1. DATA ANALYSIS FOR RESEARCH PROJECTS

2. TYPES OF DATA Quantitative data
measurements use scale with equal intervals
examples include mass (g), length (cm),
volume (mL), temperature (oC or K)
Qualitative data
non-standard scales with unequal intervals or
discrete categories
examples include gender, choice, color scales

3. Quantitative Scales of Measure
Properties
Example
Interval
(equal)
Numerical value indicates rank and meaningfully reflects relative distance between points on a scale
Temperature (oC or oF)
Ratio
Has all the properties of an interval scale, and in addition has a true zero point. (proportional scale)
Length
Weight
Temperature (K)

4. Qualitative Scales of Measure
Properties
Example
Nominal
(to name)
Data represents qualitative or equivalent categories (not numerical, cannot be rank ordered).
Eye color, hair color
Gender
Race
Ordinal
(to order)
Numerically ranked, but has no implication about how far apart ranks are.
Grades
Rating Scales

5. Sample Data
An experiment was conducted to measure the tensile strength of each of twelve pieces of two types of steel. The data from this experiment are given in the table to the right.
Is there a significant difference in tensile strength between the two types of steel?

6. Is there a better way to compare the data from these groups?
What have you used before to compare data from two different groups?

7. It is difficult to decide (consistently) whether differences between experimental groups are significant
We need a rigorous procedure that includes a clear operational definition of dissimilarity.

8. Statistics & Statistical Analysis
Statistical hypothesis-testing methods give us the ability to say with confidence that differences between groups are real and not just due to random chance, sampling errors, or other mistakes in data collection.

9. Sample data for consideration…
For the following sets of data, discuss:
What was the IV and DV tested?
How should the data be processed to determine if the IV affects the DV?
How will you decide if the IV has a significant effect on the DV?

10. Sample Data Set 1 Effect of Temperature on the pressure of a sample of gas above water
Temperature of Water (oC)
Pressure (mmHg) 50 90 55
120 60
145 65
180 70
219 75
264 80
310

11. Graphing data
Correlation coefficient gives a measure of how strong the relationship is between the graphed variables.
Multiple trials can and should all be analyzed at the same time.

12. Sample Data Set 2 Effect of Stress on the Height of Bean Plants after 30 Days
Stressed Plants (cm)
Unstressed Plants (cm)
55.0
48.0
65.0
50.0
59.0
57.0
51.0
73.0
63.0
54.0
58.0
62.0
44.0
68.0

13. Comparing levels of IV
If graphing the data is not appropriate, the different groups of the IV can be compared.
These types of statistics are called “Descriptive Statistics” since they:
describe the data sets
summarize groups of measurements

14. Descriptive Statistics:
Measure of Central Tendency
attempt to provide one value that is most typical of the entire set of data
What are some examples of measures of central tendency?
Variation
describes the spread within the data set
* two sets of data with the same mean may have quite different spread within the data

15. Frequency Distribution
Appropriate Measures of Central Tendency and Variations for Types of Data
QUANTITATIVE
DATA
QUALITATIVE
Central
Tendency
Measurement
Mean, Median
or Mode
Nominal
Ordinal
Mode
Median
Variation
Standard Deviation Or
Range
Frequency Distribution

16. What is “standard deviation”???
The standard deviation is a statistic that tells you how tightly all the various examples are clustered around the mean in a set of data. This relates the variation in a set of data.
When the data points are pretty precise (close to the mean, little variation), the bell-shaped curve is steep, and the standard deviation is small.
When there is greater variation in the data, the bell curve is relatively flat. that tells you you have a relatively large standard deviation.

17. Displaying variation: Box-and-Whisker Plot
SMALLEST
VALUE
FIRST
QUARTILE
MEDIAN
THIRD QUARTILE
LARGEST
First Quartile (Q1) – smaller than 75% of ranked values
Median (Q2) – smaller than 50% and larger than 50%
Third Quartile (Q3) – smaller than 25% of ranked values

18. Illustrating Distributions for qualitative data: Histograms
Symmetrical – mean equals median
Left-skewed – mean < median
Right-skewed – mean > median

19. Statistical Hypothesis Testing
“A trend is apparent in the graph of the data, is this trend significant?”
“So the means of the groups are different, is the difference significant?”
Statistical hypothesis testing is needed to determine the significance in the results of your data analysis.
The results of these tests provide “Inferential Statistics.” We make inferential decisions based on the data we collect from a sample population.

20. Sample Data Effect of Stress on the Height of Bean Plants after 30 Days
Stressed Plants (cm)
Unstressed Plants (cm)
55.0
48.0
65.0
50.0
59.0
57.0
51.0
73.0
63.0
54.0
58.0
62.0
44.0
68.0

21. Example for comparing means: t Test for Quantitative Data
Equal Sample Size
t =
= mean of Group 1
= mean of Group 2
= variance of Group 1
= variance of Group 2
= number of items or measurements

22. Statistical calculations
Use the TI-84 or TI-83 calculator OR
Use Microsoft Excel Data Analysis
Calculate the t-test for the stressed plants data on the next slide, using the graphing calculator

23. Level of Significance Establish a level of significance
In this class, use 0.05.
this means the probability of error in
rejecting the null hypothesis is 5/100 OR
we can be 95% confident that the null
hypothesis may be rejected

24. Results from the calculator
t: value for the t-test
x1: mean from List 1
x2: mean from List 2
Sx1: standard deviation for List 1
Sx2: standard deviation for List 2
df: degrees of freedom
n1: number of values in List 1
n2: number of values in List 2

25. t-Test Results from Excel

26. Statistical Hypotheses (different from your research hypothesis)
Null Hypothesis
suggests any observed difference between two sample means occurred by chance and is NOT significant
state that there is no relationship between variables: i.e. two means are equal OR they are not statistically different
Claim / Alternative Hypothesis
derived from literature, research hypothesis
suggests outcome of experiment if I.V. affects D.V.

27. Null Hypothesis
What would be the null hypothesis for this set of data?
The mean height of stressed plants is not significantly different from the mean height of unstressed plants.

28. Confidence Levels Probability that findings are repeatable
Infers that results of sample are the same as results of the whole population
If we reject the null hypothesis at 95% confidence level:
95% certainty that difference between groups is NOT due to chance
95% certainty that results will be the same with further testing

29. Confidence levels
Probablity of error: Error that occurs if null hypothesis is rejected when it is true and should not be rejected
Identified by Greek lowercase alpha, a
Researchers usually select a < 0.05
If confidence level is 95%, then probability of error (a) is 5%, or 0.05

30. Statistical Tests: Test Values and Critical Values
Test value – the result of a statistical test on your data.
Critical value – this is a reference value for each statistical test.
Your calculated statistical test value must exceed this value for you to reject the null hypothesis
You can find the critical value for each statistical test in publications and university websites. (links available on my website)
If you use Microsoft Excel for your statistics, the critical value will be given with the results.

31. Significance of t value
Determine the degrees of freedom
df = (number in experimental group – 1) + (number in control group – 1)
df = (10 – 1) + (10 – 1) = 18
Determine significance of calculated t by looking at table for critical t values
Calculated t < critical t  not significant
Calculated t > critical t  is significant
At df = 18, t = 2.101;
Calculated t of 1.24 < and is not significant at 0.05 level.

32. Rejecting Null Hypothesis
If test value is not significant 
null hypothesis is NOT REJECTED
If test value is significant 
null hypothesis is REJECTED

33. Do Statistical Findings Support the Research Hypothesis?
Null hypothesis was rejected =
Research hypothesis was supported
(unless research hypothesis IS a null hypothesis)
Null hypothesis was not rejected =
Research hypothesis was not supported

34. Summary: Steps of Hypothesis Testing
State the null hypothesis and alternative hypothesis (claim)
Choose the confidence level (95%) and sample size
Collect the data and calculate the appropriate statistics
Make the proper statistical inference

35. Populations of Study – Be careful what you claim!
Sample
specific portion of the population that is selected for the study ( 100 bean seedlings used in the study)
Sampled Population
population from which the sample was drawn (all the bean seedlings in the nursery from which the experimenter obtained their bean seedlings)
Target Population
ALL units (persons, things, experimental outcomes) of the specific group whose characteristics are being studied (all the bean seedlings of the same species)

36. Communicating Statistics Effect of Stress on the Mean Height of Bean Plants after 30 Days
Stressed Group
Unstressed Group
Mean
Variance
Standard Deviation
1SD
2SD
Number
60.0 cm
49.1 cm
7.0 cm
53.0 – 67.0 cm
46.0 – 74.0 cm 10
56.0 cm
60.7 cm
7.8 cm
48.2 – 63.8 cm
40.4 – 71.6 cm
Results of t test
t = df = 18
t of 1.3 < p > 0.10

38. Types of Tests For Quantitative Data: For Qualitative Data:
Linear Regression
One-Way Analysis of Variance (ANOVA)
t Test
For Qualitative Data:
Chi-Squared Test
Z Test

39. Linear Regression
Determines a linear relationship between two variables based on a correlation coefficient
H0: The number of yellow M&M’s is not related to the total number of M&M’s in the package.

40. ANOVA Test Compares the means of more than two groups
H0: There is no significant difference between the numbers of M&M’s in plain packages, almond packages and peanut packages

41. t-Test Compares the means of two independent groups
H0: There is no significant difference between the numbers of M&M’s in plain and peanut packages
Two-tail test determines if populations are not equal / the same (more difficult to support)
One-tail test determines if one mean is greater than the other (easier to support)

42. Chi-Squared Test
Determines if a proportion within a sample is larger than expected; can be used for more than two groups
H0: There are equal numbers of each color of M&M in a package.

43. Z-Test Compares proportions between two groups
H0: There are equal proportions of red M&M’s in plain and peanut packages

44. Selecting a Statistical Test
Things to consider:
Number of groups of data
Type of data: Quantitative or Qualitative
Type of variable – numerical or categorical
The relationship in the null hypothesis being tested

45. Statistical Tests Review
Comparison of two variables for correlation  correlation coefficient test
Comparing means of more than two groups/levels  ANOVA test
Comparing two means  t-test
Comparison of proportions within a population  X2 (chi-squared) test
Comparison of proportions between populations  Z test

46. Key Questions for your Research:
What kind of data will you need to collect to test your hypothesis? (Qualitative or Quantitative)
What kind of scale will you use?
How do you plan on analyzing this data?
Comparison of groups? What will you compare?
Look for a trend? What will you graph?
How many different levels will you need data for?
How many trials?
What relevant qualitative data will you look for that may also help you interpret results?