Statistics 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.
You rarely measure everyone. Statistics takes a random sample, describes it with graphs and numbers, then infers what the whole population is probably like.
Formal definition and notation
Statistics is the science of collecting, analyzing, interpreting and presenting data. It starts with a statistical question, one that expects the answers to vary. The population is the whole group of interest, and a sample is the part that is actually measured. A number describing the population, such as its mean μ, is a parameter; the same kind of number computed from a sample, such as x, is a statistic. Because different samples give different statistics, inferential methods report an estimate together with a measure of how far off it may be.
Symbol
Read it as
Meaning
n
n
the sample size, the number of values in the sample
A statistical question expects variability. How many hours do students in my class sleep? is statistical; How many hours did I sleep last night? is not.
Descriptive statistics summarize the data you have. Inferential statistics generalize from a sample to a population.
Data are categorical (eye color, favorite sport) or numerical (height, test score). The type decides which graphs and summaries make sense.
A random sample gives every member of the population the same chance to be chosen, which protects against bias.
A scatter plot can show that two variables move together, but correlation alone does not prove that one causes the other.
The most used statistic: the sample mean
The most used statistic: the sample mean
x = Σ xn
Σx
the sum of all the values in the sample
n
how many values are in the sample
x
the sample mean, used to estimate the population mean μ
A sample mean is an estimate, not the exact population value. A larger random sample usually gives an estimate closer to μ.
Try it: describe a sample
Interactive
These are the commute times, in minutes, of 12 sampled workers. Switch between the dot plot, box plot and histogram, edit the values, and watch how the mean, median and spread change.
This interactive needs JavaScript. The worked examples below show the same ideas step by step.
Statistics examples, step by step
Example 1: Spot a statistical question
Warm-up
Which is a statistical question? (a) How old is our school principal? (b) How many hours of sleep do seventh graders at our school get on a school night?
Question (a) has one answer, a single fact about one person.
Question (b) asks about many students, and their answers will vary, for example 7, 8, 9 or 10 hours.
A statistical question expects that variability.
Answer: Question (b) is the statistical question.
Example 2: Population, sample and statistic
Core
A city surveys 400 randomly chosen residents about building a new park, and 260 of them support it. Name the population and the sample, and find the sample statistic as a percent.
The population is every resident of the city.
The sample is the 400 residents who were surveyed.
The statistic is the sample proportion: 260400 = 0.65, or 65%.
Answer: Population: all city residents. Sample: the 400 surveyed. Statistic: 65% support the park.
Example 3: Use a sample to estimate a population
Test-style
In a random sample of 50 students at a school of 900 students, 18 walk to school. Estimate how many students at the school walk to school.
Sample proportion: 1850 = 0.36.
Apply it to the population: 0.36 × 900 = 324.
The sample was random, so the estimate is reasonable, but the true count could be somewhat higher or lower.
Answer: About 324 students walk to school.
Common mistakes
✗ MistakeAsking only your friends and reporting that 90% of the school likes a new lunch menu.
✓ InsteadChoose students at random from the whole school, for example by drawing names.
A sample that is not random can be biased: your friends may think alike, so they do not represent everyone.
✗ MistakeReporting that exactly 65% of the city supports the park because 65% of the sample did.
✓ InsteadSay that about 65% support it, give or take a margin.
A different random sample would give a slightly different percent. A sample statistic estimates the parameter; it does not equal it.
✗ MistakeConcluding that ice cream causes sunburn because sales of both rise together.
✓ InsteadBoth rise in hot, sunny weather. The weather is a third variable linked to each.
Correlation shows that two variables move together, not that one causes the other.
Probability starts from a known model and predicts data: a fair coin should land heads about half the time. Statistics starts from data and works back to the model: 62 heads in 100 flips asks whether the coin is fair.
Statisticsvs.Statistic (singular)
A statistic is one number computed from a sample, such as a sample mean of 12 minutes. Statistics is the whole subject, or a collection of such numbers.
Statisticsvs.Parameter
A parameter describes the entire population, such as the true mean μ. It is usually unknown, and a statistic from a sample is used to estimate it.
Quick check: 3 questions
Question 1 of 3
Which is a statistical question?
Show answer and explanation
Answer: B. Only the books question expects different answers from different students, so it involves variability. The other three each have one fixed answer: 11 letters, 12, and 366 days.
Question 2 of 3
In a random sample of 250 voters, 140 favor a ballot measure. What is the sample proportion?
Show answer and explanation
Answer: D.140250 = 0.56. 140 is the count, not the proportion; 0.44 is the share against; 1.79 divides 250 by 140, upside down.
Question 3 of 3
Which is an example of inferential statistics?
Show answer and explanation
Answer: C. Inference goes from a sample to a larger population, which only the screen-time choice does. The other choices describe or organize data that you already have, which is descriptive statistics.
Statistics: frequently asked questions
What are the two main branches of statistics?
Descriptive statistics summarize data with tables, graphs and numbers such as the mean, median and range; they describe only the data collected. Inferential statistics use a random sample to estimate or test something about a larger population, such as predicting an election from a poll of 1,000 voters, and they report how uncertain the conclusion is.
What is the difference between statistics and probability?
They run in opposite directions. Probability starts with a known chance process and predicts what data it will produce, such as about 50 heads in 100 flips of a fair coin. Statistics starts with the data you observed and reasons back to the process, such as asking whether 70 heads in 100 flips means the coin is unfair.
Is statistics part of math?
Statistics uses a lot of math, such as fractions, percents, averages, graphs and probability, and it is taught in math classes from grade 6 on. It is also its own science, because it adds ideas that pure math does not need: how to collect good data, how to avoid bias, and how to judge evidence when the answers vary.
DataData 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 6PopulationA 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 1SampleA 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.Statistics · Grade 7, Statistics, Algebra 1NumberA 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 2Standard DeviationStandard deviation is a number that measures how spread out data values are from their mean: roughly, the typical distance of a value from the mean. A small standard deviation means the values cluster near the mean; a large one means they are spread out. It is the square root of the variance.Statistics · Algebra 1, StatisticsMeanThe 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 7MedianThe median is the middle value when data are listed in order. If there is an even number of values, the median is the average of the two middle values. Half the data lie at or below the median and half at or above it, and outliers barely move it.Statistics · Grade 5, Grade 6, Grade 7ModeThe mode is the value that appears most often in a data set. To find it, count how many times each value occurs and pick the most frequent one. A data set can have one mode, more than one mode, or no mode at all when every value appears only once. The mode works for categories, too.Statistics · Grade 5, Grade 6, Grade 7RangeThe range of a data set is the difference between the greatest and least values: maximum minus minimum. For 2, 5, 9, the range is 9 − 2 = 7. Range is a simple measure of spread: it tells how far apart the extreme values are, though one outlier can make it misleading about the rest of the data.Statistics · Grade 5, Grade 6, Grade 7Interquartile RangeThe interquartile range (IQR) is the spread of the middle half of a data set: the third quartile minus the first quartile, IQR = Q3 − Q1. Because it ignores the lowest and highest quarters of the data, outliers barely change it. In a box plot it is the length of the box.Statistics · Grade 6, Grade 7, Algebra 1HeightHeight is how tall something is: the distance from its bottom to its top, measured straight up and down. In geometry, the height of a shape is the perpendicular distance from a base to the opposite vertex or side, so it always meets the base at a right angle. Area and volume formulas use it.Measurement · Grade 2, Grade 5, Grade 6Scatter PlotA scatter plot displays paired numerical data as points on a coordinate plane, one point for each pair. It helps show whether two quantities have a pattern or relationship: points rising from left to right show a positive association, points falling show a negative association, and a shapeless cloud shows no association.Statistics · Grade 8, Algebra 1CorrelationCorrelation is a measure of how strongly two variables move together in a straight-line pattern. In a positive correlation both rise together; in a negative correlation one rises as the other falls. The correlation coefficient r runs from −1 to 1: values near ± 1 are strong and values near 0 are weak. Correlation does not prove causation.Statistics · Grade 8, Algebra 1, StatisticsPrincipalThe principal is the original amount of money that is borrowed, deposited or invested, before any interest is added. In the simple interest formula I = Prt, the letter P stands for the principal. Deposit 2,000 dollars at 3% simple interest for 4 years: the principal is 2,000 dollars, the interest is 240 dollars, and the total is 2,240 dollars.Problem Solving · Grade 7, Algebra 1ProbabilityProbability measures how likely an event is, from 0 for impossible to 1 for certain. For equally likely outcomes it is the number of favorable outcomes divided by the total number of outcomes: for a fair coin, the probability of heads is 12. Probabilities can be written as fractions, decimals or percents.Probability · Grade 6, Grade 7, Grade 8AverageAn average is one number that stands for the typical size of the values in a data set. In everyday math, average means the mean: add all the values, then divide by how many values there are. In statistics, the median and the mode are also kinds of average.Statistics · Grade 5, Grade 6, Grade 7Bar GraphA bar graph uses bars to compare amounts in different categories. The taller or longer the bar, the larger the value it represents. Each bar stands for one category, such as a favorite sport, and its length is read against a number scale. Bars can be vertical or horizontal, with gaps between them.Statistics · Grade 3, Grade 4, Grade 5Relative 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, StatisticsHistogramA histogram uses bars to show how numerical data are grouped into intervals. The data are split into equal-width intervals, called bins, and each bar’s height is the count of values in that interval. The bars touch, because the intervals are continuous, which is what distinguishes a histogram from a bar graph.Statistics · Grade 6, Grade 7, Grade 8Box PlotA box plot displays a data set through five numbers: minimum, first quartile, median, third quartile, and maximum. The box holds the middle half of the data, a line inside it marks the median, and whiskers reach out to the smallest and largest values. It makes spread easy to compare between sets.Statistics · Grade 6, Grade 7, Grade 8OutlierAn outlier is a data value that is far away from most of the other values in a set. In 4, 5, 6, 7, 50, the value 50 is an outlier. A common test flags any value more than 1.5 times the interquartile range below the first quartile or above the third. Outliers pull the mean strongly.Statistics · Grade 6, Grade 7, Grade 8