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 :

• 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).

• 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

median

3.5

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(Continuous)

· Make histogram

Parametric(normal distribution)

Mean, standard deviation,

standard error

Nonparametric(not a normal distribution)

Median, mode, quartiles

Count Data(Discrete)

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.

• Qualitative data is not numerical and is usually subjective.

• Quantitative data is numerical and lends itself to statistical analysis.

Categories of data:

1.75 mL

• 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.

Quantitative data can be either discrete or continuous.

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.

• Used when data on both scales of the graph (the x and y axes) are continuous.

• The dots indicate measurements that were actually made.

Line Graphs

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

-or-

Study Time vs. Score on a Biology Exam in 2011

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.

Basic Traits of A Good Graph

DependentVariable

IndependentVariable

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.

m =y y2 – y1

x x2 – x1

=

Slope = RiseRun

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.

• 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.

Population vs. Sample

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?).

Parametric Data(normal data)

Nonparametric Data

Frequency Data

(counts)

Bar Graph Box-and-Whisker PlotBar Graph

orPie Chart

Association statistics look for associations between variables (How

are A and B correlated?).

Parametric Dataand

Nonparametric Data

Scatterplot

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® BiologyInvestigative Labs:

An Inquiry-Based Approachand

AP® BiologyQuantitative Skills:

A Guide for Teachers