Estimation from Statistical Data (Edexcel GCSE Statistics)

Estimating Population Characteristics

How can I use samples to estimate population characteristics?

• Summary statistics calculated from a sample can be used to estimate the same statistics for the population as a whole

  • e.g. the mean, median, range, quartiles and interquartile range

• Remember that a sample is a selection of members drawn from the population, whereas the population is all the members

  • Statistics calculated from a sample will usually not be exactly the same as the statistics for the whole population (different mean, etc.)

  • Statistics calculated from two different samples will also not usually be the same

• As long as the sample is representative of the population

  • then you can assume that the statistics for the population are approximately the same as those for the sample

    • e.g. assume the population mean is about equal to the sample mean

• This can be used to make predictions about the population

  • About half (50%) of the population will be above the sample median

    • and about half will be below

  • About a quarter (25%) of the population will be below the sample lower quartile

    • and about a quarter will be above the sample upper quartile

  • About half (50%) of the population will be between the sample upper and lower quartiles

• Sample size has an impact on the reliability of estimates made about the population

  • In general a larger sample size will lead to more reliable conclusions

Petersen Capture Recapture Formula

What is the capture recapture method?

• The capture recapture method is a way to estimate the size of a population

  • It is used when it is either impossible or impractical to count the whole population

    • e.g. too expensive or too time-consuming

  • Common examples include

    • the population of fish in a river/lake/sea

    • the population of wild animals in a natural habitat

• The capture recapture method is based on proportion

  • A first sample of the population is captured

    • and each member is given an identifiable marker/tag

  • All members of the sample are then replaced 

    • i.e. released back into the population

  • At a later time, a second sample of the population is taken

  • The proportion of the second sample that is tagged is assumed to be the same as the proportion of the population that was tagged in the first sample

How do I use the Petersen capture recapture formula?

• The Petersen capture recapture formula is used to find an estimate for the population size using a capture recapture experiment

  • 

    • This formula is not on the exam formula sheet, so you need to remember it

  • A shorter version is

    • N is the size of the population

    • M is the size of the first sample

    • n is the size of the second sample

    • m is the number in the second sample that have markers/tags

• It can be easier to remember the formula if you understand where it comes from

  • The proportion of the total population captured (and marked/tagged) in the first sample is

  • The proportion of the second sample that is marked/tagged is

  • We assume that those two proportions are equal

    • 

  • Rearrange that equation to get the formula

    • 

    • 

    • 

What assumptions need to be made for the capture recapture method to be valid?

• Each member (element) of the population must have an equal chance of being selected

  • This applies to both the first and second samples

  • This means that random sampling is used

• Between the first and second samples

  • The tagged members must have had sufficient time and opportunity to mix with the rest of the population

  • The population remains the same size (broadly speaking)

    • No (significant number of) births/deaths

    • No (significant number of) members leave the population

      • e.g. migration

  • No marks/tags have been removed or destroyed

    • e.g.  taken off, worn off

• The process of tagging the members of the first sample must not affect the likelihood of them being recaptured

When is the capture recapture method reliable?

• The sample sizes used must be large enough to be representative of the population for the method to be reliable