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Year 1 Statistics and Mechanics
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Document with revision notes, explanations and examples for the whole of year 1 (year 12) applied maths - stats and mechanics.
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Statistics1. Data Collection
1. 1. Populations and samples
- Population: the whole set of items of interest
- Census: an observation or measure of 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.
- Sampling unit: individual units of a population.
- Sampling frame: the list of numbered or individually named sampling units.
Advantages Disadvantages
Census Should give a completely accurate Time consuming, expensive.
result. Can’t be used when testing destroys
the items.
Hard to process a large amount of
data.
Sample Less time consuming, cheaper. Not be as accurate.
Fewer people to respond. Sample may not be representative.
Less data to process.
1. 2. Sampling
In random sampling, every unit of the population has an equal chance of being selected.
The sample should therefore be representative of the population.
Random sampling helps remove bias. There are 3 types of random sampling:
- Simple random sampling: a sample of size n is one where every unit has an equal
chance of being selected
- To carry it out:
1. Get the sampling frame
2. You generate a random number (w/ a calculator, computer or random number
table)
3. You observe/ measure the unit with the corresponding number
OR 2. You use lottery sampling: write down all the members of the sampling frame on
hell hello you different papers and put them in a ‘hat’.
3. You observe/ measure the unit you pick out
, - Systematic sampling: the required elements are chosen at regular intervals from an
ordered list.
- To carry it out:
1. Get the sampling frame
2. Find out at what interval you need to take samples
Divide the population number by sampling size
3. Generate a random number and take the first sample here
4. Observe/measure the next samples at the regular interval
- Stratified sampling: the population is divided into mutually exclusive strata and a
random sample is taken from each.
- To carry it out:
1. Divide the population into the stratum
To find out how many people go into each stratum use the formula:
number∈stratum
× overall sampling¿ ¿
number∈ population
2. Select random units from each stratum to observe/measure
Advantages and disadvantages:
Advantages Disadvantages
Simple random No bias. Not suitable for large populations
Easy and cheap for small pops. as it would be time consuming
Each unit has an equal chance of and expensive.
being selected. Sampling frame needed.
Systematic Simple and quick. Sampling frame needed.
Suitable for large pops. Can introduce bias if the sampling
frame is not random.
Stratified Sample accurately reflects the Pop must be divided into the
pop’s structure. stratum.
Guarantees proportional Selection in each stratum - same
representation of groups in pop. disadvantages as simple random.
,1. 3. Non-random sampling
There are 2 types:
- Quota sampling: the interviewer selects a sample that reflects the characteristics of the
whole population.
- To carry it out:
1. Divide the population into groups according to a given characteristic. The size of
each group will determine the proportion of the sample that should have that
characteristic.
2. The interviewer must meet the people, assess their group and then allocate
them into the appropriate quota.
3. He must do this until all the quotas are full.
- If someone refuses to answer or their quota is full you just ignore it.
- 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.
- To carry it out:
1. Stand at the place where you are going to carry out the study.
2. Interview the sample size of people you need who are in that place and
available
Advantages Disadvantages
Quota Allows a small sample to be Non-random - can introduce bias.
representative of pop. Pop. must be divided into groups -
No need for sampling frame. can be inaccurate and costly.
Quick, easy, cheap. Increased scope of study =
Allows for easy comparison. increased number of groups - adds
time and expense.
Non-responses are not recorded as
such.
Opportunity Easy to carry out. Unlikely to provide a representative
Inexpensive. sample.
Highly dependent on the
interviewer.
1. 4. Types of data
- Quantitative: variable/data associated with numerical observations.
- Qualitative: variable/data associated with non-numerical observations.
- Continuous: variable/data that can take any value in a given range.
- Discrete: variable/data that can only take specific values in a given range.
, When data is presented in grouped frequency tables, the values are not shown.
- Class boundaries: the maximum and minimum values that belong in each class.
- Midpoint: the average of the class boundaries.
- Class width: the difference between the upper and lower class boundaries.
1. 5. The large data set
In the exam, there will be questions based on real data, some of these will be based on
weather data from the large data set.
The data consists of weather data samples for 5 UK and 3 overseas weather stations for May
to October 1987 and May to October 2015.
It must be noted that Perth is in the southern hemisphere, meaning that when it is summer in
the northern hemisphere it will be winter there. This will cause data values to be very different
from the rest.
The large data set contains data on:
- Daily total sunshine
- Daily mean temperature
- Daily total rainfall
- Daily mean wind direction and speed
- Daily maximum gust
- Daily maximum humidity
- Daily mean cloud cover
- Daily mean visibility
- Daily mean pressure
IMPORTANT NOTES:
- Daily total rainfall: there are some data values that are presented as “tr”. This stands for
trace, meaning that in those days there was less than 0.05mm recorded. When working
with these values we use a 0.
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