How long do people wait to collect reserved books at a library desk? One visitor remembers a quick pickup and another a long wait, and both may be reporting honestly. A fair answer needs a precise question, a clear rule for whose waits count and a way to summarize many waits at once. This lesson builds that vocabulary on a small constructed example and separates what data describe from what they may or may not say about a larger group.
Key terms
Statistical question
Definition A statistical question asks about a defined collection of units and is answered with data that are expected to vary from unit to unit. Variability means that observations of the same characteristic differ, across units or across repeated measurements. It is a feature of the situation being studied and is not by itself a sign of a mistake. A question about one particular unit, answered by a single recorded value, is not a statistical question in this sense.
In plain words A statistical question expects different answers from different units and asks what the whole collection of answers shows, such as a typical value or how often something happens.
Example “How long was the wait at the third visit on Monday?” asks for one recorded value. “What was the mean waiting time across all collection-desk visits at this branch during one stated week?” and “What proportion of those visits involved a wait of more than 6 minutes?” are statistical questions, because waits differ from visit to visit. A 2-minute wait and a 9-minute wait can both be recorded correctly. The difference between them is variability, not an error.
Population and observational unit
Definition The observational unit is the individual thing, person or event about which data are recorded. A data table usually gives each unit its own row. A variable is a characteristic recorded for each unit, such as a waiting time. The population is the complete set of units that the question is about. It has to be defined precisely enough to decide for any unit whether it belongs, which usually means stating what counts as a unit, where and when. When it is finite, its size is written .
In plain words Before counting or averaging anything, say what one observation is and which observations the question covers. A population does not have to consist of people, and it does not have to be large.
Example In a constructed example, the population is all 12 completed collection-desk visits at one fictional library branch during one stated Monday-to-Saturday week, so . The unit is one visit, not one visitor. A person who collects books on Monday and again on Thursday contributes two visits, and counting distinct visitors would answer a different question. The variable of interest is the waiting time in minutes, from joining the collection queue until a staff member begins helping. Other branches and other weeks lie outside this population.
Sample
Definition A sample is a subset of the population on which data are actually collected. Its size is written , and when the population is finite, . A census collects data on every unit of the population, so the observed group is the whole population and . Calling a group a sample says only that it is the observed part of the population. Whether it can support conclusions about the rest depends on how it was chosen, which later lessons on sampling bias and random sampling take up.
In plain words The population is the group you want to know about, and the sample is the part you actually looked at. A sample is not automatically fair or typical, and a larger sample chosen the same way can repeat the same flaw.
Example Suppose the waits are recorded for six of the twelve visits, labeled V01 to V06. This is sample A, with and waits of 2, 3, 4, 5, 7 and 9 minutes. These constructed values are exact. Sample A was picked to illustrate the ideas, and nothing here claims that it was chosen at random or that it resembles the other six visits. Recording the waits of all twelve visits would be a census of this population.
Parameter and statistic
Definition A parameter is a number that describes the population, such as the population mean or the population proportion of units with some property. A statistic is a number computed from sample data, such as the sample mean or the sample proportion . For a sample with values , of which have the property,
The population mean and the population proportion are computed in the same way from all values of the population. For a fixed population a parameter is one fixed number, which is usually unknown. A statistic is known once the data are in, but its value depends on which units happen to be in the sample. When a statistic is used as a value for an unknown parameter, it is called an estimate of that parameter.
In plain words Parameters belong to the population and statistics belong to the sample. The bar and the hat mark the sample versions. An estimate does not become the parameter just because it is the best number available.
Example The waits in sample A add to 30 minutes, and two of them, 7 and 9, are strictly greater than 6, so
A wait of exactly 6 minutes would not count, because the condition is more than 6 minutes. The mean and the proportion summarize different features of the same six waits, and the mean does not say that every visit waited 5 minutes. Both are statistics. The parameters and are the mean wait and the proportion of waits over 6 minutes across all 12 visits, and at this stage they are unknown. Using 5 minutes as an estimate of does not make equal to 5 minutes.
Sampling variability
Definition Sampling variability is the fact that a statistic can take different values in different samples from the same population. For one particular sample whose values are recorded without error, the difference between a statistic and the parameter it estimates, such as , is its sampling error. Sampling error can be nonzero even when the sample is chosen by a sound random procedure. It is different from measurement error, which is a difference between a recorded value and the true value of the quantity being measured.
In plain words Different samples can give different answers even when nothing went wrong. Unless the sample is the whole population, the data alone cannot determine exactly how far a statistic is from its parameter, so inference needs a way to describe how much statistics vary from sample to sample.
Example The other six visits, V07 to V12, had waits of 1, 2, 3, 5, 6 and 7 minutes. Call them sample B. Their waits add to 24 minutes, so minutes, and only the 7-minute wait is over 6 minutes, so . Together A and B form a census of the twelve visits, which reveals the parameters
The sampling errors of the two sample means are and minutes. Because A and B are complementary halves of the population, their means average to exactly , which is why they land on opposite sides of it. Two samples chosen separately need not do that, and A and B are not two independent random samples. Over all 924 possible sets of six visits from this population, the sample mean ranges from 2.5 to 6.5 minutes.
Descriptive and inferential statistics
Definition Descriptive statistics organizes and summarizes the data at hand with tables, graphs and numbers such as means and proportions. It can describe a sample or a whole population. Inferential statistics uses sample data to draw conclusions about a larger population, a process or a cause, together with a statement of how uncertain those conclusions are. That step relies on how the data were produced, for example by random sampling from the population or random assignment of treatments, and on the assumptions of the method. A calculation alone does not justify it.
In plain words Description reports what was observed. Inference reaches beyond it and has to account for sampling error, measurement error and the way units were selected. A study that recorded only quiet hours, for example, can give a misleading picture of a whole week, however carefully its numbers are computed.
Example Reporting minutes for the six visits of sample A is descriptive, and so is reporting minutes after the census. Claiming from sample A alone that the mean wait for the week is close to 5 minutes would be inference, and nothing here supports it, because A was not selected by a random procedure and its selection gives no basis for judging how far off it might be. The census describes the recorded waits of this branch in this week exactly, but it does not give next week’s mean. Another week, another branch or a claim about causes needs further data and a suitable design.
Check yourself
- A school wants the mean duration of all completed trips to school made by its enrolled pupils during one stated school week. A trip starts when a pupil leaves home and ends when the pupil reaches the school entrance. The school times five of these trips and records 8, 12, 15, 10 and 20 minutes. Name the population, the observational unit, the variable, the sample, the parameter and the statistic, and compute the statistic.
Show the answer
The population is all completed trips to school by the enrolled pupils during that week. The unit is one trip, and the variable is its duration in minutes, timed from leaving home to reaching the entrance. The sample is the five timed trips, so . The parameter is the mean duration of all trips in the population, which is unknown. The five durations add to 65 minutes, so the statistic is the sample mean
It can serve as an estimate of , but how good an estimate it is depends on how the five trips were chosen and how they were timed.
- One pupil makes five of the trips in that population during the week. Does this pupil contribute one unit or five? A classmate says that the five trips are five unrelated pieces of information. What is the problem with that claim?
Show the answer
The pupil contributes five units, because the unit is a trip, not a pupil. For the same reason, the number of rows in the data need not equal the number of pupils. The five trips may share a route, a means of travel and daily habits, so their durations can be related to each other. They are five observations, but they need not carry as much information as five trips by five different pupils, and an analysis that reaches beyond the data has to take that into account.
- In the library example, someone looks at visits V01, V03, V05, V07, V09 and V11, whose waits are 2, 4, 7, 1, 3 and 6 minutes. Find the sample mean and the sample proportion of waits over 6 minutes, and compare them with the parameters minutes and . Does a difference mean that a calculation went wrong? Does the census value of 4.5 minutes give the mean wait for next week?
Show the answer
The six waits add to 23 minutes, so the sample mean is minutes. Only the 7-minute wait is strictly greater than 6, and the 6-minute wait of V11 does not count, so the sample proportion is . Both differ from the parameters, and neither difference signals a mistake. A different set of six visits can give different statistics, which is sampling variability. The census value of 4.5 minutes describes the twelve recorded visits of that one week. Next week’s visits form a different population, and their mean can be different.