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Quantitative Skills:
Data Analysis and Graphing.
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.
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
range.
Count data are recordings of
qualitative, or discrete, data.
Number of leaf stomata
Number of white eyed
individuals
When an investigation involves
measurement data, one of the first steps is
to construct a histogram, or frequency
diagram, to represent the data’s
distribution
If the data show an approximate
normal distribution on a histogram,
then they are parametric data
(normal).
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 :
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•
•
•
•
the mean (average)
sample size
variance
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
population.
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.
Variance
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
mean.
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
measurement.
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
outliers).
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•
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median
mode
quartiles
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
median
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
3.5
median
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
number.
A bimodal distribution
Data Analysis Flowchart:
Type of Data
Measurement Data
Count Data
(Continuous)
(Discrete)
· Make histogram
Parametric
(normal distribution)
Mean,
standard deviation,
standard error
Nonparametric
(not a normal
distribution)
Median, mode,
quartiles
Example of Data Analysis:
Do shady English ivy leaves
have a larger surface area
than sunny English ivy
leaves?
Since the data collected is in centimeters, it
is measurement data, not count data. So
the first step is to make a:
HISTOGRAM
Does the data resemble a normal
curve?
(Close enough, with possible differences due to sampling error)
Next, the appropriate statistical tools are
applied:
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?
(No.)
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:
HISTOGRAM
Does the data resemble a normal
curve?
(Close Enough)
Next, the appropriate statistical tools are
applied:
*Note
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.
Categories of data:
• Qualitative data is
not numerical and
is usually
subjective.
• Quantitative data is
numerical and
lends itself to
statistical analysis.
1.75 mL
Quantitative data can be either discrete
or continuous.
• Discrete data has
finite values, such as
integers, or bucket
categories such as
“red” or “tall”.
• Continuous data has
an infinite number of
values and forms a
continuum.
Which graph shows continuous data
and which graph shows discrete data?
Graph A
Graph B
One of the first steps in data analysis is to
create graphical displays of the data. Visual
displays can make it easy to see patterns
and can clarify how two variables affect
each other.
Line Graphs
• Used when data on both
scales of the graph (the
x and y axes) are
continuous.
• The dots indicate
measurements that
were actually made.
Basic Traits of A Good Graph
1. A Good Title
• A good title is one
that tells exactly
what information
the author is
trying to present
with the graph.
Relation Between Study Time and
Score on a Biology Exam in 2011
-orStudy Time vs. Score on a Biology
Exam in 2011
Basic Traits of A Good Graph
2. Axes should be
consistently
numbered.
3. Axes should
contain labels,
including units.
Basic Traits of A Good Graph
6. The
independent
variable is
always shown
on the x axis.
7. The dependent
variable is
always shown
on the y axis.
Dependent
Variable
Independent
Variable
Extrapolation is a prediction of what the
chart might look like beyond the measured
set of data. A broken line is used,
indicating this a prediction and not data
actually collected.
The slope of a line indicates the rate at
which the variables being graphed are
changing.
y
m=
x
=
y2 – y1
x 2 – x1
Rise
Slope =
Run
Positive Slope
Negative Slope
Zero Slope
Rate Increasing
Rate Decreasing
Constant Rate
Indicates
some values
were skipped
Line charts can be plotted with multiple
data sets, allowing for better comparison.
Makes
use of a
legend
Effective graphs use statistics as an
essential part of the display. Statistics is
the study of the collection, organization,
analysis, interpretation and presentation of
data.
Population vs. Sample
• Often, researchers want
to know things about a
population (N), but it
may not be feasible to
obtain data for every
member of an entire
population.
• A sample (n) is a
smaller group of
members of a
population selected to
represent the
population. The sample
must be random.
Descriptive statistics and graphical
displays allow us to estimate how well
sample data represent the true
population.
If a sample is not collected randomly, it
may not closely reflect the original
population. This is called sampling bias.
A normal distribution, also known as a
“bell curve” or “normal curve”, can be
formed with continuous data.
The type of data being collected during an
investigation should be determined before
performing the actual experiment. The
type of data will determine the statistical
analyses that can be used.
Three Types of Data:
• Parametric data: data that fit
a normal curve
• Nonparametric data: data
that do not fit a normal curve
• Frequency or count data:
generated by counting
Normal or parametric data
• Measurement data that fit a normal curve or
distribution.
• Data is continuous, generally in decimal form.
Nonparametric data
• Do not fit a normal distribution, may include
large outliers, or may be count data that can
be ordered.
• Can be qualitative data.
Frequency or count data
• Generated by counting how many of an item
fit into a category.
• Can be data that are collected as percentages.
Two Types of Descriptive Statistics:
• Comparative statistics: compare
variables
• Association statistics: look for
correlations between variables
Comparative statistics compare
phenomena, events, or populations (Is
A different from B?).
Bar Graph
Box-and-Whisker Plot
Parametric Data
(normal data)
Nonparametric
Data
Bar Graph
or
Pie Chart
Frequency
Data
(counts)
Association statistics look for
associations between variables (How
are A and B correlated?).
Scatterplot
Parametric Data
and
Nonparametric
Data
Types of graphs commonly used with
the three data types and suggested
statistical tests:
Bar Graphs
• Used to visually compare two samples of
categorical or count data.
• Are also used to visually compare the
calculated means with error bars of normal
data .
Sample standard error bars (also known as
the sample error of the sample mean) are
the notations at the top of each shaded bar
that shows the sample standard error (SE).
Scatterplots
• Used when
comparing one
measured variable
against another.
• Used when looking
for trends.
If the relationship is thought to be linear, a
linear regression line or best fit line can
be plotted to help define the pattern.
Box-and-Whisker Plots
• Allow graphical comparison of two samples of
nonparametric data (data that do not fit a
normal distribution).
In a box-and-whisker graph, the ticks at the tops and
bottoms of the vertical lines show the highest and
lowest values in the dataset, respectively. The top of
each box shows the upper quartile, the bottom of
each box shows the lower quartile, and the horizontal
line represents the median.
Histograms (Frequency Diagrams)
• Used to display the distribution of data,
providing a representation of the central
tendencies and the spread of data.
Creating a histogram requires setting up
bins — uniform range intervals that cover
the entire range of the data. Then the
number of measurements that fit in each
bin are counted and graphed.
If the data on a histogram show an
approximate normal distribution, then
these are parametric data. If the data do
not approximate a normal distribution then
they are nonparametric data.
References:
AP® Biology
Investigative Labs:
An Inquiry-Based Approach
and
AP® Biology
Quantitative Skills:
A Guide for Teachers