elementary statistics for foresters lecture 2 socrates/erasmus program @ wau spring semester...
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Elementary statistics for foresters
Lecture 2
Socrates/Erasmus Program @ WAU
Spring semester 2005/2006
Descriptive statistics
Descriptive statistics
• Data grouping (frequency distribution)
• Graphical data presentation (histogram, polygon, cumulative histogram, cumulative histogram)
• Measures of location (mean, quadratic mean, weighted mean, median, mode)
• Measures of dispersion (range, variance, standard deviation, coefficient of variation)
• Measures of asymmetry
Descriptive statistics
• Descriptive statistics are used to summarize or describe characteristics of a known set of data.
• Used if we want to describe or summarize data in a clear and concise way using graphical and/or numerical methods.
Descriptive statistics
• For example: we can consider everybody in the class as a group to be described. Each person can be a source of data for such an analysis.
• A characteristic of this data may be for example age, weight, height, sex, country of origin, etc.
Descriptive statistics
• Closer-to-forestry example: we can consider all pine stands in central Poland as a group to be characterized.
• Each stand can be described by its area, age, site index, average height, QMD, volume per hectare, volume increment per hectare per year, amount of carbon sequestered, species composition, damage index, ...
Frequency distribution
xi nini pi
pi
468
10121416
238273452421
23105178223247249250
0,0920,3280,2920,1800,0960,0080,004
0,0920,4200,7120,8920,9880,9961,000
250 1,000
Frequency distribution
• Frequency distribution is an ordered statistical material (measurements) in classes (bins) built according to the investigated variable values
Frequency distribution
• How to build it? – determine classes (values/mid-points and class
limits), depending on variable type– classify each unit/measurement to the
appropriate class– sum units in each class
Frequency distribution
• Practical issues:– number of classes should be between 6 and 16– classes should have identical widths– middle-class values/class mid-points should be
chosen in such a way, that they are easy to manipulate
Frequency distribution
xi nini pi
pi
468
10121416
238273452421
23105178223247249250
0,0920,3280,2920,1800,0960,0080,004
0,0920,4200,7120,8920,9880,9961,000
250 1,000
Graphical description of data
• Pictures are very informative and can tell the entire story about the data.
• We can use different plots for different sorts of variables. We can use for example bar plots (histograms), pie charts, box plots, ... .
Graphical description of data
Histogram for dk
dk
freq
uenc
y
0 3 6 9 12 15 180
20
40
60
80
100
Graphical description of data
polygon
dk
freq
uenc
y
0 3 6 9 12 15 180
20
40
60
80
100
Graphical description of data
cumulative histogram
dk
freq
uenc
y
0 3 6 9 12 15 180
50
100
150
200
250
Graphical description of data
Numerical data description
Sums and their properties
ncc
xccx
yxyx
xx
ii
iiii
i
ni
ii
)(
1
1.
2.
3.
Measures of location
• Arithmetic mean
• Quadratic mean
• Weighted mean
• Median
• Mode
• other
Arithmetic mean
Quadratic mean
Properties of the mean
Weighted mean
...
Median
• If observations of a variable are ordered by value, the median value corresponds to the middle observation in that ordered list.
• The median value corresponds to a cumulative percentage of 50% (i.e., 50% of the values are below the median and 50% of the values are above the median).
Median
• The position of the median is calculated by the following formula:
Median
• How to calculate it?
• If the detailed values are available, sort the data file and find an appropriate value
• If the frequecy distribution is available, use the following formula:
Mode
• The mode is the most frequently observed data value.
• There may be no mode if no value appears more than any other.
• There may also be two (bimodal), three (trimodal), or more modes (multimodal).
• In the case of grouped frequency distributions, the modal class is the class with the largest frequency.
Mode
• If there is no exact mode available in the data file, you can calculate its value by using:– an approximate Pearson formula
– by using an interpolation
Relationship between measures
x
f(x)
μμe
μo
Relationship between measuresf(x)
μo μe μ
c3c
Relationship between measures
μ μe μo
c
3c
x
f(x)
Sample calculations
Sample calculations
Measures of dispersion
• Range
• Variance
• Standard deviation
• Coefficient of variation
Range and variance
• Range is a difference between the lowest and the highest value in the data set
• Variance– average squared differences between data
values and arithmetic mean
N
xi
2
2
Variance
N
xi
2
2
NN
xx ii
2
2
2
1
2
2
n
xxs i
1
2
2
2
nn
xx
s
ii
22222 22 Nxxxxx iiiii
N
x
N
xx
N
xNx
N
xx ii
ii
ii
i
22
2
2
2
2 22
N
xx ii
2
2
NN
xnxn iiii
2
2
2
1
2
2
2
nn
xx
s
ii
22
2
22
2 kw
iiii
N
xn
N
xn
Variance
min2 ix
222xcx c
02 c
Standard deviation and coefficient of variation
2
100
w %
Sample calculations
1950iixn 165962iixn250N
544,5250
1386
250
1521016596
250250
380250016596
250250
195016596
2
2
35,2544,5 %1,30%10080,7
35,2%100
w
Measures of asymmetry
• Skewness: is a measure of the degree of asymmetry of a distribution.
• If the left tail is more pronounced than the right tail, the function has negative skewness.
• If the reverse is true, it has positive skewness.
• If the two are equal, it has zero skewness.
Skewness
Skewness
• Skewness can be calculated as a distance between mean and mode expressed in standard deviations:
oas
Acknowledgements
• This presentation was made thanks to the support and contribution of dr Lech Wróblewski