your Summarize data!
P o p ula tio nve rs us s a m p le
Population: The entire group of
individuals in which we are
interested but cant usually assess
directly.
Example: All urban
ecosystems, all Douglas-fir
trees in Oregon, all fish
Sample: The part of the population
we actually examine and for which
we do have data.
How well the sample represents
the population depends on the
sample design.
Population
Sample
A parameter is a number
describing a characteristic of the
population.
A statistic is a number describing
a characteristic of a sample.
T o wa rd s ta tis tic a linfe re nc e
The techniques of inferential statistics allow us to draw inferences or
conclusions about a population in a sample.
Your estimate of the population is only as good as your sampling
design.
Your sample is only an estimateand if you randomly sampled
again you would probably get a somewhat different result.
Population
Sample
Population vs. Sample
Population:
Timber harvested watersheds
Sample
Sample
size
1
2
4
8
Sampling unit, randomization, independence
N=160
Measure of central tendency:
mean and median
The mean and the median are the same only if the distribution is
symmetrical. The median is a measure of center that is resistant to
skew and outliers. The mean is not.
Mean and median for
Mean
a symmetric
distribution
distribution
Mean
Median
Mean and median
for skewed
distributions
distributions
Left skew
Mean
Median
Mean
Median
Right skew
Measure of dispersion and variability
Measureofspread:
Varianceandstandard deviation
1. First calculate the variance s2.
1n
s2 =
( xi x ) 2
n 1 1
2. Then take the square root to get
the standard deviation s.
x
Mean
1 s.d.
1n
s=
( xi x ) 2
n 1 1
Measureofspread:Range
Measureofspread:Range
Uncertaintyandconfidence
Uncertaintyandconfidence
Although the sample mean, X , is a unique number for any particular
sample, if you pick a different sample you will probably get a different
sample mean.
In you fact, could get many different values for the sample mean, and
virtually none of them would actually equal the true population mean, .
Uncertaintyandconfidence
Uncertaintyandconfidence
Standard deviation (/n) of sample means = standard error (SE) of the mean
n
95% of all sample
means will be within
roughly 2 standard
deviations (2*/n or
2SE) of the population
parameter .
Red dot: mean value
of individual sample
Implications
Implications
We dont need to take a lot of
random samples to rebuild the
sampling distribution and find
at its center.
n
All we need is one SRS of
Sample
n
Population
size n and relying on the
properties of the sample
means distribution to infer
the population mean .
Reworded
Reworded
With 95% confidence, we can say
that should be within roughly 2
standard deviations (2*/n) from
our sample mean X bar.
In 95% of all possible samples
of this size n, will indeed fall
in our confidence interval.
In only 5% of samples would
be farther from .
n
A confidence interval can be expressed as:
Mean m
m is called the margin of error
within X m
Example: 120 6
A confidence level C (in %)
indicates the probability that
the falls within the interval.
It represents the area under
the normal curve within m
of the center of the curve.
m
m
Howtocalculateconfidenceinterval(CI)?
CI=t/2,df X Standard Error of the mean
t: students t distribution
: significance level (5%) for 95%CI
df: degree of freedom (sample size-1)
Howtocalculateconfidenceinterval(CI)?
CI=t/2,df X Standard Error of the mean
Using our own data (Trees):
Mean=13.16
n=160
t=1.97
se=standard deviation/n=11.34/12.64=0.90
95% CI=1.97 X 0.9 = 0.18
95% CI: 13.16 + 0.18 (12.98, 13.34)
(1)The estimate of the population
is only as good as your sampling design
How many samples do you need?
(2) The sample is only an estimateand
if you randomly sample again you would
probably get a somewhat different result.
How well your mean is estimated?