Statistical Tests
Data Analysis
Statistics - a powerful tool for analyzing data
1. Descriptive Statistics - provide an overview of the attributes of a data set. These include measurements of central tendency (frequency histograms, mean, median, & mode) and dispersion (range, variance & standard deviation)
2. Inferential Statistics - provide measures of how well your data support your hypothesis and if your data are generalizable beyond what was tested (significance tests)
Inferential Statistics
2 4 10 4 6 8 7 10 4 3 7 9 6 7 5 2 5 8 2 107 2 3 5 2 9 3 9 6 1 4 2 6 4 9 3 4 1 8 79 1 8 1 10 10 6 4 2 7 1 1 9 10 4 4 6 6 2 59 10 2 6 8 10 1 6 10 10 4 4 4 9 2 1 4 5 9 66 2 7 8 8 6 6 10 6 6 7 5 9 2 6 4 8 6 6 105 7 1 9 1 10 8 8 5 10 1 4 8 3 6 7 1 5 2 44 10 5 8 5 1 1 4 3 6 7 3 1 5 4 3 6 2 7 83 3 6 6 2 8 6 5 9 8 4 6 3 8 3 3 10 8 10 57 5 1 4 3 2 1 10 2 10 6 10 7 9 8 8 4 9 9 103 7 6 2 1 1 10 3 5 7 4 1 2 9 10 10 6 1 3 21 3 9 9 4 2 2 2 1 8 3 1 5 9 9 8 3 2 5 44 2 3 10 8 2 3 4 1 3 3 2 10 10 5 7 3 3 10 15 7 5 1 2 5 8 7 3 8 9 2 10 8 1 1 5 3 3 76 7 9 8 8 4 9 8 4 3 10 8 10 4 10 2 3 5 6 31 9 8 1 10 2 3 1 6 3 8 9 6 2 4 4 2 7 8 44 4 4 10 8 5 9 3 10 5 3 6 9 3 7 4 2 3 10 25 1 6 8 5 6 8 1 8 5 7 6 4 1 2 7 2 9 5 38 2 3 2 9 9 1 1 5 7 8 5 6 3 8 5 4 10 6 95 1 10 10 5 1 4 3 2 3 6 9 10 2 6 3 1 2 8 61 8 7 8 5 3 7 2 4 1 8 9 10 10 5 1 3 6 5 83 3 8 8 2 7 1 6 9 8 2 10 3 7 9 2 1 9 7 73 1 9 6 8 2 6 4 6 3 7 10 9 6 1 10 7 5 3 101 6 5 4 3 2 4 4 1 5 5 10 6 2 1 1 1 5 6 38 10 8 10 9 7 7 7 8 4 8 1 3 5 8 1 8 4 4 64 7 2 4 9 1 8 5 3 3 5 10 1 4 6 3 3 8 2 2
The Population: =5.314
Population size = 500
2 4 10 4 6 8 7 10 4 3 7 9 6 7 5 2 5 8 2 10
7 2 3 5 2 9 3 9 6 1 4 2 6 4 9 3 4 1 8 7
9 1 8 1 10 10 6 4 2 7 1 1 9 10 4 4 6 6 2 5
9 10 2 6 8 10 1 6 10 10 4 4 4 9 2 1 4 5 9 66 2 7 8 8 6 6 10 6 6 7 5 9 2 6 4 8 6 6 105 7 1 9 1 10 8 8 5 10 1 4 8 3 6 7 1 5 2 44 10 5 8 5 1 1 4 3 6 7 3 1 5 4 3 6 2 7 83 3 6 6 2 8 6 5 9 8 4 6 3 8 3 3 10 8 10 5
7 5 1 4 3 2 1 10 2 10 6 10 7 9 8 8 4 9 9 10
3 7 6 2 1 1 10 3 5 7 4 1 2 9 10 10 6 1 3 21 3 9 9 4 2 2 2 1 8 3 1 5 9 9 8 3 2 5 44 2 3 10 8 2 3 4 1 3 3 2 10 10 5 7 3 3 10 1
5 7 5 1 2 5 8 7 3 8 9 2 10 8 1 1 5 3 3 7
6 7 9 8 8 4 9 8 4 3 10 8 10 4 10 2 3 5 6 31 9 8 1 10 2 3 1 6 3 8 9 6 2 4 4 2 7 8 44 4 4 10 8 5 9 3 10 5 3 6 9 3 7 4 2 3 10 25 1 6 8 5 6 8 1 8 5 7 6 4 1 2 7 2 9 5 38 2 3 2 9 9 1 1 5 7 8 5 6 3 8 5 4 10 6 9
5 1 10 10 5 1 4 3 2 3 6 9 10 2 6 3 1 2 8 6
1 8 7 8 5 3 7 2 4 1 8 9 10 10 5 1 3 6 5 8
3 3 8 8 2 7 1 6 9 8 2 10 3 7 9 2 1 9 7 73 1 9 6 8 2 6 4 6 3 7 10 9 6 1 10 7 5 3 10
1 6 5 4 3 2 4 4 1 5 5 10 6 2 1 1 1 5 6 3
8 10 8 10 9 7 7 7 8 4 8 1 3 5 8 1 8 4 4 64 7 2 4 9 1 8 5 3 3 5 10 1 4 6 3 3 8 2 2
The Sample: 7, 6, 4, 9, 8, 3, 2, 6, 1mean = 5.111
The Population: =5.314
2 4 10 4 6 8 7 10 4 3 7 9 6 7 5 2 5 8 2 107 2 3 5 2 9 3 9 6 1 4 2 6 4 9 3 4 1 8 7
9 1 8 1 10 10 6 4 2 7 1 1 9 10 4 4 6 6 2 5
9 10 2 6 8 10 1 6 10 10 4 4 4 9 2 1 4 5 9 66 2 7 8 8 6 6 10 6 6 7 5 9 2 6 4 8 6 6 10
5 7 1 9 1 10 8 8 5 10 1 4 8 3 6 7 1 5 2 4
4 10 5 8 5 1 1 4 3 6 7 3 1 5 4 3 6 2 7 83 3 6 6 2 8 6 5 9 8 4 6 3 8 3 3 10 8 10 57 5 1 4 3 2 1 10 2 10 6 10 7 9 8 8 4 9 9 10
3 7 6 2 1 1 10 3 5 7 4 1 2 9 10 10 6 1 3 2
1 3 9 9 4 2 2 2 1 8 3 1 5 9 9 8 3 2 5 44 2 3 10 8 2 3 4 1 3 3 2 10 10 5 7 3 3 10 15 7 5 1 2 5 8 7 3 8 9 2 10 8 1 1 5 3 3 7
6 7 9 8 8 4 9 8 4 3 10 8 10 4 10 2 3 5 6 3
1 9 8 1 10 2 3 1 6 3 8 9 6 2 4 4 2 7 8 4
4 4 4 10 8 5 9 3 10 5 3 6 9 3 7 4 2 3 10 2
5 1 6 8 5 6 8 1 8 5 7 6 4 1 2 7 2 9 5 3
8 2 3 2 9 9 1 1 5 7 8 5 6 3 8 5 4 10 6 95 1 10 10 5 1 4 3 2 3 6 9 10 2 6 3 1 2 8 61 8 7 8 5 3 7 2 4 1 8 9 10 10 5 1 3 6 5 83 3 8 8 2 7 1 6 9 8 2 10 3 7 9 2 1 9 7 73 1 9 6 8 2 6 4 6 3 7 10 9 6 1 10 7 5 3 10
1 6 5 4 3 2 4 4 1 5 5 10 6 2 1 1 1 5 6 3
8 10 8 10 9 7 7 7 8 4 8 1 3 5 8 1 8 4 4 6
4 7 2 4 9 1 8 5 3 3 5 10 1 4 6 3 3 8 2 2
The Sample: 1, 5, 8, 7, 4, 1, 6, 6mean = 4.75
The Population: =5.314
Parametric or Non-parametric?
•Parametric tests are restricted to data that: 1) show a normal distribution 2) * are independent of one another 3) * are on the same continuous scale of measurement
•Non-parametric tests are used on data that: 1) show an other-than normal distribution 2) are dependent or conditional on one another 3) in general, do not have a continuous scale of measurement
e.g., the length and weight of something –> parametric vs. did the bacteria grow or not grow –> non-parametric
The First Question
After examining your data, ask: does what you're testingseem to be a question of relatedness or a question ofdifference?
If relatedness (between your control and your experimentalsamples or between you dependent and independent variable), you will be using tests for correlation (positive or negative) or regression.
If difference (your control differs from your experimental),you will be testing for independence between distributions,means or variances. Different tests will be employed ifyour data show parametric or non-parametric properties.
See Flow Chart on page 50 of HBI.
Tests for Differences
• Between Means - t-Test - P - ANOVA - P - Friedman Test - Kruskal-Wallis Test - Sign Test - Rank Sum Test• Between Distributions - Chi-square for goodness of fit - Chi-square for independence• Between Variances - F-Test – PP – parametric tests
Differences Between Means
Asks whether samples come from populations with different means
Null Hypothesis Alternative Hypothesis
A
Y
B CA
Y
B C
There are different tests if you have 2 vs more than 2 samples
Differences Between Means – Parametric Data
t-Tests compare the means of two parametric samples
E.g. Is there a difference in the mean height of men and women?
HBI: t-Test
Excel: t-Test (paired and unpaired) – in Tools – Data Analysis
A researcher compared the height of plants grown in high and low light levels. Her results are shown below. Use a T-test to determine whether there is a statistically significant difference in the heights of the two groups
Low Light High Light49 4531 4043 5931 5840 5544 5049 4648 5333 43
Differences Between Means – Parametric Data
ANOVA (Analysis of Variance) compares the means of two or more parametric samples.
E.g. Is there a difference in the mean height of plants grown under red, green and blue light?
HBI: ANOVA
Excel: ANOVA – check type under Tools – Data Analysis
weight of pigs fed different foods
food 1 food 2 food 3 food 4
60.8 68.7 102.6 87.9
57.0 67.7 102.1 84.2
65.0 74.0 100.2 83.1
58.6 66.3 96.5 85.7
61.7 69.8 90.3
A researcher fed pigs on four different foods. At the endof a month feeding, he weighed the pigs. Use an ANOVAtest to determine if the different foods resulted indifferences in growth of the pigs.
Aplysia punctata – the sea hare
Aplysia parts
Differences Between Means – Non-Parametric Data
The Sign Test compares the means of two “paired”, non-parametric samples
E.g. Is there a difference in the gill withdrawal response of Aplysia in night versus day? Each subject has been tested once at night and once during the day –> paired data.
HBI: Sign Test
Excel: N/A
SubjectNight
ResponseDay
Response1 2 52 1 33 2 2
The Friedman Test is like the Sign test, (compares the means of “paired”, non-parametric samples) for more than two samples.
E.g. Is there a difference in the gill withdrawal response of Aplysia between morning, afternoon and evening? Each subject has been tested once during each time period –> paired data
HBI: Friedman Test
Excel: N/A
SubjectMorning
ResponseAfternoon Response
Evening. Response
1 4 3 22 5 2 13 3 4 3
Differences Between Means – Non-Parametric Data
The Rank Sum test compares the means of two non-parametric samples
E.g. Is there a difference in the gill withdrawal response of Aplysia in night versus day? Each subject has been tested once, either during the night or during the day –> unpaired data.
HBI: Rank Sum
Excel: N/A
SubjectNight
ResponseDay
Response1 52 13 24 35 46 17 5
Differences Between Means – Non-Parametric Data
The Kruskal-Wallis Test compares the means of more than two non-parametric, non-paired samples
E.g. Is there a difference in the gill withdrawal response of Aplysia in night versus day? Each subject has been tested once, either during the morning, afternoon or evening –> unpaired data.
HBI: Kruskal-Wallis Test
Excel: N/A
Differences Between Means – Non-Parametric Data
SubjectMorning
ResponseAfternoon Response
Evening. Response
1 42 53 44 35 26 3
Chi square tests compare observed frequency distributions, either to theoretical expectations or to other observed frequency distributions.
Differences Between Distributions
Differences Between Distributions
E.g. The F2 generation of a cross between a round pea and a wrinkled pea produced 72 round individuals and 20 wrinkled individuals. Does this differ from the expected 3:1 round:wrinkled ratio of a simple dominant trait?
HBI: Chi-Square One Sample Test (goodness of fit)Excel: Chitest – under Function Key – Statistical
Smooth
Fre
quen
cy
Wrinkled
E
E
E.g. 67 out of 100 seeds placed in plain water germinated while 36 out of 100 seeds placed in “acid rain” water germinated. Is there a difference in the germination rate?
HBI: Chi-Square Two or More Sample Test (independence)Excel: Chitest – under Function key - Statistical
Plain Acid PlainP
ropo
rtio
nG
erm
inat
ion
Acid
Pro
port
ion
Ger
min
atio
n
Null HypothesisAlternative Hypothesis
Differences Between Distributions
Correlations look for relationships between two variables which may not be functionally related. The variables may be ordinal, interval, or ratio scale data. Remember, correlation does not prove causation; thus there may not be a cause and effect relationship between the variables.
E.g. Do species of birds with longer wings also have longer necks?
HBI: Spearman’s Rank Correlation (NP)
Excel: Correlation (P)
Correlation
Question – is there a relationship between students aptitude for mathemathics and for biology?
Student Math score Math Rank Biol. score Biology rank
1 57 3 83 7
2 45 1 37 1
3 72 7 41 2
4 78 8 84 8
5 53 2 56 3
6 63 5 85 9
7 86 9 77 6
8 98 10 87 10
9 59 4 70 5
10 71 6 59 4
Regressions look for functional relationships between two continuous variables. A regression assumes that a change in X causes a change in Y.
E.g. Does an increase in light intensity cause an increase in plant growth?
HBI: Regression Analysis (P)
Excel: Regression (P)
Regression
Correlation & Regression
Looks for relationships between two continuous variables
Null Hypothesis Alternative Hypothesis
X
Y
X
Y
Is there a relationship between wing length and tail length in songbirds?
wing length cm tail length cm
10.4 7.4
10.8 7.6
11.1 7.9
10.2 7.2
10.3 7.4
10.2 7.1
10.7 7.4
10.5 7.2
10.8 7.8
11.2 7.7
10.6 7.8
11.4 8.3
Is there a relationship between age and systolic blood pressure?
Age (yr)
systolic blood pressuremm hg
30 108
30 110
30 106
40 125
40 120
40 118
40 119
50 132
50 137
50 134
60 148
60 151
60 146
60 147
60 144
70 162
70 156
70 164
70 158
70 159