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Quantitative Skills:
Data Analysis
Data analysis is one of the first steps
toward determining whether an
observed pattern has validity. Data
analysis also helps distinguish among
multiple working hypotheses.
Descriptive statistics serves to
summarize the data. It helps show the
variation in the data, standard errors,
best-fit functions, and confidence that
sufficient data have been collected.
Inferential statistics involves inferring
parameters in the natural population
from a sample.
Most of the data you will collect will fit into
two categories: measurements or counts.
Measurement data
Count data
Most measurements are continuous,
meaning there is an infinite number of
potential measurements over a given
Count data are recordings of
qualitative, or discrete, data.
Number of leaf stomata
Number of white eyed
How much is good enough?
• How much data should a researcher collect to
make a claim with confidence? How big
should the size of the sample be?
• Is it possible the results were due to chance
instead of the manipulation of the variable
being tested?
Conducting Data Analysis
When an investigation involves
measurement data, one of the first steps is
to construct a histogram, or frequency
diagram, to represent the data’s
If the data show an approximate
normal distribution on a histogram,
then they are parametric data.
If the data do not show an approximate
normal distribution on a histogram, then
they are nonparametric data. Different
descriptive statistics and tests need to be
applied to those data.
Sometimes, due to
sampling bias, data
might not fit a normal
distribution even when
the actual population
could be normally
distributed. In this
case, a larger sample
size might be needed.
For parametric data (a normal
distribution), the appropriate
descriptive statistics include :
the mean (average)
sample size
standard deviation
standard error
The mean (x)of the sample is the average.
The mean summarizes the entire sample
and might provide an estimate of the
entire population’s true mean.
The sample size (n)
refers to how many
members of the
population are
included in the study.
Sample size is
important when
estimating how well
the sample set
represents the entire
Variance (s2) and standard deviation (s)
measure how far a data set is spread out. A
variance of zero indicates that all the values in a
data set are identical.
Distance from the mean
Because the differences from the mean are
squared to calculate variance, the units of
variance are not the same units as in the
original data set. The standard deviation is
the square root of the variance. The
standard deviation is expressed in the same
units as the original data set, which makes
it generally more useful than the variance.
A small standard deviation indicates that
the data tend to be very close to the mean.
A large standard deviation indicates that
the data are very spread out away from the
A little more than two-thirds of the data points
will fall between +1 standard deviation and −1
standard deviation from the sample mean.
More than 95% of the data falls between ±2
standard deviations from the sample mean.
68–95–99.7 Rule
In a normal distribution, 68.27% of all values lie within one
standard deviation of the mean. 95.45% of the values lie
within two standard deviations of the mean. 99.73% of the
values lie within three standard deviations of the mean.
Sample standard error (SE) is a statistic
used to make an inference about how
well the sample mean matches up to
the true population mean.
Standard error should be represented by
including error bars on graphs when
appropriate. Error bars are used on graphs to
indicate the uncertainty of a reported
Different statistical tools are used in the
case of data that does not resemble a
normal distribution (nonparametric data,
or data that is skewed or includes large
box-and-whisker plots
The median is the value separating the
higher half of a data sample from the
lower half. To find the median of a data
set, first arrange the data in order from
lowest to highest value and then select
the value in the middle.
5, 1, 3, 7, 2
1, 2, 3, 5, 7
If there are two values in the middle of
an ordered data set, the median is
found by averaging those two values.
5, 1, 3, 7, 4, 2
1, 2, 3, 4, 5, 7
The mode is the value that appears
most frequently in a data set.
3, 5, 1, 3, 7, 2
3 is the mode in this example
because it appears more
frequently than any other
A bimodal distribution
Data Analysis Flowchart:
Type of Data
Measurement Data
Count Data
· Make histogram
(normal distribution)
standard deviation,
standard error
(not a normal
Median, mode,
Example of Data Analysis:
Do shady English ivy leaves
have a larger surface area
than sunny English ivy
Since the data collected is in centimeters, it
is measurement data, not count data. So
the first step is to make a:
Does the data resemble a normal
(Close enough, with possible differences due to sampling error)
Next, the appropriate statistical tools are
A bar graph can then be produced to
compare the means:
Do the error bars for the shady leaf
mean overlap with the error bars for
the sunny leaf mean?
A more rigorous statistical test will need to
be performed, but because the error bars
do not overlap there is a high probability
that the two populations are indeed
different from each other.
Example of Data Analysis:
Is 98.6°F actually the average body
temperature for humans?
Since the data collected is in Farenheit,
it is measurement data, not count
data. So the first step is to make a:
Does the data resemble a normal
(Close Enough)
Next, the appropriate statistical tools are
that by convention, descriptive statistics rounds
the calculated results to the same number of decimal
places as the number of data points plus 1.
According to the 68–95–99.7 Rule, 68% of
all samples lie within one standard
deviation from the mean. This means that
around 68% of the temperatures should be
between 97.51 and 98.99.
Including the standard error, we can
say with a 68% confidence that the
mean human body temperature of our
sample is 98.25 ± 0.06°F.