DATA ANALYSIS FOR RESEARCH PROJECTS

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DATA ANALYSIS FOR RESEARCH PROJECTS
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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

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

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

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• 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.

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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.

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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?

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Sample Data Set 1Effect 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
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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.

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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 65.0
50.0 59.0
57.0 57.0
59.0 51.0
73.0 63.0
57.0 65.0
54.0 58.0
62.0 44.0
68.0 50.0
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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

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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

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

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Displaying variationBox-and-Whisker Plot

• 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

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Illustrating Distributions for qualitative data
Histograms

• Symmetrical mean equals median
• Left-skewed mean lt median
• Right-skewed mean gt median

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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.

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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 65.0
50.0 59.0
57.0 57.0
59.0 51.0
73.0 63.0
57.0 65.0
54.0 58.0
62.0 44.0
68.0 50.0
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Example for comparing meanst 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

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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

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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

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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

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t-Test Results from Excel
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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.

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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.
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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

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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 lt 0.05
• If confidence level is 95, then probability of
  error (a) is 5, or 0.05

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Statistical TestsTest 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.

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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 lt
critical t ? not significant Calculated t gt
critical t ? is significant
At df 18, t 2.101 Calculated t of 1.24 lt
2.101 and is not significant at 0.05 level.
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Rejecting Null Hypothesis

• If test value is not significant ?
• null hypothesis is NOT REJECTED
• If test value is significant ?
• null hypothesis is REJECTED

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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

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SummarySteps of Hypothesis Testing

1. State the null hypothesis and alternative
   hypothesis (claim)
2. Choose the confidence level (95) and sample size
3. Collect the data and calculate the appropriate
   statistics
4. Make the proper statistical inference

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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)

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Communicating StatisticsEffect 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 10
Results of t test t 1.3 df 18 t of 1.3 lt 2.101 p gt 0.10 t 1.3 df 18 t of 1.3 lt 2.101 p gt 0.10
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Types of Tests

• For Quantitative Data
• Linear Regression
• One-Way Analysis of Variance (ANOVA)
• t Test
• For Qualitative Data
• Chi-Squared Test
• Z Test

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Linear Regression

• Determines a linear relationship between two
  variables based on a correlation coefficient
• H0 The number of yellow MMs is not related to
  the total number of MMs in the package.

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ANOVA Test

• Compares the means of more than two groups
• H0 There is no significant difference between
  the numbers of MMs in plain packages, almond
  packages and peanut packages

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t-Test

• Compares the means of two independent groups
• H0 There is no significant difference between
  the numbers of MMs 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)

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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 MM
  in a package.

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Z-Test

• Compares proportions between two groups
• H0 There are equal proportions of red MMs
  in plain and peanut packages

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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

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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

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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?