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