal or a percent. If 12 of 40 students choose apples, the relative frequency of apples is 1240 = 0.3, or 30%. All the relative frequencies in a table add to 1.Statistics · Grade 7, Grade 8, Statistics