Stats

Parameter

Number that describes the whole population

Sample

Specific number of points taken from the population

Cencus

Collection of every data point within the population

Simple random sampling

• Each item allocated a unique number

• Numbers chosen at random using random number generator

Advantages of simple random sampling

- Bias free

- Fast, easy, cheap

- Each number has a known equal chance of being selected

Disadvantages of simple random sampling

- More difficult as the population size gets larger

- Full sampling frame needed

Systematic sampling

Elements are chosen at regular intervals from an ordered list. First member chosen randomly then goes up in e.g 5ths

Advantages of systematic sampling

- Simple, fast, cheap

- Suitable for large samples and large populations

Disadvantages of systematic sampling

- Full sampling frame is needed

- It can introduce bias if the sampling frame is not random

Stratified sampling

Population is split into groups and a simple random sample is carried out in each group

How to find how many people to put in each strata for stratified sampling

(actual number in group ÷ total population) × sample size

Advantages of stratified sampling

- Sample accurately reflects the population structure

- Guarantees proportional representation of groups within a population

Disadvantages of stratified sampling

- Population must be clearly classified into distinct strata

- Selection within each stratum suffers from the same disadvantages as simple random sampling

Quota sampling

• Interviewer creates groups for population to be put into and decides the proportions

• Meet each member and put them into correct group

• Continues until all quotas (groups) are filled

• If a person refuses to be interviewed or the quota is already full then ignore the answer

Advantages of quota sampling

- No sampling frame required

- Small sample can represent whole population

- Fast, easy, cheap

- Easy comparison between groups in population

Disadvantages of quota sampling

- Non-random sampling is unrepresentative, can introduce bias

- Population must be divided into groups, which can be costly or inaccurate

- Larger sample increases number of groups which adds time and expense

- Non-responses are not recorded

Opportunity sampling

Sample taken from people who happen to be available at the time who meet the criteria

Advantages of opportunity sampling

- No sampling frame required

- Fast, easy, cheap

- Inexpensive

Disadvantages of opportunity sampling

- Non-random sampling is unrepresentative, can introduce bias

- Highly dependent on individual researcher

What is cluster sampling

• Population is split into clusters where each member of population can only be in one cluster

• Sample is taken from each cluster

• The sample taken can be using any sample technique

• Often the clusters are geographic e.g taking clusters from different parts of the UK where a particular type of bird is common

• Is usually two stage

• Can be random or non-random

Types of random sampling

- Simple random

- Systematic

- Stratified

Types of non-random sampling

- Opportunity

- Quota

Comparison of spread

IQR

Comparison of central tendancy

Median

Equation for median

(n+1)/2

How to find IQR

Q3-Q1 = IQR

3(n+1)/4 - (n+1)/4 = IQR

Upper quartile equation

3(n+1)/4

Lower quartile equation

(n+1)/4

How to find mean of data in a table

1) Find midpoint of range

2) Multiply by frequency

3) Find mean of new values

Frequency density equation

Frequency density = frequency ÷ class width

How much data 1, 2, and 3 standard deviations from the mean

1 SD = 68%

2 SD = 95%

3 SD = 99.8%

For normal distribution, how many quartile is the data split into?

4: this means each quartile has probability of 0.25

Equations to test independence

P(AandB) = P(A) × P(B)

P(A|B) = P(A|B’) = P(A)

Area of bar on a histogram

Frequency

Regression line

Line of best fit

Box plot has negative skew when...

It has a left tail

Q3 - Q2 > Q2 - Q1

Box plot has positive skew when...

It has a right tail

Q2 - Q1 > Q3 - Q2

Extrapolation

Estimating a value outside the range of measured data

CAN'T DO THIS!

Interpolation

An estimation of a value within the measured data

Range of PMCC

-1 ≤ PMCC ≤ 1

P Value

Probabibility of getting the critical value

P(X = __ ) in normal distribution

P = 0, in normal distribution X can't equal a specific number

Lower quartile equation

(n+1)/4

Conditions for binomial distribution

1. A fixed number of trials, n

2. Each trial has two possible outcomes

3. The probability of success, p, is the same for each trial

4. Each observation is independent

Mutually exclusive

Events that cannot occur at the same time - on Venn diagram the circles don't overlap

Test for mutual exclusivity

P(AorB) = P(A) + P(B) if mutually exclusive

P(A|B) =

P(A ∩ B) / P(B)

Discrete data

Data that can only take certain values, e.g shoe size.

Shown using bar charts, tally charts, pie charts.

Continuous data

Data that can take any value, e.g height.

Shown using line graph.

When can you approximate binomial distribution with normal distribution

- Probability is close to 0.5

- There is a large number of trials, n > 50

Key facts about the large data set

- 3827 cars in total

- Only five makes of car are included: Ford, BMW, Vauxhal, Toyota and VW. Ford is the most frequently registered.

- Only one electric vehicle in whole data set and only gas petrol hybrid vehicle in data set

- 5 door hatchback is the most common body type

- Data is only from a few days in summer, June, in 2002 and 2016: there is more data from 2016 than 2002

- Mass of vehicle includes 75kg driver

- Emissions data is only known for approx 80% of the whole data set: CO2, CO, NOX

- Particulate emissions are applicable to diesel cars

- Doesn’t include drivers of company cars / only shows name the car is registered to, not the driver

- Doesn’t include all regions in England, only NW, SW and London

- CO2 emissions are in 10s and 100s

- CO emissions are in decimals

- Only cars, not vans, buses etc

- Some of the categories are codes ie numbers represent different types, e.g the body type is represented by a number

Which categories in LDS are codes?

• Propulsion type: petrol, diesel, electric, gas+petrol, electric+petrol

• Body type

• Owner of car: male, female, not used, unknown, company

Conditions for normal distribution

- Data must be continuous

- 95% of the data must be within 2 standard deviations of the mean

Hypothesis for PMCC

H0: 𝑝 = 0

H1: 𝑝 > 0, 𝑝 < 0 or 𝑝 ≠ 0

In normal distribution, P(X ≠ __)

1

How to find critical region for a normal?

Use the inverse normal function with the probability equal to significance level

For a normal Hypothesis test, what is the new value for standard deviation when you change it to the sample?

standard deviation² ÷ n

How many vehicles in LDS?

3827