Sampling, Inferences & Comparing Populations — Print
You cannot measure everyone, so you measure some. A random sample lets a small group speak honestly for a large one.
What the practice questions cover
These questions use 12 ideas from the course. Follow any of them for a definition, where it is taught, and what it leads to.
- Rate — A ratio of two different units, like miles per hour or dollars per pound.
- Percent — Parts per hundred. 37% = 37100 = 0.37.
- Population — Every member of the group you care about.
- Sample — The part of the population you actually measure.
- Random sample — Every member has an equal chance of being picked — the only kind you can trust.
- Mean — Two kinds of centre: the balance point, and the middle value.
- Median — Two kinds of centre: the balance point, and the middle value.
- MAD — Mean absolute deviation — the average distance from the mean; a measure of spread.
- Scale — What one grid step is worth — and where the axis starts.
- Outlier — A dot far away from the rest of the pattern.
- Mode — The value that appears most often.
- Mean absolute deviation — The average distance of the values from the mean.
A jar holds ten thousand beads. How many are orange?
Tipping them out and counting is not going to happen. So take one scoop.
The scoop holds 12 beads and 4 are orange — one third.
If the beads were well mixed, the jar is probably about one third orange too. Roughly 3,300.
Take a second scoop and you will get a slightly different answer. That wobble is not a mistake; it is what sampling is.
The whole of statistics rests on that move: measure a small part, say something careful about the whole. The catch is the phrase "well mixed" — a scoop taken only from the top proves nothing about the bottom.
Why sampling works
If you want to know the average height of every 7th grader in the district, measuring all 900 is impractical. Measure 60 chosen at random to get an estimate; another random sample will give a somewhat different result.
The key word is random. Surveying only the basketball team gives a great answer about basketball players and a terrible one about 7th graders.
Biased samples
A sampling method can be biased when it systematically over-represents part of the population without accounting for that difference. Asking people outside a gym about exercise habits is biased. So is a survey only people with strong opinions bother to answer.
Using a sample to estimate the whole
This is proportional reasoning again. If 18 of 60 sampled students walk to school, that is 1860 = 0.30, so about 30%.
Scale up: 0.30 × 900 = 270 students in the whole district, approximately.
Say "about". A sample gives an estimate, not a fact.
Centre: mean and median
The mean is the total shared out equally: add everything, divide by how many.
The median is the middle value once the data is in order. With an even count, average the two middle numbers.
A single huge value drags the mean but barely moves the median. That is why house prices and incomes are usually reported as medians.
A useful identity
Sum = mean × count. Most "the average changes when one more score arrives" problems are solved by converting averages back into sums.
Spread: why two groups can share a mean and be nothing alike
Group A: 50, 50, 50, 50. Group B: 20, 40, 60, 80. Both have mean 50, but B is far more spread out.
Mean absolute deviation measures this: find each value's distance from the mean, then average those distances. For B: distances are 30, 10, 10, 30, so MAD = 20. For A, MAD = 0.
Comparing two populations
To describe how separated two groups are, compare the difference in centres to the spread. This is a descriptive comparison, not a test of statistical significance.
If two 7th grade classes have mean scores 82 and 86 but each has a MAD of 12, the 4-point gap is small next to the variation within each class.
If the MADs were 1, that same 4-point gap would be large relative to the within-class variation. Generalising to populations also needs sample sizes and a suitable sampling design.
Worked examples
Scaling up a sample
In a random sample of 80 fish from a lake, 12 are trout. The lake holds about 2,500 fish. Estimate the number of trout.
- Sample proportion: 1280 = 0.15 15% of the sample were trout.
- Assume the lake matches the sample. This is what a random sample buys you.
- 0.15 × 2500 = 375 Apply the proportion to the whole population.
About 375 trout.
A missing score
Rosa's first four test scores average 84. What must she score on the fifth test to average 86?
- Turn averages into sums: 84 × 4 = 336 Sum = mean × count.
- The target total: 86 × 5 = 430 Five tests averaging 86.
- 430 - 336 = 94 The fifth score must close the gap.
94
Mean vs median
Six houses sell for $210k, $225k, $230k, $240k, $250k and $1,800k. Which centre describes the neighbourhood better?
- Median: average the 3rd and 4th values, 230 + 2402 = 235 Six values, so the middle is between two.
- Mean: 210+225+230+240+250+18006 = 492.5 The mansion pulls it hard.
- No ordinary house costs near $492k. The mean describes none of the actual homes.
The median, $235k, because one outlier distorts the mean.
Words to know
- Population
- Every member of the group you care about.
- Sample
- The part of the population you actually measure.
- Random sample
- Every member has an equal chance of being picked — the only kind you can trust.
- Mean / median
- Two kinds of centre: the balance point, and the middle value.
- MAD
- Mean absolute deviation — the average distance from the mean; a measure of spread.
Mistakes to watch for
Trusting a convenience sample. Asking outside a gym about exercise habits gives an answer about gym-goers, not about the town. Where you stand changes what you hear, and a bigger biased sample is not less biased.
Forgetting to order the data before finding the median. The median is the middle of the sorted list; the middle of the list as written is just whichever value happened to be typed third.
Claiming a difference is meaningful without looking at the spread. Two groups with medians 78 and 74 may overlap almost completely if both are spread wide. Compare centres and spreads.
Treating a sample result as exact. A second scoop would give a slightly different number. Say "about a third," not "exactly 3,333" — the false precision claims more than the data can support.
In 1936 the Literary Digest polled 2.4 million Americans and confidently predicted Landon would beat Roosevelt. Roosevelt won 46 of 48 states.
Their sample came from telephone directories and car registrations — in the Depression, that meant sampling the well-off.
George Gallup predicted the right result the same year from just 50,000 people, because his sample was chosen to look like the country. A small fair sample beat a huge biased one.
Problem 1
Find the mean of 4, 8, 10, 14.
Worked answer
9
4 + 8 + 10 + 14 = 36
36 ÷ 4 = 9
Problem 2
Find the median of 3, 7, 9, 12, 15.
Worked answer
9
Five values, so the middle is the 3rd
The median is 9.
Problem 3
Find the median of 4, 6, 9, 11.
Worked answer
7.5
Middle two are 6 and 9
6 + 92 = 7.5
Problem 4
In a sample of 50 students, 15 prefer pizza. What percent is that?
Worked answer
30
1550 = 0.30
= 30%
Problem 5
Find the mode of 2, 3, 3, 5, 7, 3, 8.
Worked answer
3
3 appears three times, more than any other value.
Problem 6
In a random sample of 200 voters, 84 support a measure. Estimate how many of 5,000 voters support it.
Worked answer
2100
84200 = 0.42
0.42 × 5000 = 2100
About 2,100 voters.
Problem 7
The mean of 5 numbers is 12. What is their sum?
Worked answer
60
12 × 5 = 60
Problem 8
Four tests average 78. What score on a fifth test would raise the average to 80?
Worked answer
88
Current total: 78 × 4 = 312
Target total: 80 × 5 = 400
400 - 312 = 88
Problem 9
Find the mean absolute deviation of 4, 8, 10, 14.
Worked answer
3
Distances from 9: 5, 1, 1, 5
5 + 1 + 1 + 5 = 12
12 ÷ 4 = 3
Problem 10
You want to know how much TV the average 7th grader watches. Which sample is biased?
- 60 students drawn at random from the full roster
- Every 5th student leaving the building at dismissal
- Students waiting outside the school television studio
- 60 students picked at random from each grade list
Worked answer
Students waiting outside the school television studio
Students at the TV studio are far more interested in television than a typical student.
That makes them over-represent heavy viewers — a biased sample.
The other three give every student a fair chance of selection.
Problem 11
A data set is 12, 15, 15, 18, 140. Which measure better describes a typical value?
- The mean
- The median
- They are equally good
Worked answer
The median
Mean = 2005 = 40
Median = 15
No value is near 40 — the outlier 140 dragged the mean away from the data.
Problem 12
A factory samples 80 items and finds 24 defective. Estimate the number of defective items in a batch of 1,500.
Worked answer
450
2480 = 0.30
0.30 × 1500 = 450
Problem 13
Six numbers have a mean of 20. One of them, 35, is removed. What is the mean of the remaining five?
Worked answer
17
Total: 20 × 6 = 120
After removing: 120 - 35 = 85
85 ÷ 5 = 17
Problem 14
The mean of seven numbers is 15. Every number is increased by 4. What is the new mean?
Worked answer
19
The total rises by 7 × 4 = 28
New total: 105 + 28 = 133
133 ÷ 7 = 19 — shifting every value shifts the mean by the same amount.
Problem 15
Class A has mean 84 and MAD 2. Class B has mean 88 and MAD 3. How large is the mean gap relative to the MADs?
- Twice A's MAD and about 1.33 times B's MAD
- Half of each MAD
- Exactly equal to each MAD
Worked answer
Twice A's MAD and about 1.33 times B's MAD
The gap between means is 88-84=4.
Relative to A: 4/2=2 MADs. Relative to B: 4/3≈1.33 MADs.
This describes the separation; it is not a statistical significance test.
Problem 16
A biologist tags 60 fish and releases them. Later a sample of 45 fish contains 9 tagged ones. Estimate the lake's fish population.
Worked answer
300
Sample fraction tagged: 945 = 15
So the 60 tagged fish are about 15 of the lake
60 × 5 = 300 fish