Maths A Level - Statistics - Book 1, Chapter 1

population

the whole set of items that are of interest

census

observes or measures every member of a population

sample

a selection of observations taken from a subset of the population which is used to find out information about the population as a whole

census pros and cons

* it should give a completely accurate result

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* time consuming and expensive
* cannot be used when the testing process destroys the item
* hard to process large quantity of data

sample pros and cons

* less time consuming and expensive than a census
* fewer people have to respond
* less data to process than in a census

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* the data may not be as accurate
* the sample may not be large enough to give information about small sub-groups of the population

sampling units

individual units of a population

sampling frame

a list often made of individually named or numbered sampling units of a population

what does the size of a sample depend on?

the required accuracy and available resources

when is a larger sample required?

if the population is very varied, rather than being uniform

why can different samples lead to different conclusions?

due to the natural variation in a population

random sampling points

* each member of the population has an equal chance of being selected
* the sample should therefore be representative of the population
* random sampling also helps remove bias from a sample

what are the three methods of random sampling?

* simple random sampling
* systematic sampling
* stratified sampling

what are the two methods of choosing the numbers from a sampling frame?

* generating random numbers (using a calculator, computer or random number table)
* lottery sampling

what is a simple random sample?

a simple random sample of size n is one where every sample of size n has an equal chance of being selected.

what is systematic sampling?

in systematic sampling, the required elements are chosen at regular intervals from an ordered (not necessarily) list, going through the entirety of the list

what is stratified sampling?

in stratified sampling, the population is divided into mutually exclusive strata (males and females, for example) and a random sample is taken from each

simple random sampling pros and cons

* free of bias
* easy and cheap to implement for small populations and small samples
* each sampling unit has a known and equal chance of selection

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* not suitable when the population size or the sample size is large
* a sampling frame is needed

systematic sampling pros and cons

* simple and quick to use
* suitable for large samples and large populations

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* a sampling frame is needed
* it can introduce bias if the sampling frame is not random

stratified sampling pros and cons

* sample accurately reflects the population structure
* guarantees proportional representation of groups within a population

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* population must be clearly classified into distinct strata
* selection within each stratum suffers from the same disadvantages as simple random sampling
* not suitable when the population size or the sample size is large


* a sampling frame is needed

what is quota sampling?

in quota sampling, an interviewer or researcher selects a sample that reflects the characteristics of the whole population

what is opportunity sampling?

opportunity sampling consists of taking the sample from people who are available at the time the study is carried out and who fit the criteria you are looking for

quota sampling pros and cons

* allows a small sample to still be representative of the population
* no sampling frame required
* quick, easy and inexpensive
* allows for easy comparison between different groups within a population

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* non-random sampling can introduce bias
* population must be divided into groups, which can be costly or inaccurate
* increasing scope of study increases number of groups, which adds time and expense
* non-responses are not recorded as such

opportunity sampling pros and cons

* easy to carry out
* inexpensive

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* unlikely to provide a representative sample
* highly dependent on individual researcher

outlier

• outlier defined in relation to IQR, Q1&Q3

• avoid using the term anomaly, implies something wrong with data/data collection process

what is relative humidity measured in and above what relative humidity gives rise to misty conditions?

• in %

• above 95%

what are outliers?

values that are much greater than or much less than the other values and need to be treated with caution

features of the normal distribution

• has parameters µ, the population mean and σ2, the population variance

• is symmetrical (mean = median = mode)

• has a bell-shaped curve with asymptotes at each end

• has total area under the curve equal to 1

• has points of inflection at µ + σ and µ − σ

% stats for normal distribution

• approximately 68% of the data lies within one standard deviation of the mean

• 95% of the data lies within two standard deviations of the mean

• nearly all of the data (99.7%) lies within three standard deviations of the mean

what is daily mean visibility?

the maximum horizontal distance at which an object can be seen and recognised in daylight

state one assumption you have made in deciding these values (normal distribution)

that the population follows the normal distribution over the whole range of values i.e. that there are no extreme outliers.

comment on the suitability of using cumulative frequency diagrams to compare these distributions

the cumulative frequency diagram shows us how the lengths of the badgers is spread and enables us to estimate the median and quartiles, but tells us little about individual data points