The Statistical Analysis of Data

Outline I. Types of Data A. Qualitative B. Quantitative C. Independent vs Dependent variables II. Descriptive Statistics III. Graphical Representation of Data IV. Analyzing the Data - Statistical Significance V. Statistical Analysis Methods VI. The Scientific Method at a Glance

I. Types of Data A. Qualitative Data Describes a non-numerical characteristic Examples - Yes or No Green or Red Tall or Short B. Quantitative Data – MUCH MORE USEFUL Describes data in a numerical form Examples – numbers, measurements, percentages, averages

I. Types of Data C. Types of Variables 1. Independent Variable The factor that varies independently from the experiment. Dependent Variable The factor being influenced by the independent variable. Examples ….. Experiments seek to determine if an independent variable affects a dependent variable

I. Types of Data Examples Effects of light intensity on plant growth Independent - light intensity Dependent - plant growth Effects of Color on Betta reaction Independent - Color Dependent - Betta reaction Effects of Dive height on Kestrel CE Independent - Dive height Dependent - Capture Efficiency

I. Types of Data 2. Key Principle - Test only one independent variable at a time. Extra variables are called: Confounding variables … Why? 3. When Graphing, place the independent variable on the X axis (horizontal), the dependent varable on the y axis (vertical)

II. Descriptive Statistics A. Purpose – To represent data simply, with numbers that describe characteristics of the data B. Methods Average – The mean Median – The middle number Mode – The most frequent number Range – Endpoints of the data (Lowest and highest numbers

II. Descriptive Statistics Variance – How spread out are the numbers? How much do they vary from the mean? Variance = S 2 = Summation (X-mean) 2 Total number of data points Standard Deviation – Square Root of Var. Symbolized as S This is used more often when comparing the spread of data.

III. Graphical Representation of Data A. Purpose – To represent data visually. A graph should allow one to interpret the data more quickly…. To SEE a pattern and interpret a trend. B. Data is initially recorded in a Data Table

Sample Data Tables American Kestrel Predatory Behavior

1. The Bar Graph - Great for data in categories. Often called discrete data

2. The Line Graph - Can be used for a variety of both discrete categorical data (qualitative) and continuous quantitative data

3. A Scatter Plot - Great for quantitative data. Especially continuous data with two points

Comparing Two Dependent Variables

4. A Pie Chart - Great for percentages

IV. Analyzing the Data - Introduction to Statistical Significance A. Key Question - Is the effect you discovered great enough to have only been caused by the independent variable you are testing? Statistically Significant OR - is the effect small enough to have been caused simply by chance? Statistically Insignificant

Example Coin Flips (1,000 times) 557 heads, 443 tails Probability states 50% / 50% (hypothesis) What caused this variation? Some variation will occur by chance This variation may be small enough to be explained by chance. This variation is statistically insignificant

Example Coin Flips (1,000 times) What if heads and 300 tails Variation from Probability is much greater Couldn’t be chance This variation IS Statistically Significant Another variable must be acting here

B. Why statistical tests? Statistical analysis tests will tell you How significant the difference is Is the difference significant enough that we can be reasonably sure that chance isn’t the cause?

B. Why statistical tests? Which of the four sets of data below shows a significant effect of color? Stats tests will tell you where to draw the line

C. Statistical Significance and Hypotheses Every experiment is usually comparing Two hypotheses The Null Hypothesis The treatment or Independent Variable has no significant effect The Alternate Hypothesis The treatment or Independent variable has a significant effect, and it is …..

C. Statistical Significance and Hypotheses Red = 53% reaction, Blue =51% reaction Difference insignificant Null hypothesis supported Red = 70% reaction, Blue = 19% reaction Difference IS Significant Reject Null hypothesis (no difference) Accept Alternate hypothesis that Red color is more likely to elicit a response

V. Statistical Analysis Methods A. Chi Square Analysis B. T- Test C. ANOVA (Analysis of Variance) D. Regression Analysis E. Correlation

A. Chi Square Analysis 1. Purpose - To determine if data differs significantly from expected results X 2 = (O-E) 2 E

A. Chi Square Analysis 2. Example

A. Chi Square Analysis 3. Interpretation Computer or calculation will give you a Chi Square number. If this is higher than the critical value, the difference is significant

3. Interpretation Finding the critical value Consult a Chi-square table Use a df (degrees freedom) n-1 (number of categories -1) Use a p-value The maximum probability we will allow for the difference to be caused by pure chance. Standard P=.05 Cross index df with p to get critical

3. Interpretation Chi-Square = 47.9 df = 4 p=0.05 Critical value = 9.49 Chi-square 47.9 > critical 9.49 Therefore, these differences are sign. Ie. Could not have been caused by chance alone.

B. T- Test 1. Purpose - To determine if the data from two different groups is significantly different. 2. The computer generates a T-statistic 3. Interpretation Compare T- statistic to T critical value. T stat must be > T crit. For significant difference

B. T- Test 4. Finding the T critical on a table Cross index p value (0.05) with df (N-2), Sample size Example and computer analysis

C. ANOVA - Analysis of Variance 1. Purpose - Testing for significance difference between multiple samples of data. Use for more than two sets.

ANOVA - F value is not > F critical. Therefore, the treatment did not cause a significant difference Notice - P-value = meaning 73% assurance that the differences were caused by chance

ANOVA - Does this show a significant difference?

D. Regression Analysis 1. Purpose - Fits a trend line to a scatterplot of points. The r value tells how closely the data fits the trend. The closer to 1, the stronger the trend Excel will give you an R 2 value Take the square root the get r.

E. Correlation 1. Purpose - Enables you to determine how much two variables are related. 2. Correlation coefficient is calculated 3. Represented by “r”. The closer r is to a perfect 1, the stronger the correlation.

VI. The Scientific Method at a Glance A. Observation Make initial observations Survey past studies B. Hypothesis State null hypothesis and any alternate hypotheses you will be testing

VI. The Scientific Method at a Glance C. Experiment 1. Test one independent variable at a time What dependent effect are you measuring for the variable? Test color? Keep all others the same Measure which dependent V? Reaction time? Reaction intensity?

C. Experiment 2. Be sure to have a control. A test without the independent variable, or a normal condition to compare to. If independent has one treatment Ex. The patient gets the drug Use a control Ex. A patient doesn’t get drug

C. Experiment If Independent has many treatments Color Red, Blue, black, white They are controls for eachother Compare to one another.

C. Experiment 3. Try to concentrate on quantitative data 4. Use the largest possible sample size! D. Collect Data Data tables Graph data Analyze data

E. Conclusion (Discussion of a Lab) What does the data say? Is the null hypothesis supported or an alternate hypothesis? Why or why not? What other studies are needed?