Parameters and statistics
You wanted a number about a whole group, but you measured only part of it. Is the number you worked out the number you wanted?
Where you are 6 in Sampling and study design
Why this exists
A farmer wants the mean height of the sunflowers in her field. There are thousands of them, so she measures 25 and works out the mean of those 25 heights. She now holds a number, but not the number she wanted: hers describes the 25 she measured, and the one she wanted describes every plant in the field. They are the same kind of number about the same plants, yet reporting one as the other claims more than she knows.
Every study that measures part of a group (see populations and samples) holds these two numbers. Statistics names them so they cannot be confused: the number about the whole group is a parameter, and the number from the part is a statistic.
Definition
A parameter is a number that describes a whole population. It is worked out, or could be worked out, from the values of every individual in the population. Usually nobody knows it, because nobody has measured everyone. At any one moment a population has one value of each parameter.
A statistic is a number worked out from the sample in hand. Once the sample is measured, the statistic is known. A different sample holds different individuals, so it usually gives a different value.
This lesson uses two kinds of number, and each comes as a pair.
| Kind | The parameter | The statistic |
|---|---|---|
| a Mean | population mean: the mean of the values of every individual | sample mean: the mean of the values in the sample |
| a proportion | population proportion: the share of all the individuals with some trait | sample proportion: the share of the sample with it |
A proportion is the fraction of a group with a trait, written as a fraction, a decimal or a percent: 57 out of 150 is \displaystyle \frac{57}{150} = 0.38 = 38\%.
To tell which is which, ask of each number: which group was it worked out from? Every individual in the population makes it a parameter; only the individuals measured makes it a statistic. The first letters help: parameter goes with population, statistic with sample.
A statistic is used to estimate the parameter of the same kind: a sample mean estimates the population mean, and a sample proportion the population proportion. An estimate is a value used in place of an unknown one because there is reason to expect it to be close.
Intuition
Figure 1 shows a made-up lake. In real life nobody sees every fish, but in a made-up lake we can: it holds 300 fish, and 90 are trout. So the population proportion of trout is
That is the parameter. It is one number, and it does not move.
Beside the lake are three random samples of 20 fish. Sample 1 holds 4 trout, a sample proportion of \displaystyle \frac{4}{20} = 20\%. Sample 2 holds 6, which is 30\%. Sample 3 holds 7, which is 35\%. Three samples, three statistics, one parameter.
Each sample honestly reports its own 20 fish; they differ because they hold different fish. Sample 2 happens to match the lake, but nobody holding only Sample 2 could know that. How far statistics stray, and in what pattern, is the subject of sampling variability.
Results
One parameter, many possible statistics. For a given population, variable and moment, a parameter has one value. A statistic has a value for each sample, and different samples usually give different values.
A statistic estimates the parameter of the same kind. A sample mean estimates the population mean of the same variable; a sample proportion estimates the population proportion of the same trait. To estimate a count in the population, multiply the sample proportion by the population size.
A random sample makes the estimate fair. For a simple random sample, the sample proportion and the sample mean are right on average over all the samples that could have been drawn (the first result of populations and samples). Right on average is not right every time.
A census gives the parameter. If every individual is measured correctly, the number worked out is the parameter itself, with nothing left to estimate.
Why it is true
The parameter is worked out from every individual; the statistic from the sampled individuals only. Everyone outside the sample helps decide the parameter and plays no part in the statistic. That one fact gives both halves of the lesson.
Why statistics vary. A statistic depends on which individuals were picked. Pick others and the values going in change, so the number coming out can change. The parameter depends on everyone, and everyone is the same group whichever sample was drawn.
Why a statistic is only an estimate. The values outside the sample were never seen, so as far as the sample can tell they could be anything (Example 6 puts numbers on this). What makes a statistic worth using anyway is how the sample was chosen: when chance picks it, the seen and unseen individuals have no reason to differ. That is a reason to expect the estimate to be close, not a proof that it is.
Worked examples
Naming the parameter and the statistic
A made-up town has 5{,}200 households. A survey asks 150 households chosen at random whether they have a dog, and 57 say yes. Name the population and the sample, give the statistic, and describe the parameter. Then estimate how many households in the town have a dog.
SolutionThe question is about every household in the town, so the population is the 5{,}200 households. The sample is the 150 asked.
The statistic is the sample proportion, found by dividing the part by the whole (see what percent one number is of another):
The parameter is the population proportion, the share of all 5{,}200 households with a dog. It is unknown, because 5{,}050 households were never asked.
The sample was random, so 38\% is a fair estimate, and the number of households with a dog is about
so roughly 2{,}000. A different random 150 would give a different figure.
Working out a sample mean
A made-up farmer measures 25 sunflowers chosen at random from her field, rounding each height to the nearest 10 cm. Find the sample mean, and say what it estimates.
| Height (cm) | 150 | 160 | 170 | 180 | 190 |
|---|---|---|---|---|---|
| Number of plants | 3 | 5 | 8 | 6 | 3 |
The table holds the whole sample: 3 + 5 + 8 + 6 + 3 = 25 plants. Each column adds its height once for every plant in it, so the total height is (see the mean from a frequency table)
and the sample mean is
This is a statistic: it comes from the 25 plants measured. It estimates the parameter, the population mean height of every sunflower in the field. The honest report is “about 170 cm”, not “the sunflowers in this field average 170.4 cm”.
Two samples and a known parameter
A made-up company has 1{,}360 workers, and its travel records hold every worker’s commute time. From those records, the mean commute of all 1{,}360 workers is 31.0 minutes. Before anyone looks at the records, Sabine asks 50 workers chosen at random and finds a mean of 29.2 minutes. Thorne asks a different random 50 and finds 33.0 minutes. Classify each number, and decide whether Sabine or Thorne made a mistake.
Solution31.0 minutes uses every worker’s commute, so it is a parameter, the population mean. This is the rare case where a parameter is known: the records are a census.
29.2 and 33.0 minutes each come from 50 workers, so each is a statistic, a sample mean.
Sabine’s statistic is 31.0 - 29.2 = 1.8 minutes below the parameter, and Thorne’s is 33.0 - 31.0 = 2.0 minutes above it. Neither made a mistake: their samples hold different workers, and 50 workers can easily include a few more long commutes, or short ones, than the company as a whole. Without the records, neither could tell whose figure was nearer, which is the usual position.
Sorting the numbers in a report
A made-up factory made 16{,}000 phones in March. A tester picks 40 of them at random and runs each from full charge until it goes flat. The report says: “The mean battery life of the tested phones was 21.8 hours, and 6 of the 40 lasted less than 20 hours.” Identify each statistic and the parameter it estimates, and estimate how many of March’s phones last less than 20 hours.
SolutionThe population is the 16{,}000 phones made in March. Both numbers come from the 40 tested phones, so both are statistics.
- 21.8 hours is a sample mean. It estimates the population mean, the mean battery life of all 16{,}000 phones.
- 6 of 40 is a sample proportion, \displaystyle \frac{6}{40} = 0.15 = 15\%. It estimates the population proportion, the share of all 16{,}000 phones lasting less than 20 hours.
Neither parameter is in the report, because neither is known. Using the sample proportion, about
of March’s phones last less than 20 hours.
Had the question been about the 40 tested phones themselves, those 40 would be the whole population and 21.8 hours a parameter. Which name a number gets depends on the question, not on the number.
Two nets, two statistics
Ulla wants the share of trout among the fish in a made-up lake. She sets nets at several places spread over the lake, at different depths, and catches 48 fish, of which 12 are trout. Thorne sets one net in the shallow bay by the jetty and catches 40 fish, of which 4 are trout. Find both statistics. Which is the better estimate of the parameter?
SolutionUlla’s sample proportion is \displaystyle \frac{12}{48} = 0.25 = 25\%, and Thorne’s is \displaystyle \frac{4}{40} = 0.1 = 10\%. Both are statistics.
Which to trust depends on how each sample was gathered, not on the numbers. Thorne’s net can only catch fish that swim in the bay, so his sample speaks for the bay, not the lake: it is a convenience sample. If trout keep more to deep water, the bay holds a smaller share of them than the lake, and his net will show too few however many fish it catches. Ulla’s nets give far more of the lake’s fish a chance to be caught. That is still not a simple random sample, since no net catches every fish equally well, but it stands in for the lake much better, so her 25\% is the better estimate.
Calling a number a statistic says where it came from, not whether it is a good estimate.
What one sample can and cannot show
A made-up warehouse holds 200 crates of mangoes. An inspector opens 50 crates chosen at random and finds bruised fruit in 10 of them. Find the sample proportion, and the smallest and largest the population proportion of crates with bruised fruit could be, using only what the sample shows.
SolutionThe sample proportion is \displaystyle \frac{10}{50} = 0.2 = 20\%. The sample shows at least 10 crates with bruised fruit and says nothing about the 200 - 50 = 150 crates nobody opened.
- If none of those 150 has bruised fruit, the warehouse holds 10 such crates: \displaystyle \frac{10}{200} = 5\%.
- If all 150 have it, it holds 10 + 150 = 160: \displaystyle \frac{160}{200} = 80\%.
Both are possible, and so is every value between, so from the sample alone the parameter could be anything from 5\% to 80\%. The 20\% is not forced by the evidence. It is a sensible estimate because chance chose which 50 crates were opened, so the unopened 150 have no reason to be better or worse. That makes 20\% a good estimate of the parameter. It does not make it the parameter.
Boundaries
The question decides which is which. The same number is a parameter for a question about the group it came from and a statistic for a question about a larger group. Until the population is named, “parameter or statistic?” has no answer.
Same kind, same variable. A sample mean of commute times estimates the population mean of commute times. It says nothing about the share of workers who cycle.
Every number from a sample is a statistic, good sample or bad. A random sample gives a fair estimate; a sample chosen for convenience can lean far off however large it is (see bias).
A census gives the parameter only if it is complete and correct. Records with gaps or errors give a number that is itself only close.
Means and proportions only. Medians and spreads also come in both versions; they are not treated here.
Where this goes next
Simple random samples show how to draw a sample by chance, which is what makes a statistic a fair estimate. Sampling variability draws many samples, as Figure 1 began to, and finds that the spread of their statistics follows a predictable pattern. That pattern lets statistical inference say how far from 38\% the parameter could reasonably be, and the claims tested in hypothesis tests are always claims about parameters. Even a standard deviation comes in a population version and a sample version.
Exercises
Parameter or statistic
- A made-up bird reserve has 640 tern nests. A warden checks 32 nests chosen at random and finds that 24 of them hold eggs. Find the share of the checked nests that hold eggs, and say whether it is a parameter or a statistic. What parameter does it estimate, and is that known?
- A made-up cycling club records the finishing time of every one of its 85 riders in a race, and the mean is 134 minutes. The question is about the club’s riders in that race. Is 134 minutes a parameter or a statistic?
- Valentijn measures the stem width of 30 sunflowers chosen at random from a made-up field and finds a mean of 2.6 cm. Describe in words the parameter that 2.6 cm estimates.
- Ulla’s net catches 64 fish from a made-up lake. Say whether each is a parameter or a statistic: (a) the share of trout among the 64 fish; (b) the mean length of all the trout in the lake; (c) the mean length of the trout in the net.
Working out a statistic and using it
- A random sample of 120 of the 4{,}800 households in a made-up town finds 42 with a cat. Find the sample proportion, and estimate how many of the town’s households have a cat.
- A random sample of 8 phones from a made-up factory had battery lives of 20, 23, 22, 19, 24, 21, 22 and 25 hours. Find the sample mean, and say what it estimates.
- In a random sample of sunflowers from a made-up field, 2 plants are 140 cm tall, 4 are 155 cm, 6 are 165 cm, 5 are 175 cm and 3 are 185 cm. Find the sample mean.
- A random sample of 60 of the 1{,}360 workers at the company of Example 3 finds that 18 cycle to work. Find the sample proportion, and estimate how many of the company’s workers cycle.
- Yestin asks 40 workers chosen at random how long their commute is. The 40 times add up to 1{,}260 minutes. Find the sample mean.
Different samples, one parameter
- Sabine and Wren-Lily each take a random sample of 100 households from the same made-up town. Sabine finds 41 with a dog and Wren-Lily finds 35. Find both statistics. Can both be fair estimates of the same parameter?
- Two random samples of 25 sunflowers from one made-up field have sample means of 171 cm and 168 cm, and no plant is in both. Put together, they make one sample of 50. Find its mean. Is that mean the parameter?
- A made-up plant nursery has 120 seedlings. A random sample of 30 of them holds 9 with yellow leaves. Find the sample proportion, and the smallest and largest the population proportion could be using only what the sample shows.
Reasoning
- Explain why a statistic is called an estimate of the parameter, and not the parameter itself.
- A report says: “The mean commute of the 60 workers we asked was 28.5 minutes, so the mean commute at the company is 28.5 minutes.” Which part is a statistic and which a parameter? Rewrite the sentence so that it claims only what the sample shows.
- Yestin says: “The 40 workers I asked are a population too, so their mean commute is a parameter.” When is he right, and when is he wrong?
- Thorne says of his 10\% from the shallow bay: “It is a statistic, so it is a good estimate of the lake’s share of trout.” What is right and what is wrong in this?
- In Figure 1, Sample 2 shows 30\% trout, exactly the lake’s share. A person who drew only Sample 2 says she has found the parameter. What has she shown, and what has she not?
Answers
- 75\%, a statistic. \displaystyle \frac{24}{32} = 0.75, worked out from the 32 checked nests only. It estimates the share of all 640 nests that hold eggs, a parameter, which is unknown because only 32 were checked.
- A parameter. It uses the time of every rider in the population the question names.
- The mean stem width of all the sunflowers in the field, the population mean. The 2.6 cm is the sample mean.
- (a) A statistic. (b) A parameter. (c) A statistic. Only (b) is about every trout in the lake.
- 35\%, and about 1{,}680 households. \displaystyle \frac{42}{120} = 0.35, and 0.35 \times 4{,}800 = 1{,}680.
- 22 hours. The total is 20 + 23 + 22 + 19 + 24 + 21 + 22 + 25 = 176, and 176 \div 8 = 22. It is a statistic estimating the mean battery life of all the phones the sample came from.
- 166 cm. There are 2 + 4 + 6 + 5 + 3 = 20 plants with total height 280 + 620 + 990 + 875 + 555 = 3{,}320 cm, and 3{,}320 \div 20 = 166.
- 30\%, and about 408 workers. \displaystyle \frac{18}{60} = 0.3, and 0.3 \times 1{,}360 = 408.
- 31.5 minutes. 1{,}260 \div 40 = 31.5.
- 41\% and 35\%. Yes: both are random samples of the same town, so both fairly estimate the one parameter, the share of all its households with a dog. They differ because they hold different households.
- 169.5 cm. The total is 25 \times 171 + 25 \times 168 = 4{,}275 + 4{,}200 = 8{,}475, and 8{,}475 \div 50 = 169.5. It is still a statistic, from 50 plants rather than every plant.
- 30\%, and anything from 7.5\% to 82.5\%. \displaystyle \frac{9}{30} = 0.3. The 90 unseen seedlings could include none with yellow leaves, giving \displaystyle \frac{9}{120} = 7.5\%, or all of them, giving \displaystyle \frac{99}{120} = 82.5\%.
- The statistic uses only the sampled individuals; those outside could have any values, so the sample cannot fix the parameter, and another sample would usually give another statistic. It is used in place of the unknown parameter because, for a random sample, it is right on average and so expected to be close. That is what an estimate is.
- The 28.5 minutes of the 60 workers is a statistic; “the mean commute at the company” is a parameter, and the sentence treats the one as the other. Better: “The mean commute of 60 workers chosen at random was 28.5 minutes, so the company’s mean commute is probably about 28.5 minutes.” (Any version that calls the company figure an estimate works.)
- He is right if the question is about those 40 workers only; then they are the population. He is wrong if the question is about the company’s workers, the usual reason to ask 40 of them: then the 40 are a sample and their mean is a statistic.
- Right: 10\% is a statistic, worked out from his catch. Wrong: being a statistic does not make it a good estimate. Every number from a sample is a statistic. His net reached only the bay, so his 10\% speaks for the bay and can be far from the lake’s share.
- She has a statistic, 30\%, which is a fair estimate if her sample was random. She has not shown it equals the parameter: from her side, the lake’s share could have been 20\% or 40\% and her sample still come out at 30\%. The match is luck she cannot detect.