Estimates from Samples and How Much They Vary | Grade 7 Math | LevelKite

Suppose you want to know how many of the 600 students at a school walk there each morning. Asking every student would take a long time. You already know a better plan: choose a random sample, a smaller group picked so that every member of the population (the whole group you want to know about) has the same chance of being chosen. A random sample is likely to be representative, which means it looks like a small copy of the population.

In this lesson you will use data from a sample like that to make an estimate: a reasoned, approximate value for the whole population. Then you will find out how much that estimate could change if chance had given you a different sample.

Here is the idea behind an estimate. In a random, representative sample, the fraction of the sample with some trait, such as walking to school, is usually close to the fraction of the population with that trait. If 36 of every 100 sampled students walk, then about 36 of every 100 students in the whole school probably walk too.

A conclusion about a population that you draw from a sample is called an inference. It is valid only when the sample is random and representative. A biased sample (one picked in a way that favors some members) tends to lean one way, so scaling it up gives a number you cannot trust.

For the walking question, this idea gives two ratios (comparisons of two quantities by division) that should be nearly equal. The symbol $\approx$ means is approximately equal to. Each ratio compares positive counts of students, and both keep the same order: walkers on top, all students on the bottom.

\frac{\text{walkers in sample}}{\text{students in sample}} \approx \frac{\text{walkers in school}}{\text{students in school}}

Key concept

A random sample gives an approximate estimate

A random, representative sample usually behaves like a small copy of the population. So an amount per member (for each one member) of the sample, such as the fraction of students who walk or the number of books per student, is usually close to the same amount per member of the population. Multiply that per-member amount by the population size to get an estimate.

Use this with positive quantities whose units are kept and matched, a sample size (the divisor) that is not zero, and matching quantities in matching positions.

Another random sample would likely give a somewhat different result, so every estimate is approximate. Looking at many samples of the same size shows how much it varies. An estimate from a biased sample is not valid.

\text{estimate} = \frac{\text{amount found in sample}}{\text{sample size}} \times \text{population size}

Worked example

Estimating a count from one random sample

A school has 600 students. You load all 600 names into a random name picker and draw 50 of them. Of those 50 students, 18 say they walk to school. Estimate how many of the 600 students walk to school.

Check the sample first. Every one of the 600 students had the same chance of being drawn, so the sample is random, and a random sample is likely to be representative. That makes it fair to use for an estimate about the whole school.
Name the unknown. Let w be the estimated number of students in the whole school who walk. Write the sample ratio and the school ratio in the same order, walkers over students, so matching quantities sit in matching positions. The estimate is the value of w that makes the two ratios equal.
$\frac{18 \text{ walkers}}{50 \text{ students}} = \frac{w \text{ walkers}}{600 \text{ students}}$
Find the sample fraction, which is the walkers in the sample divided by the students in the sample. Both 18 and 50 are positive counts of students, and the divisor 50 is not zero. Doubling the top and the bottom gives hundredths without changing the value. The result means 0.36 walkers for each 1 student sampled.
$\frac{18}{50} = \frac{18 \times 2}{50 \times 2} = \frac{36}{100} = 0.36$
Multiply both sides of the equation by 600. Multiplying both sides by the same nonzero number keeps them equal, and it leaves w by itself, because w divided by 600 and then multiplied by 600 is w again. To multiply 0.36 by 600, treat 600 as 100 times 6: 0.36 times 100 is 36, and 36 times 6 is 216.
$w = \frac{18}{50} \times 600 = 0.36 \times 600 = 36 \times 6 = 216$
Check and interpret. The school is 12 times the size of the sample, and scaling the 18 walkers by 12 also gives 216. That value is a whole number between 0 and 600, so it can be a count of students. It rests on one sample of 50, so it is approximate: another random sample would likely give a somewhat different number.
$600 \div 50 = 12, \qquad 18 \times 12 = 216$

Answer

About $216$ of the $600$ students walk to school. This is an estimate from one random sample, not an exact count.

Worked example

Estimating a total from a per-student rate

At the same school of 600 students, the librarian uses a random number generator to choose 40 students from the full student list. Together, those 40 students checked out 138 library books this term. Estimate how many library books all 600 students checked out this term.

The generator gave every student the same chance, so the sample is random and likely to be representative. Let b be the estimated number of library books checked out by all 600 students this term.
Find the sample rate, the number of books for each 1 student. Both quantities are positive, and the per-one quantity, 40 students, is the divisor and is not zero. Halve both parts, then multiply both by 5. Changing both parts by the same nonzero factor keeps the ratio equal.
$\frac{138 \text{ books}}{40 \text{ students}} = \frac{69 \text{ books}}{20 \text{ students}} = \frac{345 \text{ books}}{100 \text{ students}} = 3.45 \text{ books per student}$
Apply that rate to all 600 students, because a random sample's rate is usually close to the school's rate. Books per student times students gives books, so the unit matches the question. Split 3.45 into 3 and 0.45 to multiply.
$b = 3.45 \times 600 = 3 \times 600 + 0.45 \times 600 = 1{,}800 + 270 = 2{,}070$
Round sensibly. Another random sample of 40 would likely give a somewhat different total, so the last digits of 2,070 are not dependable. To the nearest hundred, 2,070 is 2,100.
$2{,}070 \approx 2{,}100$

Answer

All 600 students checked out about 2,100 library books this term. This is an approximate value based on one random sample of 40 students.

Go back to your estimate of 216 walkers. It came from one random sample of 50 students, and chance decided which 50. A second random draw would almost certainly pick a different group of students, and probably a different number of walkers. Then the estimate would come out different too. This change from one sample to the next is called variation.

To judge an estimate, you need a sense of how large that variation is. You could survey many more real samples, but that takes time. Instead you can run a simulation: a model, or stand-in for the real situation, that imitates the random draw. Each sample the model produces is a simulated sample.

Here is a simulation for the walking question. A computer builds a model school of 600 students in which exactly 216 are marked as walkers, to match your estimate. It draws 50 students at random, counts the walkers, puts all 50 back, and repeats until it has 10 samples.

Every simulated sample has 50 students, the same size as your real sample, so the estimates can be compared fairly. For each one you make an estimate just as before: divide the walkers drawn by the 50 students drawn, then multiply by 600. The table shows every quantity and every calculation, including how far each estimate lands from the model's 216.

Ten simulated random samples of 50 students from a model school with 216 walkers among 600 students

Simulated sample	Students drawn	Walkers drawn	Sample fraction (walkers divided by students)	Estimated walkers in school (fraction times 600)	Distance from 216 (larger minus smaller)
1	$50$	$20$	$\frac{20}{50} = 0.40$	$0.40 \times 600 = 240$	$240 - 216 = 24$
2	$50$	$17$	$\frac{17}{50} = 0.34$	$0.34 \times 600 = 204$	$216 - 204 = 12$
3	$50$	$22$	$\frac{22}{50} = 0.44$	$0.44 \times 600 = 264$	$264 - 216 = 48$
4	$50$	$15$	$\frac{15}{50} = 0.30$	$0.30 \times 600 = 180$	$216 - 180 = 36$
5	$50$	$18$	$\frac{18}{50} = 0.36$	$0.36 \times 600 = 216$	$216 - 216 = 0$
6	$50$	$19$	$\frac{19}{50} = 0.38$	$0.38 \times 600 = 228$	$228 - 216 = 12$
7	$50$	$14$	$\frac{14}{50} = 0.28$	$0.28 \times 600 = 168$	$216 - 168 = 48$
8	$50$	$21$	$\frac{21}{50} = 0.42$	$0.42 \times 600 = 252$	$252 - 216 = 36$
9	$50$	$16$	$\frac{16}{50} = 0.32$	$0.32 \times 600 = 192$	$216 - 192 = 24$
10	$50$	$18$	$\frac{18}{50} = 0.36$	$0.36 \times 600 = 216$	$216 - 216 = 0$

Worked example

Gauging variation with simulated samples

Use the table of 10 simulated samples above. Describe how much an estimate from one random sample of 50 students can vary, and explain what that means for your estimate that 216 students walk to school.

Confirm how a row is built, using sample 3. Divide the 22 walkers drawn by the 50 students drawn, multiply by the 600 students in the school, then subtract to find the distance from 216. The counts are positive numbers of students and the divisor 50 is not zero, so the fraction is defined.
$\frac{22}{50} = 0.44, \qquad 0.44 \times 600 = 264, \qquad 264 - 216 = 48$
Scan the estimate column for the lowest and highest values. They mark the edges of what these 10 samples produced. The lowest comes from sample 7 and the highest from sample 3.
$\text{lowest} = 168, \qquad \text{highest} = 264$
Find the range, which is the highest value minus the lowest value. The range tells you how spread out the estimates are, measured in students.
$264 - 168 = 96$
See where most of the estimates fall. In the distance column, 8 of the 10 distances are 36 or less. Those 8 estimates lie from 180 to 252. Only 168 and 264 are farther out.
$216 - 36 = 180, \qquad 216 + 36 = 252$
Find the biggest miss. The largest distance in the table is 48, and it appears twice: once below 216 and once above. Each time the smaller number is subtracted from the larger, so the distance is positive.
$216 - 168 = 48, \qquad 264 - 216 = 48$
Say what this suggests for your real estimate. Random samples of 50 from a school like this one gave estimates up to 48 students away from the model's true count of 216. So one sample of 50 cannot pin the number down exactly. This is an informal judgment from 10 samples, not a guarantee.

Answer

In this simulation the estimates ran from $168$ to $264$ students, a range of $96$ , and $8$ of the $10$ were within $36$ of $216$ . So one sample of $50$ supports saying that about $216$ students walk, with the true number possibly around $50$ higher or lower, not exactly $216$ .

Takeaway. A random, representative sample lets you estimate a value for a whole population: find the sample's amount per member, then multiply by the population size. The result is approximate, because another random sample would likely give a different estimate. Simulated samples of the same size show how large that variation is, so you can say roughly how far off your estimate might be.

Next, you will compare estimates from several random samples and explain why random sampling is what makes these inferences valid.