1
A critical reader's brief guide to statistics
Gregory S. Gilbert
Environmental Studies, UC Santa Cruz
Statistics are simply formal, mathematical ways to (1)
DESCRIBE
observations about
groups of things (e.g., to describe the size of a population of animals), and to (2)
TEST
for
trends, patterns, or differences among groups. (e.g., to see if plants grow larger as soils
become more fertile (a trend) or to see if more rats died after eating a particular chemical
than after eating a placebo (a difference)).
Let’s say we want to describe
the frog population in Walden
Pond. We go out and catch 60
frogs and weigh them (Table 1).
Some frogs are bigger, some
smaller. We can illustrate the
variability in the population
looking at the frequency
DISTRIBUTION
of the frog’s
weights (Table 2) and making a
graph (Figure 1). This is
usually a frequency graph, or
histogram, showing the number
of frogs in each weight class (a
weight class might be all the
frogs from 00.4 g, or from 2.0
2.4 g, and so on) distribution,
that is, the population will tend
to have most of its members
clustered around some particular
weight, with some a little heavier and some a little
lighter, and even fewer much heavier or much
lighter. We can describe this central tendency in
several ways. One common statistic is the
MEDIAN
 this is the value where one half (50%)
of the frogs are heavier and 50% of the frogs are
lighter (median =1.8 g). Another common statistic
is the
MEAN
(in common usage often called the
average). The mean is calculated by adding up the
weights of all the frogs and dividing by the
number of frogs (mean = 1.75 g). This can be
thought of as the weight of the “typical” frog.
When the distribution of the population
approximates a bellshaped (“normal”) curve, the
median and the mean are very similar. Sometimes
the distribution has a longer tail in one direction
than the other  in these cases the median and
mean can be very different. But the weight of the
“typical” frog is only part of the useful
information. We also want to know how variable
the population is. One measure of the spread of the distribution is the
VARIANCE
, which
is a measure of how far away the members of the population are from the mean. The square
Table 1. Weights of
Table 2. Frequency
frogs in Walden pond.
of
weights of frogs.
Frog #
Weight
(g)
Weight
class (g)
Num.
of frogs
1
2.1
00.4
2
2
3.0
0.40.8
3
3
1.8
0.81.2
7
4
2.5
1.21.6
11
5
3.2
1.62.0
15
6
1.4
2.02.4
10
7
1.8
2.42.8
8
.
.
2.83.2
4
.
.
.
.
57
2.5
58
2.2
59
1.9
60
3.1
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Fig. 2. Number of frogs by weight class in Walden
and Crystal Ponds (
N
=60 in each).
root of the variance is called the
STANDARD DEVIATION
, and is very often presented
along with the mean (and the sample size,
n
, (number of frogs caught)) to describe the
population (mean = 1.75 ± 0.7 g,
n
=60. Translation: Of the 60 frogs measured, the mean
frog weighed 1.75 grams, with a variation (standard deviation) around that mean of 0.7 g).
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 Fall '08
 Zavaleta,E
 Statistics, Null hypothesis, Statistical hypothesis testing, Walden Pond

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