Download Standard Error of the Mean % 95% Confidence Interval

Descriptive Statistics
and
Standard Error
Standard Deviation
s=
å( x - x )
2
i
n -1
This is on the formula chart!
Let’s find the standard deviation and other
descriptive statistics for this list of numbers which
represent the density of trichomes on fast plants.
8,11,9,10,8,6
Standard Deviation= Average distance from the mean
s=
å( x - x )
2
i
n -1
8,11,9,10,8,6
First we need the mean!
Mean
8.7
Some other descriptive stats
8,11,9,10,8,6
s=
å( x - x )
2
i
n -1
So the mean is 8.7
What is the median?8.5
Put the numbers in order. It’s the middle
one. Average the two middle ones if even.
6, 8, 8, 9, 10, 11
Still Descriptive Stats
8,11,9,10,8,6
s=
å( x - x )
2
i
n -1
So the mean is 8.7
The median is 8.5
Mode? 8 (Most common number)
Range? 11-6= 5 (One number)
6, 8, 8, 9, 10, 11
Standard Deviation (s)
• Standard deviation is a measure of how the
data points are spread around the mean.
• This is the average distance from the mean.
Standard Deviation
SD is small
Data is close to
mean
SD is large
Data is spread out
widely from the mean
Standard Deviation
• In a normal distribution, 68% of all values lie within +/- 1 SD
of the mean
• In a normal distribution 95% of all values lie within +/- 2 SD of
the mean
Mean
68%
95%
Descriptive Stats: Standard Deviation
8,11,9,10,8,6
Mean 8.7- let’s
round up to 9.
s=
å( x - x )
2
i
n -1
Xi is each of the values (6 here).
Σ is the “sum of.”
s is the standard deviation of the sample.
n is the number of data points (6)
Calculating Standard Deviation
8,11,9,10,8,6
Mean is 9
s=
å( x - x )
Standard Deviation “s” = 1.7 Round to 2
i
n -1
2
Complete for this set “Pop 2”
12, 6, 15, 9, 13, 8
s=
å( x - x )
Mean?
Median?
Mode?
Range?
Standard Deviation?
i
n -1
2
Answers for “Pop 2”
12, 6, 15, 9, 13, 8
s=
å( x - x )
2
i
n -1
Mean: 10.5
Sameness is
coincidence
Median:10.5
Mode: None
Range:15-6= 9
Standard Deviation: 3.4 (round to 3)
Pop 3:
13,17,9,14,12,16
From Penny Smeltzer
Westwood High School
Let’s graph the means of the 3 populations
Mean number of trichomes/cm2
16
14
What about error
bars?
12
10
Trichomes
per cm2
These graphically
show how variable the
data is
8
6
They could show
range, standard
deviation and more.
4
2
0
Pop 1
Pop 2
Pop 3
Mean number of trichomes/cm2
Showing +/-1
Standard Deviation
Standard Error (Inference)
• What does it mean?
• If this is +/- 1 SE then there is
a 68% chance the true mean
lies within the SE bars
• If the error bars are +/- 2SE
there is a 95% chance the true
mean lies within the SE bars.
Mean number of trichomes/cm2
For SE
+/-1 SE
Trichomes
per cm2
• Most common way to report a sample mean is
the mean +/- SE
• This gives an idea of how well the sample
mean estimates the population mean.
No overlap between SE of Pop1 and 3
Mean number of trichomes/cm2
+/- 2 SE
Trichomes
per cm2
There is no overlap
between the 95%
confidence intervals
for Pop 1 and 3
CI=+/-2SE
This is a significant
difference: Likely that
the population means
are different
Standard Error of the Mean
&
95% Confidence Interval
Standard Error of the Mean
*Remember: Standard deviation is a measure of
the spread (or average) of the data from the
mean.
• The sample mean is not necessarily identical
to the mean of the entire population. Means
will vary with different samples from the same
population.
• This variability can be expressed by calculating
the standard error of the mean (SEM)
Example
• Assume that there is a population of a species
of geckos living on an island of the Caribbean.
If you were able to measure the length of the
hind limbs of every individual in this
population and then calculate the mean, you
would know the value of the population
mean.
However
• There are thousands of individuals, so you
take a sample of 10 geckos and calculate the
mean hind limb length for that sample.
Another researcher working on that island
might catch another sample of 10 geckos and
calculate the mean hind limb length for this
sample, and so on. The sample means of
many different samples would be normally
distributed. So……
The standard error of the
mean represents the
standard deviation of such
a distribution and
estimates how close the
sample mean is to the
population mean.
What does the SEM tell you?
• SEM tells you that about two-thirds (68.3%) of
the sample means would be within +1
Standard error of the population mean and
95.4% would be within +2 standard errors.
95% Confidence Interval (CI)
• Used for large sample sizes
2s
√n
***If error bars are +2 SE, there is a 95% chance
that the true mean lies within the error bars.
AP Test Connection
• If you have to graph the mean of a data set,
you should always use error bars!!!