Estimating from Samples (Higher)
Estimating from samples means using a measurement from part of a population, such as a sample mean or proportion, to predict a value for the whole population, for example scaling a sample proportion up to a population total. Reliable estimates need a representative sample; larger samples usually give more trustworthy results. This underpins topics such as capture-recapture on GCSE Statistics Higher tier.
Before you start
Make sure you're comfortable with these topics first:
Method
- Take a representative sample from the population, using an appropriate sampling method so the sample reflects the population's structure.
- Calculate the required statistic from the sample, such as a sample mean, a sample proportion, or a sample frequency for a category.
- Scale the sample statistic up to the population: population estimate = sample statistic x (population size / sample size), or apply the sample proportion directly to the population size.
- State any assumptions made, such as that the sample is representative and that the population has not changed between sampling and estimating.
- Comment on reliability: larger samples and more representative sampling methods generally give more reliable estimates, but cost more time and money to collect.
Worked example
A conservation charity wants to estimate the total number of oak trees in a 400-hectare forest. They randomly sample 25 one-hectare plots and count 340 oak trees across those plots in total. Estimate the total number of oak trees in the forest.
- Total oak trees counted across the sampled plots = 340, over a sampled area of 25 hectares.
- Mean number of oak trees per hectare in the sample = 340 / 25 = 13.6 trees per hectare.
- Scale this rate up to the whole 400-hectare forest: estimate = 13.6 x 400 = 5440 trees.
- This estimate assumes the 25 sampled hectares are representative of tree density across the whole forest.
- Final answer: the estimated number of oak trees in the forest is 5440.
Practice questions
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Q1A researcher samples 50 out of 2000 fish in a lake and finds 8 have a particular fin marking. Estimate the number of fish in the lake with that marking.Show answer
Answer: 320 fish (proportion 8/50 = 0.16; 0.16 x 2000 = 320)
Q2In a sample of 60 shoppers, 15 said they preferred own-brand products. A shopping centre has 3000 visitors a day. Estimate how many of the day's visitors prefer own-brand products.Show answer
Answer: 750 visitors (proportion 15/60 = 0.25; 0.25 x 3000 = 750)
Q3A quality-control sample of 40 bottles from a batch of 5000 finds the mean volume per bottle is 498 ml. Estimate the total volume, in litres, of the whole batch.Show answer
Answer: 2490 litres (498 x 5000 = 2,490,000 ml = 2490 litres)
Q4Explain why a sample of 10 students' shoe sizes is unlikely to give a reliable estimate of the mean shoe size of a school of 1200 students.Show answer
Answer: The sample is very small compared to the population, so it may not be representative and could be badly affected by one or two unusual values, giving an unreliable estimate
Q5A wildlife park has 3 enclosures with 200, 350 and 450 birds (1000 birds in total). A stratified sample of 40 birds is taken proportionally across the enclosures. In the sample taken from the largest enclosure, 12 of the birds sampled are male. Estimate the number of male birds in that enclosure.Show answer
Answer: 300 male birds (sample size for that enclosure = 450/1000 x 40 = 18; proportion male = 12/18 = 2/3; 2/3 x 450 = 300)
Q6Student A estimates the dandelions on a 500 square metre field using five 1 square metre quadrats, counting 3, 5, 4, 6 and 2 dandelions. Student B uses twenty 1 square metre quadrats with a mean of 4.2 dandelions per quadrat. Whose estimate is likely to be more reliable, and what is Student B's estimate for the whole field?Show answer
Answer: Student B is likely more reliable, as 20 quadrats is a much larger sample than 5; Student B's estimate = 4.2 x 500 = 2100 dandelions
Exam-style questions
Written in the style of a GCSE Statistics exam paper, with a full mark scheme.
A sample of 80 out of 4000 members of a gym is taken. In the sample, 28 members use the gym at least 3 times a week. Estimate the number of gym members overall who use the gym at least 3 times a week.
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A council wants to estimate the total number of potholes on 250 km of road. Inspectors survey a random sample of 15 km of road and find 42 potholes. (a) Estimate the total number of potholes on the full 250 km of road. (b) State one assumption needed for your estimate to be reliable.
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A survey samples 120 out of 15000 local residents and finds that 54 support a new bike lane. Estimate how many of the 15000 residents support the new bike lane.
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See real GCSE Statistics past-paper questions, with official mark schemes →
Free printable worksheet
Want more practice on paper? Download the estimating from samples (higher) worksheet pack - 16 pages of exam-style questions with a full mark scheme. One email opens every download in this browser for 14 days - no account, no card. Print it for personal and classroom use.
This topic is chapter 3 of GCSE Statistics Higher Workbook 1, the whole course as one free printable PDF.
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