Populations, samples and census
You want a fact about a whole group, but you can only look at part of it. When does the part tell you the truth about the whole?
Where you are 6 in Sampling and study design
Why this exists
Some questions are about a group far too large to look at one member at a time. What percent of the fish in a lake carry a parasite? How long do the light bulbs in a shipment last? What do the adults of a city think of a plan for a new park?
To answer exactly, you would have to check every fish, burn out every bulb and ask every adult. Measuring every member of a group like this is called a census, and often it cannot be done. It costs too much, it takes too long, the group changes while you are counting, or measuring destroys the thing measured: a bulb tested until it burns out can no longer be sold.
So you measure part of the group and use what you find to say something about the whole. That move is the starting point of statistics, and it raises the question this lesson answers: when can the part stand in for the whole, and when does it mislead? The answer depends first on how the part was chosen, and only after that on how big it is.
Definition
A statistical question is always about some group. That group is the population.
The population is the entire set of individuals that a statistical question is about. Each member of it is an individual. Individuals are often people, but they can just as well be fish, light bulbs, days or coin tosses (see individuals and variables). The number of individuals in the population is the population size, written N.
A sample is a set of individuals taken from the population, the ones whose data are actually collected. The number of individuals in the sample is the sample size, written n. Since a sample is part of the population, n \le N. When data are collected from every individual in the population, the study is a census rather than a sample (see Census and sample, at the end of this section).
The population is fixed by the question, not by who happens to be easy to reach. “How many hours do the seventh graders at this school sleep on a school night?” makes the population every seventh grader at the school, even if only 30 of them are asked. Those 30 are the sample.
Two kinds of number go with the two groups.
A parameter is a number that describes the population. It is computed, or could be computed, from the values of every individual in the population: the percent of all the lake’s fish that carry the parasite, or the mean lifetime of all the bulbs in the shipment.
A statistic is a number computed from the data in a sample: the percent of the fish caught that carry the parasite, or the mean lifetime of the bulbs that were tested.
The first letters line up: parameter with population, statistic with sample. The parameter is usually the number you want and cannot get. The statistic is the number you can get, and you use it to estimate the parameter. Parameters and statistics takes the pair further.
Finally, a sample of size n is a simple random sample when a chance process picks it so that every possible group of n individuals from the population is equally likely to be the sample. No individual can be picked twice (the sample is drawn without replacement), and every individual has the same chance of being chosen. Drawing n names from a well-mixed hat holding one slip per individual does exactly this. In this lesson “random sample” always means a simple random sample; other chance methods are the subject of random sampling.
Census and sample
A census collects data from every individual in the population, so the group measured is the whole population and n = N. A sample survey, or simply a sample, collects data from only some of the individuals, so n < N.
The two give different kinds of number. A number worked out from a census uses every individual, so it is the parameter itself, not an estimate of it, provided every individual really is reached and every value is recorded correctly. A number worked out from a sample is a statistic: it estimates the parameter and can miss it.
That makes a census sound always better. Often it is impossible, or a poor use of effort, for one of these reasons.
- Cost and time. Asking every one of a city’s 42{,}000 adults costs far more than asking 600 of them, and the answer arrives much later.
- Change. A large population changes while it is being counted. Fish are born and die during a count of a lake, and by the time the last one is counted the first count is out of date.
- Destruction. When measuring uses up the thing measured, a census leaves nothing behind. A bulb tested until it burns out, a match struck to see whether it lights, a cake cut open to check the middle.
- No list, or no end. Some populations cannot be listed at all. Nobody knows how many fish are in an ocean, and the tosses a coin could ever make have no last one.
- Care. A sample is small enough to be done well: trained people, measurements checked twice, every person who does not reply followed up. A census spreads the same effort over everyone, and a census that misses many people can land further from the truth than a careful sample does (see A census that misses people, under Intuition).
A census is the natural choice in other cases.
- The population is small and easy to reach. For the 28 students in one class, measuring all 28 costs hardly more than measuring 10, and gives the parameter exactly.
- The data already exist for everyone. A club’s membership records, a library’s loan records or a school’s register hold a value for every individual, so a computer can run a census of the records at almost no cost. The mean age of 41.5 years in Example 2 comes from such a census.
- The question is about each individual, not a summary. An election counts every ballot cast, because who wins depends on the exact total, not an estimate of it.
- A law requires it.
National governments run the best-known censuses. The Constitution of the United States (Article I, Section 2) orders an “actual Enumeration” of the people every ten years, and the counts decide how many seats each state has in the House of Representatives. The same government gathers its detailed questions about income, work and housing by a sample instead: the American Community Survey goes to about 3.5 million addresses each year, a small part of the country’s homes. One government runs both, a census where the exact count is the point and a sample where an estimate is enough. The sources for these facts are listed with this lesson’s references.
Intuition
Figure 1 shows a made-up school of 1{,}000 students, one dot per student. 400 of them play a musical instrument, so the parameter, the percent of all the students who play, is 400/1000 = 40\%. 150 of the students eat lunch in the music wing, and 120 of those 150 play.
In the top grid, 50 students were chosen at random from the whole school. 18 of them play, so the statistic is 18/50 = 36\%: not 40\%, but near it. In the bottom grid, 50 students were chosen at random from the music wing only, which is where someone with a clipboard might go at lunchtime. 41 of them play, so the statistic is 41/50 = 82\%, more than twice the true figure.
Both samples hold 50 students. What separates them is where they were drawn from. The first had the whole school to draw from, so it holds a mix of students much like the school’s mix. The second could only hold wing students, and the wing is not like the school.
One sample on its own proves little, since any single sample can be lucky or unlucky. Figure 2 repeats each kind of sample 40 times and plots the percent from each one.
Three things show.
- Random samples from the whole school land on both sides of 40\%. They scatter around it rather than leaning to one side.
- Larger random samples land closer. Samples of 10 spread across much of the scale; samples of 200 crowd near the line.
- Samples from the music wing crowd around 80\%, the wing’s own percent. They are bunched tightly, but around the wrong number.
The table in the next section puts counts on this, from 1{,}000 samples of each kind.
A census that misses people
Now try a census of the same school. The music teacher hands a form to every one of the 1{,}000 students: “Do you play a musical instrument?” Students who play care more about the question, so more of them hand the form back. In Figure 3, 300 of the 400 players reply and 200 of the 600 others do.
Every student was asked and half of them replied, 500 forms in all. Of those, 300 say they play, so the replies give 300/500 = 60\%. That is 20 points above the true 40\%, far worse than the random sample of 50 in Figure 1, which gave 36\%.
The replies are a sample after all, one the students chose for themselves, and the players chose to be in it more often. Asking everyone does not make a census. Getting an answer from everyone does.
Results
Simple random samples are right on average. Take a population of N individuals, Y of whom have some trait, and a fixed sample size n. Suppose the sample is a simple random sample, so every possible group of n individuals is equally likely to be the one chosen. Then the average of the sample percents, taken over all the possible samples, is exactly the population percent:
The same holds for a mean: if each individual has a numerical value, the average of the sample means over all the possible samples is exactly the population mean. The result is about the average; it says nothing about how close any one sample lands.
A random sample from part of the population is right on average for that part. If the sample is a simple random sample of a group inside the population, the result above applies with that group as the population. The sample percents then average the group’s percent, whatever the rest of the population is like; they match the whole population’s percent only if the group happens to share it. Taking a larger sample from the same group does not move that average.
What larger random samples do. How close a single sample tends to land depends on its size. The table comes from a computer simulation of the made-up school in Figure 1. Each row is 1{,}000 simple random samples of size n, each drawn afresh either from the whole school or from the music wing only. “Within 10 points” counts the samples, out of 1{,}000, whose percent was between 30\% and 50\% inclusive, that is, within 10 percentage points of the school’s 40\%. “Range” gives the smallest and largest sample percents among the 1{,}000.
| From | n | Within 10 points | Range |
|---|---|---|---|
| school | 10 | 668 | 0 to 90 |
| school | 50 | 896 | 20 to 60 |
| school | 200 | 998 | 28.5 to 50.5 |
| wing | 50 | 0 | 66 to 92 |
| wing | 100 | 0 | 72 to 87 |
Twenty separate runs of the same simulation, the table being one of them, gave counts between 635 and 685 for n = 10, between 871 and 912 for n = 50, at least 997 for n = 200, and 0 every time for the music wing. The exact counts move from run to run; the pattern does not.
A census gives the parameter; a census with gaps gives only bounds. If a census measures every one of the N individuals correctly, the percent it finds is the population percent, with nothing left to estimate. If instead only r of the N individuals reply, and k of those r have the trait, then with nothing known about the N - r who did not reply, the population percent can be anything between these two ends:
and both ends are possible. The gap between the two ends is (N - r)/N, the fraction who did not reply, so it closes only as the replies approach everyone. For the census in Figure 3, k = 300, r = 500 and N = 1{,}000, so all the replies can promise is a figure from 30\% to 80\%.
Why it is true
The second result is the first one applied to a smaller group, so only the first and the last need an argument. The first is a count, done two ways over the same list.
Write down every possible sample of size n, and say there are S of them. Every individual appears in the same number of these samples; call that number T. The reason is symmetry: swap the names of any two individuals, and every sample containing the first turns into a sample containing the second, and the other way round. So the two appear in equally many samples.
Now count the places in the list, one place for each individual in each sample, in two ways. Each of the S samples has n places, so there are nS places. Each of the N individuals fills T of them, so there are NT places. Therefore
Next count only the places filled by individuals with the trait. Each of the Y such individuals fills T places, so the trait turns up YT times across the whole list. The average number with the trait per sample is
using NT = nS. Each sample’s fraction is its count divided by the same n, so the average fraction is the average count divided by n, which is Y/N. For a mean, count each individual’s value T times instead of counting the trait, and the same steps give the population mean.
The hypothesis enters in one place. “Every possible group is equally likely” is what makes the plain average over the list the right average for the sample you will actually get. If some groups were more likely than others, the list would need weights, and the symmetry would be lost.
Example 6 carries out the whole argument on a population small enough to list.
The bounds for a census with gaps need no symmetry, only counting. The individuals with the trait are the k repliers who have it together with some number j of the N - r who did not reply, where 0 \le j \le N - r. So the number with the trait is k + j, which lies between k and k + (N - r), and dividing by N gives the bounds. Both ends can happen: j = 0 if none of the silent individuals has the trait, and j = N - r if all of them do. The replies cannot rule out either, because they carry no information about anyone who did not reply.
Worked examples
Naming the population and the sample
A made-up juice plant fills 6{,}000 bottles on Monday. An inspector takes 40 bottles from the line at random times during the day and measures the juice in each. The question is how much juice the bottles filled on Monday hold on average. Name the population, an individual, the sample, N and n.
SolutionThe question is about all the bottles filled on Monday, so the population is those 6{,}000 bottles and N = 6{,}000. Each bottle is an individual.
The sample is the 40 bottles the inspector measured, so n = 40.
The population is not “every bottle the plant has ever filled”, because the question names Monday.
Parameter or statistic
A made-up hiking club has 2{,}400 members. Its records list every member’s age, and the mean age of all members is 41.5 years. The club also surveys 120 members chosen at random: 78 of them say they hike at least once a month, and the mean age of the 120 is 39.8 years. Classify each number as a parameter or a statistic: (a) 41.5 years, (b) 39.8 years, (c) the percent of the 120 who hike monthly. Find (c).
Solution(a) 41.5 years describes all 2{,}400 members and comes from the records of every one of them, so it is a parameter.
(b) 39.8 years is computed from the 120 surveyed members only, so it is a statistic. It estimates the parameter in (a) and misses it by 1.7 years, which is how a statistic usually behaves: close, not equal.
(c) \displaystyle \frac{78}{120} = 0.65 = 65\%. It comes from the sample, so it is a statistic. The matching parameter, the percent of all 2{,}400 members who hike monthly, is unknown, because nobody asked the other 2{,}280.
Census or sample
For each made-up study, say whether a census or a sample is the better plan, and why.
(a) A swimming coach wants the mean 50-meter time of the 18 swimmers on her team.
(b) A match factory wants the percent of the 2 million matches it made this month that light on the first strike.
(c) A library wants the percent of its 40{,}000 books that are overdue, and its computer holds the due date of every book on loan.
(d) A national park wants the mean weight of the trout in its rivers.
Solution(a) A census. The population is 18 swimmers, all at the same practice, so timing every one costs little more than timing a few, and the mean of all 18 times is the parameter itself.
(b) A sample. A match struck to see whether it lights is used up, so a census would leave the factory with nothing to sell; and 2 million is far too many to strike in any case.
(c) A census, of the records. The due date of every book is already stored, so the computer can check all 40{,}000 at almost no cost and give the exact percent. Taking a sample would save nothing.
(d) A sample. No one can catch every trout in a river system, no one knows how many there are, and fish are born and die while the count goes on. A sample of caught fish is the only plan available.
Estimating a count from a sample
A made-up orchard has 3{,}200 apple trees. A random sample of 80 trees is inspected, and 12 of them have leaf spot. Estimate how many trees in the orchard have leaf spot.
SolutionThe statistic is the percent of the sampled trees with leaf spot:
The sample was chosen at random from the whole orchard, so it is reasonable to use this statistic as an estimate of the parameter, the percent of all 3{,}200 trees with leaf spot. If about 15\% of the orchard has it, the number of trees with it is about
So about 480 trees have leaf spot. The word “about” matters: a different random sample of 80 would very likely give a different percent, and so a different estimate.
Which plan can stand in for the town
A made-up town has 9{,}000 adults, and its library wants to know what percent of them hold a library card. Two plans are put forward.
Plan A: on one Saturday, ask the first 300 adults who walk into the library.
Plan B: take the town’s full list of adult residents and pick 300 of them at random, so that every group of 300 on the list is equally likely, then contact each one.
Which plan gives a sample that can stand in for the town’s adults?
SolutionThe population is the 9{,}000 adults of the town, so a sample that can stand in for it must be able to include any of them.
Plan A can only include adults who come to the library, and adults who come to a library are far more likely than the town as a whole to hold a card. Its sample comes from a group inside the population that differs from the rest on exactly the thing being measured, like the music wing in Figure 1. Even if its 300 were a simple random sample of library visitors, the second result would make its percent right on average for library visitors, not for the town; and the first 300 through the door on one Saturday are not a random sample of anyone. Asking 3{,}000 visitors instead of 300 would not change that.
Plan B is a simple random sample of the whole population, so by the first result its percent is right on average for the town, provided the people picked actually answer (Example 8 shows what goes wrong when many do not).
Plan B can stand in for the town’s adults. Plan A cannot.
Every possible sample from a tiny population
A made-up shelf holds 5 plants, A, B, C, D and E, and plants A and B are flowering. A sample of 2 plants is chosen so that every pair is equally likely. List every possible sample, find the percent flowering in each, and compare their average with the population percent.
SolutionThe parameter is the percent of the shelf that is flowering, \displaystyle \frac{2}{5} = 40\%.
There are 10 possible pairs. Sorted by how many flowering plants they hold:
| Samples | Flowering plants | Sample percent |
|---|---|---|
| AB | 2 | 100\% |
| AC, AD, AE, BC, BD, BE | 1 | 50\% |
| CD, CE, DE | 0 | 0\% |
The average of the 10 sample percents is
exactly the parameter. Yet not one of the ten samples shows 40\%: a sample of 2 can only show 0\%, 50\% or 100\%. Being right on average does not make any single sample right.
The count from the proof shows here too. Each plant appears in T = 4 of the S = 10 samples (plant A is in AB, AC, AD and AE), and nS = 2 \times 10 = 20 = 5 \times 4 = NT.
How much to trust a small sample
Two random samples are drawn from the made-up school of Figure 1, where the true figure is 40\%. Sample P has 10 students, 6 of whom play, which is 60\%. Sample Q has 200 students, 78 of whom play, which is 39\%. Does P’s result show that it was badly chosen? If the music teacher did not know the true 40\%, which statistic should she trust more?
SolutionA result of 60\% from a random sample of 10 is not unusual. In the simulation behind the table, 180 of the 1{,}000 random samples of 10 came out at 60\% or more. So P’s result is something random samples of 10 from this school do fairly often, and it is no evidence that P was badly chosen.
A random sample of 200 tends to land much closer: in the table, 998 of 1{,}000 such samples were within 10 points of 40\%. So without knowing the true figure, the teacher should trust Q’s 39\% far more than P’s 60\%. Both were chosen at random, and Q is much larger.
Returned forms against a random sample
A made-up town of 4{,}000 households is deciding whether to build a skate park. It posts a form to every household; 1{,}600 households send it back, and 1{,}200 of those are in favor. Separately, the town picks 200 households at random from its list of all 4{,}000 and keeps visiting each one until it answers; 104 of the 200 are in favor.
(a) What percent of the returned forms are in favor?
(b) From the forms alone, what is the smallest and the largest the percent of all 4{,}000 households in favor could be?
(c) Which figure should the town trust, the forms or the visited sample?
Solution(a) \displaystyle \frac{1200}{1600} = 0.75 = 75\%.
(b) Here N = 4{,}000, r = 1{,}600 and k = 1{,}200, so N - r = 2{,}400 households did not reply. If none of them is in favor, the number in favor is 1{,}200; if all of them are, it is 1{,}200 + 2{,}400 = 3{,}600. By the result on a census with gaps the percent of all the households in favor lies between \displaystyle \frac{1200}{4000} = 30\% and \displaystyle \frac{3600}{4000} = 90\%, and the forms cannot narrow it further.
(c) The visited sample, which gives \displaystyle \frac{104}{200} = 52\%. Every household was asked, but the 1{,}600 that replied chose themselves, and households keen on a skate park have a reason to send the form back that others lack, so the replies probably lean toward “in favor”. The 200 were picked by chance from the whole town and every one of them answered, so by the first result their percent is right on average for the town. 1{,}600 replies sounds like far more evidence than 200, but size does not repair how a group was chosen. A better use of the forms would be to visit a random sample of the 2{,}400 silent households, so that the people who did not reply are represented too.
Counting fish you cannot see
A made-up pond holds an unknown number of fish, N. A ranger catches 60 fish, marks each with a harmless tag, and puts them back. A week later, after the fish have mixed, she catches 75 fish and finds that 9 of them carry tags. Estimate N, and say what has to be true for the estimate to be sound.
SolutionIf no tagged fish has died or lost its tag, the pond still holds exactly 60 tagged fish, so the parameter “fraction of the pond’s fish that are tagged” is \displaystyle \frac{60}{N}. It is unknown only because N is.
The second catch is a sample, and its statistic is
If the second catch can stand in for the pond, the statistic estimates the parameter, so set them equal and solve for N:
Cross-multiplying is allowed because neither N nor 75 is zero.
The pond holds about 500 fish. This is the mark-and-recapture estimate: with M marked, C caught the second time and R of those marked, \displaystyle N \approx \frac{MC}{R}.
It is an estimate, and it rests on the condition this whole lesson is about: the second catch must be able to stand in for the pond. That needs the pond to be closed (no fish born, dying, arriving or leaving between the two catches), no tag to fall off or go unseen, the tagged fish to have mixed through the pond, and every fish to be equally likely to be caught (a net that reaches only the shallows samples only the fish in the shallows). If any of these fails, 500 can be far off.
Checking a census with a sample
A made-up city’s census counts 57{,}000 people. To check it, the city picks some of its blocks at random, and in those blocks only, a second team makes its own careful count, working independently of the census. In the chosen blocks the census counted 950 people. The second team found 960, and on matching names and addresses, 912 of the team’s 960 had also been counted by the census. Estimate how many people live in the chosen blocks, the percent of them the census missed, and the population of the city.
SolutionThis is the fish count of Example 9 with people. The 950 people the census counted play the part of the tagged fish. The second team’s 960 are the second catch, and 912 of them are “tagged”, that is, on the census list. If the team was just as likely to find a person whether or not the census had counted them, the fraction of the team’s people on the list estimates the fraction of all the blocks’ people on the list. With N people living in the blocks:
About 1{,}000 people live in the chosen blocks. The census counted 950 of them, so it missed about 50, which is \displaystyle \frac{50}{1000} = 5\%; it counted 95\%.
The blocks were picked at random, so it is reasonable to take them as a guide to the whole city: if the census counted about 95\% of everyone, its 57{,}000 is 0.95 times the population, so
This is a simplified version of the dual-system estimation the U.S. Census Bureau used in its Post-Enumeration Survey of the 2020 census (see Boundaries). The real method must also remove people the census counted twice or in the wrong place, which this example assumes away. It rests on the same conditions as the fish: the two counts independent, and everyone equally likely to be found by the second team.
Boundaries
A sample speaks for the group it was drawn from, and no larger. A random sample of the seventh graders at one school is a sample of that school’s seventh graders. However well it was chosen, it is not a sample of all the seventh graders in a country.
Right on average is not right. The first result is about the average over all possible samples. Any one random sample can miss, and a small one can miss by a lot: among the random samples of 10 behind the table, 4 of the 1{,}000 had no players at all. Random choice makes a sample fair; it does not make it exact.
A larger sample does not repair how a sample was chosen. Size narrows how far a random sample tends to stray. It does nothing for a sample drawn from the wrong group: none of the 1{,}000 samples of 100 from the music wing came within 10 points of 40\%, and the lowest was 72\%. The ways a sample can lean away from its population are the subject of bias.
A census is exact only if it reaches everyone and records each one correctly. A real census of a large population misses some people and counts others twice, so its total is itself only close. After the 2020 census of the United States, the Census Bureau checked it with a sample, the Post-Enumeration Survey, which asked a sample of people where they had been living on census day and matched them against the census. For the nation as a whole the net error was a fraction of a percent, too small to tell apart from zero; but the survey estimated that the census had undercounted the Hispanic or Latino population by 4.99\% and the Black or African American (alone or in combination) population by 3.30\%, and overcounted the non-Hispanic White (alone) population by 1.64\% and the Asian (alone or in combination) population by 2.62\%. A census can be close for the whole and still lean for the parts.
The population has to be named first. A statistic with no stated population answers no question. The same 30 students are a whole population for a question about their class and a sample for a question about their school.
A parameter belongs to a moment. Populations change: students join and leave, fish are born and die. A sample taken at another time may describe a different population.
Random choice settles who is asked, not what they say. A slanted question or careless answers spoil the data of a perfectly random sample, and of a census just as much: see wording a survey question.
The results assume a simple random sample of a fixed size n. Other chance methods exist, and each needs its own check before these results can be used with it.
Where this goes next
The next lessons sharpen each piece of this one. Parameters and statistics gives the pair their own symbols and treats a statistic as an estimate of a parameter. Random sampling turns “chosen at random” into methods you can carry out, and Bias names the ways a sample can lean away from its population, the music wing being one of them.
Figure 2 is the start of a larger idea. How a statistic spreads across repeated samples can itself be studied and predicted, which is sampling variability, and that is what lets statistics put an honest margin on an estimate, the aim of statistical inference.
Exercises
Naming the parts
- A made-up bakery bakes 1{,}800 rolls on Saturday. The baker weighs 25 rolls chosen at random to learn the mean weight of Saturday’s rolls. Name the population, the sample, N and n.
- A made-up city council wants to know what percent of the city’s 42{,}000 adults support a new park. It phones 600 adults chosen at random from a list of all the city’s adults. Name the population, the sample, N and n.
- A made-up shipment holds 500 light bulbs. To learn how long the bulbs last, a tester leaves 12 of them switched on until each one burns out. Give a reason to test a sample rather than every bulb.
- The question is “How many hours do the seventh graders at a made-up school sleep on a school night?”, and 30 seventh graders are asked. Which is the population: (a) all the students at the school, (b) all the seventh graders at the school, or (c) the 30 students asked?
- Write one question for which the 28 students in one made-up class are the whole population, and one for which the same 28 students are only a sample.
Parameter or statistic
- A made-up hiking club surveys 150 of its 2{,}000 members, and 42\% of those surveyed say they hike every week. Is 42\% a parameter or a statistic?
- The same club’s records show that 1{,}240 of its 2{,}000 members joined more than five years ago. Find the percent, and say whether it is a parameter or a statistic.
- A teacher at a made-up school records the test score of every one of the 28 students in her class and finds a mean of 81. The question is about her class. Is 81 a parameter or a statistic?
- A random sample of 40 of the 900 books in a made-up library has 6 books with torn pages. Find the percent of the sample with torn pages, and say whether it is a parameter or a statistic.
Census or sample
- Say whether each made-up study is a census or a sample. (a) A school nurse measures all 412 students at a school to find the mean height of the school’s students. (b) A radio station wants its listeners’ favorite song and counts the texts of the listeners who choose to send one. (c) A company’s payroll computer adds up the pay of every one of its 85 workers to find their mean pay.
- Choose a census or a sample for each made-up study, and give a reason. (a) The percent of the 24 students in one class who have a pet. (b) The mean burning time of the candles a factory makes this week. (c) The mean length of the trout in a large river.
- A made-up building has 200 flats. A form about a new playground goes to every flat, 150 flats return it, and 90 of those want the playground. Find the percent of the returned forms in favor, and the smallest and largest possible percent of all 200 flats in favor.
- A made-up town checks its census in some blocks chosen at random. There the census counted 540 people, an independent second count found 570, and 513 of those 570 were also on the census list. Estimate the number of people living in those blocks, and the percent of them the census missed.
- A census of a made-up county counts 47{,}500 people. A check like the one in Example 10 estimates that the census counted 95\% of the people living there. Estimate the county’s population.
Using a sample
- A random sample of 60 of the 1{,}500 students at a made-up school finds that 21 walk to school. Find the sample percent, and estimate how many of the 1{,}500 walk.
- A made-up farm packs 7{,}200 eggs in a week. A random sample of 90 of that week’s eggs has 4 cracked eggs. Estimate how many of the week’s eggs are cracked.
- At a made-up lake, 40 frogs are caught, tagged and released. Later 50 frogs are caught, and 8 carry tags. Estimate the number of frogs, assuming the second catch can stand in for all the frogs.
- In a made-up forest, 120 deer are tagged. Later 90 deer are seen, and 15 carry tags. Estimate the number of deer, under the same assumption.
- A made-up population has 4 members, F, G, H and J, and only F has a certain trait. List every possible sample of 2 members, the percent with the trait in each, and the average of those percents.
- A made-up population has 5 members, 3 of whom have a trait, and there are 10 possible samples of 3 members. How many of those samples hold 3, 2, 1 and 0 members with the trait? Find the average sample percent and compare it with the population percent.
- For a population with N = 6 and samples of size n = 3, there are S = 20 possible samples. Use nS = NT to find how many of them contain any one given individual.
Can this sample stand in?
- A made-up town wants the percent of its working adults who ride the bus to work. A volunteer asks 200 adults waiting at the town’s main bus station at 8 in the morning. Can this sample stand in for the town’s working adults? Explain.
- A made-up gym wants the mean number of visits per week among its 3{,}000 members. It writes every member’s number on a slip, mixes the slips well and draws 50. Can this sample stand in for the members? Will its mean equal the parameter?
- For the made-up school of Figure 1, which gives a more trustworthy percent of all students who play: a random sample of 50 from the whole school, or a random sample of 100 from the music wing? Use the table in Results.
Reasoning
- Explain why a parameter is one fixed number while a statistic can change from one sample to the next.
- Use the second result to explain why random samples of 50 from the music wing give about 80\% on average, and why random samples of 100 from the wing do too.
- A random sample of 10 students from the made-up school of Figure 1 comes out at 70\% players. Does that show the sample was not chosen at random? Explain, using the simulation.
- In the argument under Why it is true, explain in your own words why every individual appears in the same number of possible samples.
- In a census with gaps, r of the N individuals reply and k of those have the trait, with 0 < r < N. Show that the percent of the replies with the trait, \displaystyle \frac{k}{r} \times 100\%, always lies between the two bounds in Results.
- A made-up magazine writes: “We asked all 50{,}000 of our readers, and 12{,}000 replied, so our figure is far more reliable than a poll of 1{,}000 people chosen at random.” Explain what is wrong with this argument, and say how wide the gap between the bounds in Results is for the magazine’s census.
Answers
- Population: the 1{,}800 rolls baked on Saturday, so N = 1{,}800. Sample: the 25 rolls weighed, so n = 25.
- Population: the city’s 42{,}000 adults, so N = 42{,}000. Sample: the 600 adults phoned, so n = 600.
- Testing destroys what is tested. A bulb left on until it burns out cannot be sold, so testing all 500 would leave no bulbs to sell. (It would also take as long as the longest-lasting bulb.)
- (b). The question names the seventh graders at the school, so they are the population. The 30 asked are the sample, and the other grades lie outside the question.
- For example, “What is the mean height of the students in this class?” makes the 28 the population, and “What is the mean height of the students in this school?” makes them a sample. Any pair of questions of this kind works.
- A statistic. It comes from the 150 members surveyed, not from all 2{,}000.
- 62\%, a parameter. \displaystyle \frac{1240}{2000} = 0.62, and it comes from the records of every member.
- A parameter. The population is her class, and the mean uses every student in it. For a question about the whole school, the same number would be a statistic.
- 15\%, a statistic. \displaystyle \frac{6}{40} = 0.15, computed from the sample of 40 books.
- (a) A census: every student is measured, so the mean is the parameter. (b) A sample: the population is all the listeners, and only those who choose to text are counted. (It is not a random sample either, so it may not stand in for the listeners.) (c) A census of the records: the pay of all 85 workers is used.
- (a) A census: 24 students are few and easy to ask, and asking all of them gives the exact percent. (b) A sample: a candle burned to time it is used up, so a census would leave nothing to sell. (c) A sample: no one can catch every trout, their number is unknown, and fish are born and die while a count goes on.
- 60\% of the returned forms, and between 45\% and 70\% of all the flats. \displaystyle \frac{90}{150} = 0.6. The 200 - 150 = 50 silent flats could hold from 0 to 50 more in favor, so the number in favor is from 90 to 140, and \displaystyle \frac{90}{200} = 45\%, \displaystyle \frac{140}{200} = 70\%.
- About 600 people, of whom the census missed about 10\%. \displaystyle \frac{540}{N} = \frac{513}{570}, so 513N = 540 \times 570 = 307{,}800 and N = 600. The census counted 540, so it missed about 60, and \displaystyle \frac{60}{600} = 10\%.
- About 50{,}000 people. 47{,}500 = 0.95 \times \text{population}, so the population is about \displaystyle \frac{47{,}500}{0.95} = 50{,}000.
- 35\%, and about 525 students. \displaystyle \frac{21}{60} = 0.35, and 0.35 \times 1{,}500 = 525.
- About 320 eggs. \displaystyle \frac{4}{90} \times 7{,}200 = \frac{28{,}800}{90} = 320.
- About 250 frogs. \displaystyle \frac{40}{N} = \frac{8}{50}, so 8N = 2{,}000 and N = 250.
- About 720 deer. \displaystyle \frac{120}{N} = \frac{15}{90}, so 15N = 10{,}800 and N = 720.
- Average 25\%. The six samples are FG, FH and FJ, each 50\%, and GH, GJ and HJ, each 0\%. The average is \displaystyle \frac{3(50) + 3(0)}{6} = 25\%, equal to the population percent \displaystyle \frac{1}{4} = 25\%.
- 1 sample holds all 3, 6 hold 2, 3 hold 1, and 0 hold none. Call the members with the trait A, B, C and the others D, E. Only ABC holds all three. Two of A, B, C (three ways) with one of D, E (two ways) makes 6. One of A, B, C with both D and E makes 3. No sample of 3 can avoid all three. The ten percents add to 100 + 6\left(66\tfrac{2}{3}\right) + 3\left(33\tfrac{1}{3}\right), which is 100 + 400 + 100 = 600, so the average is 600 \div 10 = 60\%, equal to the population percent 3 \div 5 = 60\%.
- T = 10. From nS = NT, 3 \times 20 = 6T, so T = 10.
- No. Nearly everyone waiting at a bus station in the morning is about to ride a bus, so the sample comes from a group inside the population that differs from the rest on the very thing being measured. Its percent is all but certain to overstate the town’s percent, and asking more people at the station would not fix that.
- Yes, it can stand in. Every member’s slip is in the draw and the slips are well mixed, so every group of 50 members is equally likely: a simple random sample. Its mean will very likely not equal the parameter exactly, though: the first result says it is right on average, not that any one sample is right.
- The random sample of 50 from the whole school. In the table, 896 of 1{,}000 such samples landed within 10 points of 40\%, while 0 of 1{,}000 random samples of 100 from the wing did, the lowest of them being 72\%. The wing sample is larger, but it is drawn from the wrong group.
- A parameter is computed from every individual in the population, and at a given moment there is one population, so there is one value. A statistic is computed from whichever individuals are in the sample, and a different sample holds different individuals, so it can give a different value.
- A random sample from the wing is a random sample of the 150 wing students, a population in which 120 of 150, or 80\%, play. The first result, applied to that population, says the sample percents average exactly 80\% for any sample size, so for samples of 50 and of 100 alike. The whole school’s 40\% plays no part, because students outside the wing can never be chosen.
- No. In the simulation, 68 of the 1{,}000 random samples of 10 from this school came out at 70\% or more. Such a result is uncommon, but random sampling does produce it. Whether a sample was random has to be known from how it was collected, not read off one percent.
- Swapping the names of two individuals turns the list of all possible samples into itself: each sample containing the first becomes one containing the second, and each containing the second becomes one containing the first. So those two appear in the same number of samples. Since any two individuals can be swapped this way, all of them appear equally often.
- The left bound: r \le N, so dividing k by the larger number N gives the smaller result, \displaystyle \frac{k}{N} \le \frac{k}{r}. The right bound: multiply both sides of \displaystyle \frac{k}{r} \le \frac{k + N - r}{N} by rN, which is positive. It becomes kN \le rk + r(N - r), that is, k(N - r) \le r(N - r). This holds because k \le r (only repliers are counted in k) and N - r > 0. Each step can be run backwards, so the original inequality holds too, and the reply percent always lies between the bounds.
- Size cannot repair how the group was chosen. The 12{,}000 readers who replied chose themselves, and readers who feel strongly about the question are more likely to reply, so their percent can lean far from the readers’ true percent, while the poll’s 1{,}000 were picked by chance and are right on average. The gap between the magazine’s bounds is the fraction who did not reply, \displaystyle \frac{50{,}000 - 12{,}000}{50{,}000} = \frac{38{,}000}{50{,}000} = 76\%: from the replies alone, the true percent is pinned only to a range 76 percentage points wide.