Statistics - YEAR1

What is a census?

This observes or measures every member of a population.

What is a sample?

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

Advantages of census

• It should give a completely accurate result.

Advantages of sample

• Less time consuming and expensive than a census.

• Fewer people have to respond.

• Less data to process than in a census.

Disadvantages of census

• Time consuming and expensive.

• Cannot be used when the testing process destroys the item.

• Hard to process large quantity of data

Disadvantages of sample

• The data may not be as accurate.

• The sample may not be large enough to give information about small sub-groups of the population.

What are sampling units?

Individual units of a population

What is a sampling frame?

When sampling units are individually named or numbered to form a list.

What is random sampling?

This is when every member of the population has an equal chance of being selected. The sample should therefore be representative of the population. This helps to remove bias from a sample.

Three methods of random sampling

• Simple random sampling

• Systematic sampling

• Stratified sampling

What is simple random sampling

This is where every sample of the ‘item’ has an equal chance of being selected

How to carry out simple random sampling

• You need a sampling frame, usually a list of people/things.

• Each person/thing is allocated a unique number and a selection of these numbers is chosen at random.

• There are two methods of choosing the numbers: generating random numbers and lottery sampling.

• In lottery sampling, the members of the sampling frame could be written on tickets and placed into a ‘hat’. The required number of tickets would then be drawn out.

advantages of simple random sampling

• 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

disadvantages of simple random sampling

• A sampling frame is needed.

• Not suitable when the population size or the sample size is large as it is potentially time consuming, disruptive and expensive.

What is systematic sampling

This is where the required elements are chosen at regular intervals from an ordered list.

advantages of systematic sampling

• Simple and quick to use

• suitable for large samples and large populations

disadvantages of systematic sampling

• a sampling frame is needed.

• it can introduce bias if the sampling frame is not random

What is stratified sampling?

This is where the population is divided into mutually exclusive strata (like males and females) and a random sample is taken from each.

the number sampled in a stratum = (number in stratum/number in population) x overall 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

Two types of non-random sampling

• Quota sampling

• Opportunity sampling

What is quota sampling

This is when an interviewer or researcher selects a sample that reflects the characteristics of the whole population

advantages of quota sampling

• 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

disadvantages of quota sampling

• 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

What is opportunity sample

This 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.

advantages of opportunity sampling

• easy to carry out

• inexpensive

disadvantages of opportunity sampling

• unlikely to provide a representative sample

• highly dependent on individual researcher

What is quantitative data/variables

variables/data associated with numerical observations. (shoe size)

what is qualitative data/variables

variables/data associated with non-numerical observations. (hair colour)

what is a continuous variable

a variable that can take any value in a given range (time)

what is a discrete variable

a variable that can take only specific values in a given range (number of girls in a family)

What are classes

when data is presented in a grouped frequency table, the groups are called classes.

What does:

• Class boundaries

• The midpoint

• The class width

tell you?

• class boundaries tell you the maximum and minimum values that belong in each class

• the midpoint is the average of the class boundaries

• the class width is the difference between the upper and lower class boundaries

what is a measure of location

a single value which describes a position in a data set.

when is mode the best measure to use?

This is used when data is qualitative or quantitative with either a single mode or two modes. it is not very informative if each value occurs only once

when is median the best measure to use?

this is used for quantitative data and it is usually used when there are extreme values as they do not affect it

when is mean the best measure to use?

this is used for quantitative data and uses all the pieces of data. it therefore gives a true measure of the data. however, it is affected by extreme values.

what is a measure of spread

this is a measure of how spread out the data is

what is coding

this is a way of simplifying statistical calculations. each data value is coded to make a new set of data values which are easier to work with

what is an anomaly

this is when an outlier should be removed from the data since it is clearly an error and it would be misleading to keep it in.

they can be a result of experimental or recording error, or it could be data values which are not relevant to the investigation

what is a histogram and what does it represent

it gives a good picture of how the data is distributed and it enabled you to see a rough location, the general shape and how spread out the data is

it represents grouped continuous data

what is bivariate data

data which has pairs of values for two variables and can be represented on a scatter diagram

what is correlation

this describes the nature of the linear relationship between two variables

what is an experiment

a repeatable process that gives rise to the number of outcomes

what is an event

a collection of one or more outcomes

what is a sample space

a set of all possible outcomes

what is a mutually exclusive event

when events have no outcomes in common

what is a tree diagram used for

showing the outcomes of two or more events happening in succession

what is a random variable

a variable whose value depends on the outcome of a random event

what is a sample space

the range of values that a random variable can take

what is a discrete variable

the variable is discrete if it can only take certain numerical values

what is a random variable

the variable is random if the outcome is not known until the experiment is carried out

what is a probability distribution

this describes the probability of any outcome in the sample space

what is a hypothesis

this is a statement made about the value of a population parameter

• the null hypothesis is the hypothesis you assume to be correct

• the alternative hypothesis tells you about the parameter if your assumption is shown to be wrong

how to carry out a hypothesis test

• you assume the null hypothesis is true then consider how likely the observed value of the test statistic was to occur

• if this likelihood is less than a given threshold, the significance level of the test, then you reject the null hypothesis

what is a test statistic

this is the result of the experiment or the statistic that is calculated from the sample

what is a critical region

a region of the probability distribution which, if the test statistic falls within it, would cause you to reject the null hypothesis

what is a critical value

the first value to fall inside of the critical region

when is a one tailed test used

used to test when it is claimed that the probability has either gone up, or gone down

when is a two tailed test used

used when it is thought that the probability has changed in either direction