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. AP Biology Quantitative Skills Manual

Descriptive statistics serves to summarize the data 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. AP Biology Quantitative Skills Manual

Inferential statistics involves inferring parameters in the natural population from a sample. AP Biology Quantitative Skills Manual

Most of the data you will collect will fit into two categories: measurements or counts. AP Biology Quantitative Skills Manual Measurement data Count data

Most measurements are continuous, meaning there is an infinite number of potential measurements over a given range. AP Biology Quantitative Skills Manual

Count data are recordings of qualitative, or discrete, data. AP Biology Quantitative Skills Manual Number of leaf stomata Number of white eyed individuals

Conducting Data Analysis AP Biology Quantitative Skills Manual

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 AP Biology Quantitative Skills Manual

If the data show an approximate normal distribution on a histogram, then they are parametric data. AP Biology Quantitative Skills Manual

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. AP Biology Quantitative Skills Manual

For parametric data (a normal distribution), the appropriate descriptive statistics include : sample size the mean (average) variance standard deviation standard error AP Biology Quantitative Skills Manual

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. AP Biology Quantitative Skills Manual

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. AP Biology Quantitative Skills Manual

The mean (x)of the sample is the average 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. AP Biology Quantitative Skills Manual

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. AP Biology Quantitative Skills Manual Distance from the mean Variance

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. AP Biology Quantitative Skills Manual

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. AP Biology Quantitative Skills Manual

We can use standard deviations to predict locations of data along a normal distribution. A little more than two-thirds (68%) of the data points will fall between +1 standard deviation and −1 standard deviation from the sample mean. AP Biology Quantitative Skills Manual

68–95–99.7 Rule http://en.wikipedia.org/wiki/68%E2%80%9395%E2%80%9399.7_rule AP Biology Quantitative Skills Manual 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. AP Biology Quantitative Skills Manual

Standard error should be represented by error bars on graphs Standard error should be represented by error bars on graphs. Error bars are used on graphs to indicate the uncertainty of a reported measurement. Watch out for overlap. This indicates data that are very similar. AP Biology Quantitative Skills Manual

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 AP Biology Quantitative Skills Manual

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. AP Biology Quantitative Skills Manual 5, 1, 7, 3, 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 AP Biology Quantitative Skills Manual 3.5 median

The mode is the value that appears most frequently in a data set. 3, 5, 1, 3, 7, 2 AP Biology Quantitative Skills Manual

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)

Let’s apply this: Question- Do shady English ivy leaves have a larger surface area than sunny English ivy leaves? AP Biology Quantitative Skills Manual

Since the data collected is in centimeters, it is measurement data, not count data. So the first step is to make a: AP Biology Quantitative Skills Manual HISTOGRAM

Do the data resemble a normal curve? AP Biology Quantitative Skills Manual (Close enough, with possible differences due to sampling error)

Next, the appropriate statistical tools are applied: AP Biology Quantitative Skills Manual

A bar graph can then be produced to compare the means: AP Biology Quantitative Skills Manual

Do the error bars for the shady leaf mean overlap with the error bars for the sunny leaf mean? AP Biology Quantitative Skills Manual

Because the error bars do not overlap, there is a high probability that the two populations are indeed different from each other. AP Biology Quantitative Skills Manual

Another Example of Data Analysis: Question- Is 98 Another Example of Data Analysis: Question- Is 98.6°F actually the average body temperature for humans? The data are actually from a sample data set prepared by Allen Shoemaker (Shoemaker, 1996). This particular data set has been modified from the results of a study published in the Journal of American Medical Association (Mackowiak, Wasserman, and Levine, 1992).

Since the data collected are in Farenheit, they are measurement data, not count data. So the first step is to make a: AP Biology Quantitative Skills Manual HISTOGRAM

Do the data resemble a normal curve? AP Biology Quantitative Skills Manual (Close Enough)

Next, the appropriate statistical tools are applied: AP Biology Quantitative Skills Manual *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 (plus or minus 0.73 degrees). AP Biology Quantitative Skills Manual

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. AP Biology Quantitative Skills Manual

Now you try it. Complete the practice sheet and then collect and analyze your own class data. AP Biology Quantitative Skills Manual