Background statistics
ANU

 Background statistics
Forest Inventory.

Statistics may be defined as numerical facts that are systematically collected. It may also be used to refer to the science of collecting, classifying, summarising and using these numerical facts.

Statistics can be used to describe a forest population in a consistant and repeatable manner - these are called descriptive statistics. When numerical facts are collected via a probability-based system (ie the probability of selecting an object for counting or measuring is known) then the science of statistics also allows predictions about the forest population to be made - these are called inferential statistics. Inferential statistics are generally based on a measurements from a subset (or sample) from a population of interest.
Mean The mean is descriptive statistic that summarises the centre of a range of values. An unbiased estimate of the mean size or value of a population is one of the most common objectives in a forest inventory.

A mean is defined as:

where:
  • xi denotes the size of individual i and
  • n the number of individuals measured. If all the individuals in a population were measured, then the mean is called the population mean, but if only a subset of individuals were measured, then it is a sample mean.

The mean is sensitive to extreme (minimum and maximum) values, but is probably the most common statistic reported from a forest inventory.
Mode The mode is a descriptive statistic that identifies the value (in a list of values) with the highest frequency. It is commonly used when the variables observed are ordinal or nominal (eg presence or absence, classifications of damage or habitat type).

The mode may not be a unique value, for example when two or more values occur the same number of times.

Median The median is descriptive statistic that summarises the middle of a range of values - half of the values are less than the median while half are above.

The data values are sorted into ascending (or descending order) and then the median is selected:
  • If there are n observations, and n is an odd number, then the median is (n+1)/2 ordered value.

  • If there are n observations, and n is an even number, then the median is the average of the n/2 and (n+1)/2 ordered values.
The median is an effective statistic to represent the middle of a distribution and it is not very sensitive to extreme (minimum or maximum) values.
Variance
Standard Deviation 		The variance and the standard deviation are descriptive statistics that indicate the spread of individual observations around the mean. In forest inventory, the usual measure of this spread is the standard deviation:


where:
  • xi denotes the size of individual i and
  • n the number of individuals measured.
The standard deviation is the square root of the variance, ie variance =s^2.

The sample estimate of the mean and standard deviation can be used to make inferences about a population. For example, in a normal population, 68% of all individuals will be within s of the mean, i.e. 68% of the population lie within the range of most of the time. 95% of individuals will be within 2 times the standard deviation of the mean, almost all (99%) will be within 3 times the standard deviation of the mean.

If the population is not normally distributed, then at least the fraction will lie within k*s of the mean value.

Coefficient of variation The coefficient of variation is a description of the relative variation in a population. You would expect, for example, that the range of heights in a stand of tall trees would be greater than the range in a stand of small trees. The coefficient of variation (CV) removes this noise by expressing the variation as a percentage of the mean value:


The CV for volume/ha or stand basal area in a well managed plantation may be about 40%, while in a natural / native forest, the CV may exceed 100%.
Sample size The required output from many inventories is an unbiased estimate of the mean size or value of a parameter with an indication of the precision or reliability of that estimate. Thus, the inventory designer must be able to provide an estimate of the mean value and an estimate of the possible range of values that would occur if another sample of the same size were taken. The range due entirely to the sample taken, is termed the sampling error. Often the desired precision of the inventory is determined in terms of the sampling error. The number of samples required to meet this sampling error is calculated as:

where:
  • E denotes the maximum sampling error desired,
  • t the student t value (about 1 or 2 for relatively large samples and with probability levels p=0.33 and 0.05 respectively), and
  • n the number of samples required.


Standard Error (of an estimate)

The standard error is a descriptive statistic that indicates the spread of population variables around their own [Mean]. That is, if you are estimating the mean of a population using a sample of n (randomly chosen) individuals, then you could get a range of possible samples means from different samples of n. The standard error of the mean is estimated as

Se^2 = S^2 / n


where:


Just as the sample estimate of the mean and standard deviation can be used to make inferences about the spread os individuals around the population mean, the standard error can be used to make inferences about the spread of all possible sample means around the true mean.. For example, for large sample sizes (n>30), 68% of all possible sample means will be withing Se of the population mean. Again, 95% of the possible samples will be within 2 times the standard error of the population mean, and almost all (99%) will be within 3 times the standard error of the mean.
Activity
Try your skill at interpreting these basic statistics for real forest populations.

[stats.htm] Revision: 6/2000
Cris.Brack@anu.edu.au