Statistical basics

Marian ScottDept of Statistics, University of Glasgow

August 2008

What shall we cover?

• Why might we need some statistical skills

• Statistical inference- what is it?

• how to handle variation

• exploring data

• probability models

• inferential tools- hypothesis tests and confidence intervals

Why bother with Statistics

We need statistical skills to:

•  Make sense of numerical information, 

• Summarise data,

•  Present results (graphically),

•  Test hypotheses

•   Construct models

statistical language

• variable- a single aspect of interest

• population- a large group of ‘individuals’

• sample- a subset of the population

• parameter- a single number summarising the variable in the population

• statistic- a single number summarising the variable in the sample

statistical language- Radiation protection- C-14 in fish

• variable- radiocarbon level (Bq/KgC)

• population- all fish caught for human consumption in W Scotland

• sample- 20 fish bought in local markets

• parameter- population mean C-14 level

• statistic- sample mean C-14 level

Variables- number and type

• Univariate: there is one variable of interest measured on the individuals in the sample. We may ask:

• What is the distribution of results-this may be further resolved into questions concerning the mean or average value of the variable and the scatter or variability in the results?

Bivariate

• Bivariate two variables of interest are measured on each member of the sample. We may ask :

• How are the two variables related?• If one variable is time, how does the other

variable change?• How can we model the dependence of

one variable on the other?

Multivariate

Multivariate many variables of interest are measured on the individuals in the sample, we might ask:

• What relationships exist between the variables?• Is it possible to reduce the number of variables,

but still retain 'all' the information?

Can we identify any grouping of the individuals on the basis of the variables?

Data types

• Numerical: a variable may be either continuous or discrete.

• For a discrete variable, the values taken are whole numbers (e.g. number of invertebrates, numbers of eggs).

• For a continuous variable, values taken are real numbers (positive or negative and including fractional parts) (e.g. pH, alkalinity, DOC, temperature).

categorical

• Categorical: a limited number of categories or classes exist, each member of the sample belongs to one and only one of the classes e.g. compliance status is categorical.

• Compliance is a nominal categorical variable since the categories are unordered.

• Level of diluent (eg recorded as low, medium ,high) would be an ordinal categorical variable since the different classes are ordered

Inference and Statistical Significance

Sample Population

inference

• Is the sample representative?

• Is the population homogeneous?

Since only a sample has been taken from the population we cannot be 100% certain

Significance testing

what are your objectives?

 describing a characteristic of interest (usually the average, but could also be the variability or a high percentile),

describing spatial patterns of a characteristic,mapping the spatial distribution,

quantifying contamination above a background or specified intervention level

  detecting temporal or spatial trends,

  assessing environmental impacts of specific facilities, or of events such as accidental releases,

the statistical process

• A process that allows inferences about properties of a large collection of things (the population) to be made based on observations on a small number of individuals belonging to the population (the sample).

• The use of valid statistical sampling techniques increases the chance that a set of specimens (the sample, in the collective sense) is collected in a manner that is representative of the population.

Variation

• soil or sediment samples taken side-by-side, from different parts of the same plant, or from different animals in the same environment, exhibit different activity densities of a given radionuclide.

• The distribution of values observed will provide an estimate of the variability inherent in the population of samples that, theoretically, could be taken.

Representativeness

An essential concept is that the taking of a sufficient number of individual samples should reflect the population.

Representativeness of environmental samples is difficult to demonstrate.

Usually, representativeness is considered justified by the procedure used to select the samples

What is the population?

• The population is the set of all items that could be sampled, such as all fish in a lake, all people living in the UK, all trees in a spatially defined forest, or all 20-g soil samples from a field. Appropriate specification of the population includes a description of its spatial extent and perhaps its temporal stability

What are the sampling units?

In some cases, sampling units are discrete entities (i.e., animals, trees), but in others, the sampling unit might be investigator-defined, and arbitrarily sized.

Example- technetium in shellfishThe objective here is to provide a measure (the average)

of technetium in shellfish (eg lobsters for human consumption) for the west coast of Scotland.

• Population is all lobsters on the west coast• Sampling unit is an individual animal.

• Summarising data- means, medians and other such statistics

• plotting data- histograms, boxplots, stem and leaf plots, scatterplots

median

lower quartile

upper quartile

Preliminary Analysis

• There is considerable variation – Across different sites – Within the same site

across different years• Distribution of data is

highly skewed with evidence of outliers and in some cases bimodality 2004 2005 2006 2007

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Boxplots of FS: 114567

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