Transcript normal distribution - Livingston Public Schools

Basic Statistics for
Scientific Research
Statistics involves math and calculations of numbers. It also guides
how numbers are chosen, interpreted, and compared to other numbers.
Consider the following three scenarios and the interpretations based
upon the presented statistics. You will find that the numbers may be
right, but the interpretation may be wrong. Try to identify a major flaw
with each interpretation before we describe it.
1) A new advertisement for Ben and Jerry's ice cream
introduced in late May of last year resulted in a 30%
increase in ice cream sales for the following three months.
Thus, the advertisement was effective.
A major flaw is that other things also affect ice cream consumption
and these other variables were not controlled or accounted for. Ice
cream consumption generally increases in the months of June, July,
and August regardless of advertisements. This effect is called a
history effect and leads people to interpret outcomes as the result of
one variable when another variable (in this case, the passage of
time) is actually responsible.
Statistics involves math and calculations of numbers. It also guides
how numbers are chosen, interpreted, and compared to other numbers.
Consider the following three scenarios and the interpretations based
upon the presented statistics. You will find that the numbers may be
right, but the interpretation may be wrong. Try to identify a major flaw
with each interpretation before we describe it.
2) The more churches in a city, the more crime there is.
Thus, churches lead to crime.
A major flaw is that other things also affect crime rates and
these other variables were not controlled or accounted for. Both
increased churches and increased crime rates can be explained
by larger populations. In bigger cities, there are both more
churches and more crime. This problem is an eg of the thirdvariable problem. Namely, a third variable can cause both
situations; however, people erroneously believe that there is a
causal relationship between the two primary variables rather
than recognize that a third variable can cause both.
Statistics involves math and calculations of numbers. It also guides
how numbers are chosen, interpreted, and compared to other numbers.
Consider the following three scenarios and the interpretations based
upon the presented statistics. You will find that the numbers may be
right, but the interpretation may be wrong. Try to identify a major flaw
with each interpretation before we describe it.
3) You measured the fall time of an object repeatedly and
experimentally determined the acceleration due to gravity
to be 9.3 m/s2. Does your result agree with the accepted
value of 9.8 m/s2?
NO MEASUREMENT IS EXACT because there are random
errors in every measurement. Therefore, every measured
value really represents a range of values (9.3 ± uncertainty).
Without knowledge of the uncertainty in the average value,
no comparison between the 2 results can be made. How
close is close enough for agreement?
Statistics is not just
mathematics and calculations
of numbers. It is also a guide
to interpreting those numbers.
VALIDITY OF A CONCLUSION
Almost every research problem investigates a
relationship between variables. There are basically 2
possible conclusions (and either could be wrong)
1. There is NO relationship (NULL hypothesis):
(alternatively, there is a relationship but it could have been
missed or not seen because it is so weak/infrequent, not
enough data collected)
2. There is a relationship: (alternatively, there is no
relationship, but things could have been seen that were not
really there)
Things that improve conclusion validity
- Statistics (collect more info, larger sample size)
- Be aware of assumptions made in data analysis
- Make more precise, less noisy measurements
How is Science Practiced?
Ask questions (science) OR define a problem (engineering)
Develop or use a model
(expected)
Plan and carry out an
experimental investigation
Analyze and interpret data using
mathematical and computational
techniques
Compare
experimental results and
expected model to
construct explanations
(science)
OR design solutions
(engineering)
Engage in argument (make conclusions) from evidence
Communicate information
Experiment to do in-class
Question – Does stretching before
exercising improve performance?
Research Question – Does
stretching out legs before a wall sit
increase the amount of time it can
be done for?
Wall-Sit Experiment: Experimental Design
Research Question: Does stretching legs before a wall-sit
increase the length of time that it can be done for?
Variables (what and how measured)
Independent variable: stretching legs vs not stretching legs.
This is a categorical variable. It is not measured with a
continuous number value. Stretching treatment is leg and
quad stretches done for 2min.
Dependent Variable: Length of time in a wall sit measured
with a stop watch. A “wall-sit” is done like this: stand with
your back against a wall and lower yourself until your
thighs are parallel to the floor. Hold the position
as long as possible.ust design e
Time is called when the subject stands or sits.
Wall-Sit: Experimental Design
CONTROLS – How will we know if the treatment works?
For this experiment, we are going to divide the class into two
groups:
- Control group (no PreTreatment of stretching)
- PreTreatment group (leg stretching before wall sit).
• How can we minimize bias in the allocation of groups?
• Should we let people pick their own group?
CONSTANTS – How do we eliminate other
sources of variation?
- Both groups should start the experiment at
the same time
Wall-Sit Experiment: Analysis
Research Question:
Does stretching legs before a wallsit increase the length of time that
it can be done for?
http://www.timeanddate.com/stopwatch
• How could we display the data?
• How can we quantify the effect of the
treatment?
Descriptive Statistics
Descriptive statistics can be used to summarize
and describe a single variable at a time (UNIvariate
analysis). There are 3 major characteristics of a
single variable (found by repeating the measurement)
1. The distribution (frequencies, percentages)
2. The central tendency (mean, median)
3. The dispersion or spread around the central
tendency (standard deviation, range)
One Way to Summarize Data is a BOX AND WHISKERS PLOT
Class 1
Class 2
Test scores Test scores
95
80
100
77
90
65
70
75
85
65
68
84
80
93
81
65
77 Order
70 the data
72
90
60
65
73
84
62
100
82
95
75
91
73
Class 2
Class 1
100
95
93
90
85
84
81
80
80
77
75
70
68
65
65
Max
Q3: median
of upper
part
Median
Q1: median
of lower part
Min
100
95
91
90
84
82
77
75
73
73
72
70
65
65
62
60
Max
Q3: median
of upper
part
Median
Q1: median
of lower part
Min
Class 2
scores
100
95
93
90
85
84
81
80
80
77
75
70
68
65
65
Max
100
95
91
90
Q3
84
75%
82
77
Median 75
50% 73
73
72
70
Q1
65
25%
65
62
Min
60
BOX and Whiskers Plots
Max
Q3
75%
Median
Q1
25%
Min
Interquartile Range (Q1 to Q3)
(50% of data is in this range)
Class1
scores
100
Max
90
Q3
80
Median
70
Q1
Min
60
Class1 Class2
100
Max
90
Q3
80
Median
70
Q1
Min
60
Class1 Class2
BOX and Whiskers
Plots of Test
Performance in Two
Classes
How confident are we in
saying that Class 1 did better
than Class2?
Is the 6% difference due to
chance or is it due to some
factor that is different in the
two classes?
In other words, are the
means statistically different
from each other?
Wall-Sit Experiment
Research Question:
Does stretching legs before a wall-sit
increase the length of time that it can
be done for?
• Summarize the data by calculating the max, Q3, median,
Q1 and min values for the wall sit times with and without
stretch pretreatment. Display in a well organized table
• Make a box and whiskers plot of wall times with and
without stretch pretreatment.
• Does stretching preTreatment increase wall sit time?
• Do you think the increase with preTreatment is
significant? Explain.
In most research, data analysis involves three major
steps, done in roughly this order:
• Data Preparation - Logging, cleaning and
organizing the data for analysis
• Descriptive Statistics – numbers used to
summarize and describe data. Together with
graphical analysis, they form the basis of almost
every quantitative analysis of data. With descriptive
stats, you are simply describing what the data shows
• Inferential Statistics - Testing Hypotheses and
Models. Conclusions from inferential stats extends
beyond the immediate data (sample) and tries to
infer more general conclusions (population)
Types of Statistics/Analyses
Descriptive Statistics – describes basic features of data.
Simplifies large amounts of data with single indicators
How many? How much?
– Means (central tendency)
How variable? How
– standard deviation/variance
uncertain?
(dispersion around mean)
BP, HR, BMI, IQ, etc.
– Frequencies/percentages
(distribution)
Inferential Statistics – goes beyond the immediate
data to make inferences about population phenomena
–
–
–
–
–
Hypothesis Testing
Correlation
Confidence Intervals / T-tests
Significance Testing
Prediction
Proving or disproving theories
Associations between
phenomena
If sample relates to the larger
population
E.g., Diet and health
Descriptive Statistics used to present quantitative descriptions
in a manageable form. In a research study there may be lots
of measures. Descriptive statistics helps simplify large amounts
of data in a sensible way. Each descriptive statistic reduces lots
of data into a simpler summary.
Using a single indicator for
Examples of Descriptive Stats a large set of observations
can distort the original data
or loose important detail
- Batting average –summarizes
how well batter performs. The
single number describes a large
number of discrete events.
- Grade Point Average (GPA)-single
number that describes the
general performance of a student
across a potentially wide range
of course experiences.
doesn’t tell whether hits are
HRs or singles, whether
slump or streak.
doesn’t tell whether difficult
or easy courses, major field
or others
Descriptive Statistics
Descriptive statistics can be used to summarize
and describe a single variable at a time (UNIvariate
analysis). There are 3 major characteristics of a
single variable (found by repeating the measurement)
1. The distribution (frequencies, percentages)
2. The central tendency (mean, median, mode)
3. The dispersion or spread around the central
tendency (standard deviation, range)
Types of Variables
QUALITATIVE
DISCRETE or
TYPE
DEFINITION
EXAMPLES
Categorical Variable that can take on one of a
or nominal limited number of possible values,
groups or categories. There is no
intrinsic ordering to the categories.
Gender
haircolor
Bloodtype (A, B, AB, O)
State resident lives in
Ordinal
Economic class
Similar to categorical, but there is a
clear ordering of the variables
Educational experience
QUANTITATIVE
CONTINUOUS or
Likert scales or ratings
Interval
Similar to ordinal except the intervals
between the values of the interval
variable are equally spaced so they
can be measured along a continuum
and given numerical values
Annual income
Temperature
Reaction time
BP
Ratio
interval variables, but with the added
condition that 0 of the measurement
indicates that there is none of that
variable. You can use ratios to relate
these measurements.
Height
Mass
Distance
weight
Temp only in Kelvin
Distributions of Discrete Variables
use frequencies (counts) and percentages
examples of discrete variables: Levels, types, groupings, yes/no, Drug
A vs. Drug B
Different ways to display frequencies and percentages for M&M data:
2
4
Table 1. Frequencies in the
Bag of M&M's
Color Frequency
Red
Brown
Yellow
Green
Orange
Blue
18
17
7
7
4
2
Total
55
Table
%
30.9
32.7
12.7
12.7
3.6
7.3
Pie chart
17
7
7
18
FREQUENCY
DISTRIBUTIONS
good if more
than 20
observations
Bar chart
Distributions of Ordinal Level Data
Frequencies and percentages can also be computed
for ordinal data (which is also discrete)
– Examples: Likert Scales (Strongly Disagree to Strongly Agree);
High School/Some College/College Graduate/Graduate School
60
50
40
30
20
10
0
Strongly
Agree
Agree
Disagree Strongly
Disagree
Distribution of Continuous
(Interval/Ratio) Data
We can compute frequencies and percentages for
continuous (interval and ratio level) data as well
– Examples: Age, Temperature, Height, Weight, Many
Clinical Serum Levels
Distribution of Injury Severity
Score in a population of patients
Distributions of a Continuous or Quantitative Variable
There is natural variability when measuring any single variable in a
sample or population. The distribution of scores or values can be
displayed using Frequency Histograms and Box and Whiskers Plots.
Hemoglobin Levels of 70 women
Frequency Histogram
Box and Whiskers Plot
BOXPLOT
(BOX AND WHISKER PLOT)
MAX: 97.5th Centile
12
10
Q3:
Pain (VAS)
8
6
MEDIAN
4
(50th centile)
2
Q1:
0
Inter-quartile
range: Q3-Q1
(50% of the data
are in this range)
-2
N=
75th Centile
74
27
Female
Male
25th Centile
MIN: 1.5th Centile
NO MEASUREMENT IS EXACT.
There is unavoidable variability when repeating the
measurement of a single variable due to random errors.
The variability due to random errors is called uncertainty.
Since random errors are by nature, erratic, they are subject to
the laws of probability or chance so individual measurements
will be scattered or distributed around the average.
The amount of spread
or dispersion of
individual
measurements around
the average value is a
measure of the
uncertainty of the
measurement.
STANDARD DEVIATION AND SAMPLE SIZE
The uncertainty of a measurement can be reduced by
repeating the measurement more.
As sample size, n,
increases, so
SD or dispersion around
the mean decreases
Degrees of freedom, df,
is related to sample size
n=150
n=50
n=10
Experiment: What is the effect of release
height on the horizontal landing distance.
Here we characterize ONE
VARIABLE - the distance (for one
height)
MEAN = 65.36 cm
62
63
64
65
66
67
68
Larger spread or
uncertainty (less precise)
Same average
values
Smaller spread or
uncertainty (more precise)
The average spread of the multiple measurements around a
mean value is an estimate of the uncertainty or the
precision and is called the standard deviation, SD.
Mean
68% of all the
measurements fall within
1 SD from the mean
95% of all the
measurements fall within
2 SDs from the mean
99% of all the
measurements fall within
3 SDs from the mean
The average spread of the multiple measurements around a
mean value represents the uncertainty of the variable and
is called the standard deviation, SD.
NORMAL DISTRIBUTION of a Variable
Frequency or probability
In a normal distribution, points are distributed symmetrically
around the mean. Many naturally-occurring phenomena can be
approximated surprisingly well by this distribution. Standard
Deviation is a measure of the dispersion around the mean.
Mean (central tendency)
p<0.05
SD
Dispersion
measured by S.D.
SD
SD
+
SD
+
SD
+
SD
Only 3 points in
1000 will fall
outside 3 S.D.
from mean
normal distrib interactive
Sum of Squares (SS)
trial
measurement
spread
(x –
xavr)2
x
x – xavr
1
9
2
4
2
4
-3
9
3
7
0
0
4
6
-1
1
5
10
3
9
6
5
-2
4
7
5
-2
4
8
7
0
0
9
8
1
1
10
8
1
1
total
mean
7
33
SS is the TOTAL dispersion of the dataset
SD is the AVERAGE dispersion of the dataset

SD 
n 1
( x x )2
 
2
SD is also known as the
square root of the
variance, 2
33
SD 
 1.9 (27%)
9
mean  7  1.9 ( SD)
trial
measurement
Spread
x
x – xavr
(x – xavr)2
1
9
2
4
2
4
-3
9
3
7
0
0
4
6
-1
1
5
10
3
9
6
5
-2
4
7
5
-2
4
8
7
0
0
9
8
1
1
10
8
1
1
total
mean
7
33

SD 
( x  xavr ) 2
n 1
mean  7  1.9 ( SD )
68% confidence that another
measurement would be within one
SD of the average value.
(between 5.1-8.9)
95% confidence that another
measurement would be within two
SDs of the average value.
(between 3.2-10.8)
99% confidence that another
measurement would be within three
SDs of the average value.
(between 1.3-12.7)
Example
measured average  7  0.3 ( SD )
accepted value  8
Does your measurement agree with the accepted value?
NO since the range within 2 STDs of the mean (95%
confidence interval between 6.4 – 7.6) does not overlap with
the accepted value. Therefore there is less than 5%
probability (p<0.05) that they agree and there is a
statistically significant difference between the two values.
What is the relative error of your measurement?
Re l Error (%) 

measurement  accepted
accepted
7 8
8
100  12.5%
100
Example
measured average  7  0.6 ( SD )
accepted value  8
Does your measurement agree with the accepted value?
YES because within 2 STDs of uncertainty, your measured
average agrees with the accepted value. You have 95%
confidence that the measured average value is within two
STDs (between 5.8 – 8.2). There is less than 5% probability
(p<0.05) that the two values agree by random chance.
What is the relative error of your measurement?
Re l Error (%) 

measurement  accepted
accepted
7 8
8
100  12.5%
100
OTHER DISTRIBUTIONS of a single variable
Normal or
symmetric
Right Skewed
Bimodal
Rare
Left Skewed
Fat tailed
DESCRIBING DATA – Central Tendency
MEAN
Average or arithmetic mean of the data
MEDIAN
The value which comes half way when
the data are ranked in order
MODE
Most common value observed
• In a normal distribution, mean and median are the
same
• If median and mean are different, indicates that
the data are not normally distributed
• The mode is of little if any practical use
SKEWED DISTRIBUTION
MEAN
MEDIAN – 50% OF
VALUES WILL LIE ON
EITHER SIDE OF THE
MEDIAN
DISTRIBUTIONS: EXAMPLES
NORMAL
DISTRIBUTION
SKEWED DISTRIBUTION
•
•
•
•
•
•
•
•
•
•
Bankers’ bonuses (+)
•
Age at death in developed
countries (-)
•
Number of fingers (-)
Height
Weight
Haemoglobin
IQ
BP
Test scores
Speed on HW at a
spot
Number of marriages (+)
Mileage on used cars for
sale (+)
HOW TO TELL IF A VARIABLE FOLLOWS
A NORMAL DISTRIBUTION
• Important because parametric statistics
assume normal distributions
• Statistics packages can test normality
• Distribution unlikely to be normal if:
– Mean is very different from the median
– Two SDs below the mean give an
impossible answer (eg height <0 cm)
Descriptive Statistics
Descriptive statistics can be used to summarize and
describe a single variable at a time (UNIvariate
analysis). There are 3 major characteristics of a single
variable (repeatedly measured)
1. The distribution (frequencies, percentages)
2. The central tendency (mean, median)
3. The dispersion or spread around the central
tendency (standard deviation, variance, range).
Large STD means that there is a lot of uncertainty
to the mean and that the measurement is not
precise.
By taking multiple measurements, a good estimate
of these 3 properties of the measured variable
fully characterizes the uncertainty of the variable.
INFERENTIAL STATISTICS
With inferential statistics, you are trying to reach
conclusions that extend beyond the immediate data
(sample) to a population. Inferential statistics can be
used to prove or disprove theories, determine
associations between variables, and determine if findings
are significant and whether or not we can generalize
from our sample to the entire population
The types of inferential statistics we will go over:
•
•
•
•
T-tests
ANOVA Tests
Correlation
Logistic Regression
Types of Variables
QUALITATIVE
DISCRETE or
TYPE
DEFINITION
EXAMPLES
Categorical Variable that can take on one of a
or nominal limited number of possible values,
groups or categories. There is no
intrinsic ordering to the categories.
Gender
haircolor
Bloodtype (A, B, AB, O)
State resident lives in
Ordinal
Economic class
Similar to categorical, but there is a
clear ordering of the variables
Educational experience
QUANTITATIVE
CONTINUOUS or
Likert scales or ratings
Interval
Similar to ordinal except the intervals
between the values of the interval
variable are equally spaced so they
can be measured along a continuum
and given numerical values
Annual income
Temperature
Reaction time
BP
Ratio
interval variables, but with the added
condition that 0 of the measurement
indicates that there is none of that
variable. You can use ratios to relate
these measurements.
Height
Mass
Distance
weight
Temp only in Kelvin
Inferential Statistics
• Comparisons of ONE VARIABLE between 2 or more
groups (UNIvariate analysis)
– Correlation T-tests
– T-tests (comparison of the mean of one variable
between 2 groups)
– ANOVA (comparison of the mean of one variable
between 3 or more groups)
• Relating how TWO VARIABLES vary together (BIvariate)
– Correlations
– Logistic Regression
T-Tests
One of the simplest inferential tests. Used to compare the
average of ONE VARIABLE between two groups to see if
there is a statistical difference.
eg. Do 8th-grade boys and girls differ in math test scores?
or Does the outcome measure differ from a control group?
T-test gives the probability of
the null hypothesis – that there
is no difference between the
means of a variable in two
groups.
If p <0.05, the null hypothesis is
rejected and the means of two
groups of data are statistically
different from each other.
T-Tests
What does it mean to say that the averages for
two groups are statistically different?
To judge the difference between two groups, we must consider the
difference between their means relative to the spread or uncertainty of the
group. The t-test does just this.
SD
SE 
n
In these 2 examples, the difference between the means is the same.
signal Difference between means
t value 

noise
SE of the groups
Lower
variability
Higher t-value
Higher
variability,
Lower tvalue
The two distributions
are widely separated so
their means are clearly
different
The distributions overlap,
so it is unclear whether
the samples come from
the same population
T-Test
tests the Null Hypothesis
t-value is signal to
noise or the size of
the difference in
means relative to
the total variation.
Larger the t-value,
the larger the
likelihood that there
is a difference
between the two
averages.
p-value that corresponds
to the t-value. It is the
probability of the null
hypothesis, that there is
no difference between
the variables.
High p – likely that there
is no difference between
the averages of the 2
groups
Low p (<0.05) likely that
there is a statistically
significant difference
between the averages of
the two groups.
T-Tests
What does it mean to say that the averages for
two groups are statistically different?
DEPENDS ON
t-value is signal to noise or the size of the difference in means
relative to the total variation. By itself, it doesn’t tell us much
because it is the value from one data set and would vary if we
repeated the experiment to get a better idea of the population
value. NEED probability of t-value as well. Can get that from a
t-distribution centered at 0 (assuming the null hypothesis)
p-value is the probability that the null hypothesis is true
(probability the t-value is different than 0) or that the average
of the variables in the two groups are NOT different. (can look
up in a table)
Depends on degrees of freedom, df = n-2
Significance Level is probability level considered to be statistically
significant and not due to chance. Usually it is chosen to be p < 0.05
2 types of T-tests
1. Paired t-tests: Used to compare the MEANS of a continuous
variable in two non-independent or related samples
Examples
a) measurements on the same people before and after a treatment
b) Is diet X effective in lowering serum cholesterol levels in a sample of
12 people?
c) Do patients who receive drug X have lower blood pressure after
treatment then they did before treatment?
2. Independent samples t-tests: Used to compare the MEANS of a
continuous variable in two independent samples
Examples
a) Measurements from two different groups of people
b) Do people with diabetes have the same Systolic Blood Pressure as
people without diabetes?
a) Do patients who receive a new drug treatment have lower blood
pressure than those who receive a placebo?
Tip: if you have > 2 different groups, you use ANOVA, which compares the means of
EXAMPLE: Sam
A scores B scores
Sleepresearcher hypothesizes
5
8
that people who sleep for
7
1
only 4 hours will score
5
4
significantly lower than
3
6
people who sleep for 8 hours
5
6
on a cognitive skills test. He
3
4
brings 16 participants into his
3
1
sleep lab and randomly
9
2
T-test Calculator
assigns them to one of two
(click on it)
Avr = 5
Avr = 4
groups. Group A sleeps 8 hrs,
sd = 2.14 sd = 2.56
group B sleeps 4 hrs. The
Aavr  Bavr
54
next morning he administers
t

 0.847
2
2
2
2
the SCAT to all participants.
sd A sd B
2.14 2.56


(Scores on SCAT range from
nA
nB
8
8
1-9 with high scores
according to the t-table, with df = 14, t must
representing better
be at least 2.145 to reach p <0.05, so this
performance).
difference is not statistically significant
EXAMPLE: Sam Sleepresearcher
hypothesizes that people who sleep
for only 4 hours will score significantly
lower than people who sleep for 8
hours on a cognitive skills test.
T-test Calculator
(click on it)
t
Aavr  Bavr
sd A2 sd B2

nA
nB

54
2.14 2 2.56 2

8
8
A scores
B scores
(sleeps 8hrs) (sleeps 4 hrs)
5
8
7
1
5
4
3
6
5
6
3
4
3
1
9
2
Avr = 5
sd = 2.14
Avr = 4
sd = 2.56
t  0.847
p = 0.4111 according to the t-table, with df n-2 = 14
p is the probability of the null hypothesis, that the 2 scores are not
different (41% chance). p must be <0.05 to reject the null
hypothesis and conclude that the scores are statistically different.
Therefore, the testing scores are not statistically different.
What can you conclude if there is no significant difference
between the averages of a variable in two groups?
YOU CANNOT PROVE A NEGATIVE RESULT
If t-test indicates no significant difference between 2 groups, that
does NOT prove that there is no difference. You can only say that
the probability or confidence level is less than a particular p-value
If there is no statistically significant difference between groups, it
could indicate EITHER
- The difference is very small and not enough data was collected
to see the difference (need a larger sample size)
- There really is no difference
A low probability value indicates that a difference between the
groups would be very unlikely.
How do you report t-tests results?
To display the results of a study
that compares the mean value
of ONE VARIABLE in two
categorical groups,
make a histogram of the mean
values in each group with
error bars of SD.
In the figure legend and text,
cite the t-value and associated
p value.
* As can be seen in Figure 1, specialty candidates had significantly
higher scores on questions dealing with treatment than residency
candidates (t = [insert t-value from stats output], p < .001).
SUMMARY THUS FAR …
ONE-SAMPLE t-Test
Used to compare a variable mean
with a hypothesized value
Independent sample t-TEST Used to compare variable means
of two independent samples
(Unpaired test)
PAIRED t-TEST
(MATCHED PAIR)
t-value is signal to
noise or the size of the
difference relative to
the variation.
Used to compare two (repeated)
measures from the same subjects
p-value is the
probability of the null
hypothesis, that there
is no difference
between the variables.
• What does a t-test tell you?
T-tests
The probability of the null hypothesis (that the means of the same
variable over two groups is the same). If p < 0.05, then the null
hypothesis can be rejected and there is a statistically significant
difference between means of the two groups. The 2 groups or
variables being compared can be either independent OR related (ie
before and after, control and experimental)
• What do the results look like?
You get a t-value and associated p-value
• How do you interpret it?
T-test Calculator
(click on it)
By looking at the p-value that corresponds to the t-value
• p < 0.05 means the 2 groups are significantly different from each other
• p > 0.05 means the difference between the groups is NOT statistically
significant. It does NOT mean the 2 groups are statistically the same.
(either the difference is small and more measurements are required to
detect it or there is no difference between the groups)
Research Question:
Does stretching legs before a wall-sit increase
the length of time that it can be done for?
• Apply the T-test to the Wall Sit data to determine
whether the average wall sit time was different after
stretching pretreatment
• State the null hypothesis to be tested
• Which t-test? Paired t-test or independent t-test?
• Determine
t-value = ________
df = _________
p = ________
• Report result by graphing a histogram of the average
wall sit times with SD error bars
Example of Wall-Sit Experiment Writeup
Introduction
Research Question: Does stretching legs before a wall-sit
increase the length of time that it can be done for?
Hypothesis: __________________
Experimental Design
Describe what the indep variable is and how its measured
Describe what the dep variable is and how its measured
Describe controls (how will you know if treatment or condition made a difference?)
Describe constants (how did you eliminate nonspecific causes of variation?)
Data
Summarized with descriptive statistics
- Present distribution of variable dataset for each group
- Present mean ± SD of each variable dataset for each group
Example of Wall-Sit Experiment Writeup
Analysis
Independent T-test of Null Hypothesis : There is NO difference
between the wall sit times between the groups that had no
pretreatment and those that had a stretching pretreatment.
Independent T-test
t-value = 0.6431
df = 28
p = 0.5254
Conclusion: The difference between the mean wall sit times for the
groups with and without stretch pretreatment was not statistically
significant. According to t-test analysis, there is a 53% probability that
there is no difference between them.
Alphabet Practice Experiment
• For this experiment, we use one
group, who will perform an
experiment six times.
• You will be issued with a piece
of paper with the task on it. Do
not turn it over until instructed.
• You will need a pen to write on
your recording sheet.
• Each attempt needs to be
timed, and the time recorded.
Alphabet Practice Experiment
Time how long it takes to find the letters A to Z in order.
Alphabet Practice Experiment
Recording times to find the letters of the alphabet in order
1. What do you think the research question is?
2. What is the treatment (independent variable)?
What is the response variable (dependent variable)?
3. What do you think the outcome will be? (Hypothesis)
Alphabet Practice Experiment
DATA
- Record and display class data in table
- Show Descriptive Statistics of the dependent variable for
2 groups
a) show the distribution (with box and whiskers plots
or frequency histograms)
b) Determine the mean value of each dataset
c) Quantify the average dispersion of each variable
dataset with SD
Alphabet Practice Experiment
ANALYSIS
• Apply the T-test to the Alphabet Practice Experiment
• State the null hypothesis to be tested
• Which t-test? Paired t-test or independent t-test?
• Determine
t-value = ________
df = _________
p = ________
• Report result by graphing a histogram of the average
values with SD error bars
• State your conclusions to the experiment and support
with evidence.
ANOVA - Comparison of one variable between THREE
OR MORE Samples
• Extends the t-test to more than 2 groups
• ANalysis Of VAriance (ANOVA) gives the probability of the null
hypothesis – that there is no difference between the means of a
variable in three or more groups. If p <significance level (0.05),
the null hypothesis is rejected and the means of the groups of
data are statistically different from each other.
• ANOVA involves dividing the variance of the groups into:
– Variance BETWEEN groups (variance of averages)
– Variance WITHIN groups
Measure of BETWEEN Groups variance
F
Measure of WITHIN Groups variance
The greater F, the more statistically significant the difference is
(values of F in standard tables)
ANOVA
Remember variance is the variability around the
mean. It is related to the average dispersion.
BETWEENGroup Variance
WITHINGroup Variance
F
BETWEEN Groups variance
WITHIN Groups variance
Here, the BETWEEN-group variance
is large relative to the WITHIN-group
variance, so F will be LARGE
ANOVA
BETWEENGroup Variance
WITHINGroup Variance
Here, the WITHIN-group variance is
larger, and the BETWEEN-group
variance smaller, so F will be
smaller (reflecting the likelihood of
no significant differences between
these three sample means
BETWEEN variance
F
WITHIN variance
F=0 if the group means
are identical
F>0 if not
F could be >0 by chance.
So how large is F
to be statistically
significant?
ANOVA
What does it mean to say that the averages for
three or more groups are statistically different?
DEPENDS ON
F-value ratio of between group variance to within group
variance. By itself it doesn’t tell us much because it is the value
from one data set and would vary if we repeated the experiment
to get a better idea of the population value. NEED probability of
F value as well. Can get that from a distribution of F-values
centered at 0 (assuming the null hypothesis)
p-value is the probability that the null hypothesis is true
(probability the F-value is different than 0) or that the means of
the variable in the multiple groups are NOT different. (can look
up in a table)
Depends on degrees of freedom, df:
(dfbetw=ngroup-1;
dfwithin = ntotal-1-dfbetw)
Significance Level is probability level considered to be statistically
significant and not due to chance. Usually it is chosen to be p < 0.05
ANOVA – AN EXAMPLE
A marketing research firm tests the
effectiveness of three new flavorings
for a leading beverage using a sample
of 30 people, divided randomly into
three groups of 10 people each.
Group 1 tastes flavor 1, group 2 tastes
flavor 2 and group 3 tastes flavor 3.
Each person is then given a
questionnaire which evaluates how
enjoyable the beverage was. The
scores are shown below. Determine
whether there is a perceived
significant difference between the
three flavorings.
Sum of Squares is related to
the total variance (variation)
of a dataset around the mean
SSW   ( xi  xavr ) 2  SD 2 (n  1)
Perceived good score
Within
group
mean
SS
Flavor
1
Flavor
2
Flavor
3
13
12
7
17
8
19
19
6
15
11
16
14
20
12
10
15
14
16
18
10
18
9
18
11
12
4
14
16
11
11
Between
Group
Mean and
SS
13.5
13.2
126.5
77.4
15.0 11.1
120
168.9
ANOVA Table
Source of
Variation
Sum of
Squares
SS   ( xi  x )
BETWEEN Groups
(3 groups)
77.4
WITHIN Groups
(30 participants)
415.4
Total
492.8
Degrees of
Freedom
2
(df)
2
(dfn=ngroups-1)
27
(dfd=ntotal-1-dfbetween)
29
Variance
(SS/df)
F
BETWEEN Var
WITHIN Var
p-value
38.7
15.39
2.51
0.0913
• BETWEEN Groups Sum of Squares is related to the variance (or variability) of the
average values of the groups. Thus we treat the collection of the 3 group means
as data.
• WITHIN Groups Sum of Squares concerns the variance within the groups
• The degrees of freedom (df) represent the number of independent data points
required to define each value calculated.
Does the F-value represent a statistically significant difference????
Depends on the sample size (df), the associated p-value and the
significance level
Reporting ANOVA Results
Perceived Flavor score
Perceived goodness score
20
18
16
14
12
10
8
6
4
2
0
Flavor1 Flavor2 Flavor3
To display the results of a
study that measures the
value of ONE VARIABLE in
multiple categorical
groups
make a histogram of the
mean values in each
group with error bars of
SD. In the figure legend
and text, cite the F-value
and associated p value.
Conclusion: There is no statistically significant difference in the how
enjoyable the flavor was perceived to be the three flavor groups
(F=2.51; p=0.091). There is a 9% probability that any difference in
the perceived goodness was due to chance.
T-test and ANOVA Assumptions
There are basic assumptions used in T-tests and
ANOVA
1. the variances within each group are equal to
each other
2. The data are normally distributed
Inferential Statistics
Correlation and Regression
(BIVARIATE Analysis)
T-tests and ANOVA compare one variable
between 2 or more groups
eg. Comparison of BP in females and males or
Comparison of BP before and after a drug
Correlation and regression analyses are used to
see how two variables vary together.
Eg. How are the variables BP and Heart Rate
related
Correlation
One of the most common and most useful statistics. A correlation is a
single number that describes the degree of relationship between two
variables. It is NOT related to cause and effect.
• When to use it? When you want to know about the association
or relationship between two continuous variables
egs. Relationship between food intake and weight;
drug dosage and blood pressure
air temp and metabolic rate
• What does it tell you? If a linear relationship exists between two
variables, and how strong that relationship is.
• What do the results look like?
– The correlation coefficient = Pearson’s r
can use CORREL() function in Excel to find r
– Pearson’s coefficient, r, ranges from -1 to +1
Correlation
Variable 2
Guide for interpreting
strength of correlations:
Variable 1
r = 0 – 0.25:
Little / no relationship
r = 0.25 – 0.50:
Fair relationship
r = 0.50 - 0.75:
Moderate relationship
r = 0.75 – 1.0:
Strong relationship
r = 1.0:
perfect correlation
Correlation
• How do you interpret it?
a) If r is positive, high values of one variable are associated with high values of
the other variable (both go in SAME direction: ↑↑ OR ↓↓)
Eg. Diastolic blood pressure tends to rise with age, thus the two variables are
positively correlated
b) If r is negative, low values of one variable are associated with high values of
the other variable (variables change in opposite direction: ↑↓ OR ↓ ↑)
Eg. Heart rate tends to be lower in persons who exercise frequently, the two
variables correlate negatively
c) r=0: Correlation of 0 indicates NO linear relationship
• How do you report it?
“Diastolic blood pressure was positively correlated with age (r = .75, p < . 05).”
Tip: Correlation does NOT equal causation!!! If two variables are highly
correlated, this does NOT mean that one CAUSES the other!!!
NO Correlation does NOT mean no relationship
Correlation Example
Let's assume that we want to look at the relationship between two
variables, ice cream sales and temperature
Ice Cream Sales vs
Temperature for 12 days
Temperature
°C
Ice Cream
Sales
14.2
$215
16.4
$325
11.9
$185
15.2
$332
18.5
$406
22.1
$522
19.4
$412
25.1
$614
23.4
$544
We can easily see that warmer weather leads to
more sales, the relationship is good but not perfect.
r = 0.9575
18.1
$421
Correlation does NOT mean causation
22.6
$445
17.2
$408
Correlation Significance
How statistically significant is the correlation value?
Depends on
- significance level (usually 0.05 which is the probability that the
correlation occurred by chance. p<5% is defined as statistically
significant and not due to chance)
- df or sample size (degrees of freedom)
you can look in a table of r values to see if the r is large
enough to say that the difference between the groups is
significant and not likely to have been due to chance
(p<0.05)
Correlation Example
Correlation is not good at curves. It only works well for relationships
that are linear.
Here is the ice cream example again, but there has been a heat wave
r = 0 so there is no
correlation between ice
cream sales and temp.
That does NOT mean
that there is no
relationship between
the variables
There is a nice peak at around 25oC. Correlation analysis cannot
“see” this non linear relationship.
Need a Regression analysis of scatter plot
Regression
Regression analysis is a statistical process for estimating the
relationships among variables. It includes many techniques for modeling
and analyzing several variables, when the focus is on the relationship
between a dependent variable and one or more independent variables .
• When to use it? When you want to mathematically model the
relationship between any continuous independent and
dependent variable
• What does it tell you? The mathematical relationship between
two continuous variables.
• What do the results look like?
– The two variables are graphed on a scatter plot
– The data is fit with a curve of best fit
– The mathematical function that describes the curve of best fit
is a mathematical model of the relationship between the
indep and dep variable and the R2 value gives goodness of fit
Displaying or Reporting Regression Analysis
Here is the ice cream example again,
but there has been a heat wave.
Ice Cream Sales vs Daily temperature
700
Ice Cream Sales ($)
600
500
400
300
200
Sales = -1.7253T2 + 94.3T - 730.59
R² = 0.8848
100
0
10
15
20
25
30
Temp (oC)
35
40
45
Regression Analysis:
The best fit curve to
the data is shown
along with the
equation that
describes the curve.
This equation is a
mathematical model
of the relationship
between the two
variables, ice cream
sales and the
temperature.
The R2 value is the
goodness of the fit.