Transcript normal distribution - Livingston Public Schools

Basic Statistics for
Scientific Research
Outline
• Descriptive Statistics
– Frequencies & percentages
– Means & standard deviations
• Inferential Statistics
– Correlation
– T-tests
– Chi-square
– Logistic Regression
Types of Statistics/Analyses
Descriptive Statistics
– Frequencies
– Basic measurements
Inferential Statistics
–
–
–
–
–
Hypothesis Testing
Correlation
Confidence Intervals
Significance Testing
Prediction
Describing a phenomena
How many? How much?
BP, HR, BMI, IQ, etc.
Inferences about a phenomena
Proving or disproving theories
Associations between phenomena
If sample relates to the larger
population
E.g., Diet and health
Descriptive Statistics
Descriptive statistics can be used to summarize
and describe a single variable (aka, UNIvariate)
• Frequencies (counts) & Percentages
– Use with categorical (nominal) data
• Levels, types, groupings, yes/no, Drug A vs. Drug B
• Means & Standard Deviations
– Use with continuous (interval/ratio) data
• Height, weight, cholesterol, scores on a test
Frequencies & Percentages
Look at the different ways we can display frequencies and
percentages for this data:
Pie chart
Table
AKA frequency
distributions –
good if more
than 20
observations
Good if more
than 20
observations
Bar chart
Distributions
The distribution of scores or values can also be
displayed using Box and Whiskers Plots and Histograms
Ordinal Level Data
Frequencies and percentages can be computed
for ordinal data
– 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
Interval/Ratio Data
We can compute frequencies and percentages
for 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
Interval/Ratio Distributions
The distribution of interval/ratio data often
forms a “bell shaped” curve.
– Many phenomena in life are normally
distributed (age, height, weight, IQ).
Interval & Ratio Data
Measures of central tendency and measures of dispersion are often
computed with interval/ratio data
• Measures of Central Tendency (aka, the “Middle Point”)
– Mean, Median, Mode
– If your frequency distribution shows outliers, you might want to use
the median instead of the mean
• Measures of Dispersion (aka, How “spread out” the data are)
― Variance, standard deviation, standard error of the mean
― Describe how “spread out” a distribution of scores is
― High numbers for variance and standard deviation may mean that
scores are “all over the place” and do not necessarily fall close to the
mean
In research, means are usually presented along with standard deviations or
standard errors.
INFERENTIAL STATISTICS
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:
• Correlation
• T-tests/ANOVA
• Chi-square
• Logistic Regression
Type of Data & Analysis
• Analysis of Categorical/Nominal Data
– Correlation T-tests
– T-tests
• Analysis of Continuous Data
– Chi-square
– Logistic Regression
Correlation
• When to use it?
– When you want to know about the association or relationship
between two continuous variables
• Ex) food intake and weight; drug dosage and blood pressure; air temperature and
metabolic rate, etc.
• 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
– Ranges from -1 to +1
– See next slide for examples of correlation results
Correlation
Guide for interpreting
strength of correlations:
 0 – 0.25 = Little or no
relationship
 0.25 – 0.50 = Fair degree of
relationship
 0.50 - 0.75 = Moderate
degree of relationship
 0.75 – 1.0 = Strong
relationship
 1.0 = perfect correlation
Correlation
• How do you interpret it?
– If r is positive, high values of one variable are associated with high values
of the other variable (both go in SAME direction - ↑↑ OR ↓↓)
• Ex) Diastolic blood pressure tends to rise with age, thus the two variables are
positively correlated
– If r is negative, low values of one variable are associated with high values
of the other variable (opposite direction - ↑↓ OR ↓ ↑)
• Ex) Heart rate tends to be lower in persons who exercise frequently,
the two variables correlate negatively
– 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!!! Just because two variables are highly correlated, this
does NOT mean that one CAUSES the other!!!
T-tests
• When to use them?
– Paired t-tests: When comparing the MEANS of a continuous variable in
two non-independent samples (i.e., measurements on the same people
before and after a treatment)
• Ex) Is diet X effective in lowering serum cholesterol levels in a sample of 12
people?
• Ex) Do patients who receive drug X have lower blood pressure after
treatment then they did before treatment?
– Independent samples t-tests: To compare the MEANS of a
continuous variable in TWO independent samples (i.e., two different
groups of people)
• Ex) Do people with diabetes have the same Systolic Blood Pressure as
people without diabetes?
• Ex) 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 3 or more groups
T-tests
• What does a t-test tell you?
– If there is a statistically significant difference between
the mean score (or value) of two groups (either the
same group of people before and after or two
different groups of people)
• What do the results look like?
– Student’s t
• How do you interpret it?
– By looking at corresponding p-value
• If p < .05, means are significantly different from each other
• If p > 0.05, means are not significantly different from each
other
How do you report t-tests results?
“As can be seen in Figure 1, children’s mean reading
performance was significantly higher on the post-tests in
all four grades, ( t = [insert from stats output], p < .05)”
“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).
Chi-square
• When to use it?
– When you want to know if there is an association between two
categorical (nominal) variables (i.e., between an exposure and
outcome)
• Ex) Smoking (yes/no) and lung cancer (yes/no)
• Ex) Obesity (yes/no) and diabetes (yes/no)
• What does a chi-square test tell you?
– If the observed frequencies of occurrence in each group are
significantly different from expected frequencies (i.e., a
difference of proportions)
Chi-square
• What do the results look like?
– Chi-square test statistics = X2
• How do you interpret it?
– Usually, the higher the chi-square statistic, the
greater likelihood the finding is significant, but you
must look at the corresponding p-value to
determine significance
Tip: Chi square requires that there be 5 or more in each cell of a 2x2 table and 5 or more in 80% of
cells in larger tables. No cells can have a zero count.
How do you report chi-square?
“248 (56.4%) of women and 52
(16.6%) of men had abdominal
obesity (Fig-2). The Chi square
test shows that these differences
are statistically significant
(p<0.001).”
“Distribution of obesity by gender showed
that 171 (38.9%) and 75 (17%) of women
were overweight and obese (Type I &II),
respectively. Whilst 118 (37.3%) and 12
(3.8%) of men were overweight and obese
(Type I & II), respectively (Table-II).
The Chi square test shows that these
differences are statistically significant
(p<0.001).”
NORMAL DISTRIBUTION
THE EXTENT OF THE ‘SPREAD’
OF DATA AROUND THE MEAN
– MEASURED BY THE
STANDARD DEVIATION
MEAN
CASES DISTRIBUTED
SYMMETRICALLY ABOUT THE
MEAN
AREA BEYOND TWO STANDARD
DEVIATIONS ABOVE THE MEAN
DESCRIBING DATA
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
BOXPLOT
(BOX AND WHISKER PLOT)
97.5th Centile
12
10
75th Centile
8
6
MEDIAN
(50th centile)
4
2
25th Centile
0
-2
N=
74
27
Female
Male
Inter-quartile range
2.5th Centile
9/14/2010
Photo courtesy of Judy Davidson, DNP, RN
25
STANDARD DEVIATION – MEASURE OF
THE SPREAD OF VALUES OF A SAMPLE
AROUND THE MEAN
THE SQUARE OF THE SD IS
KNOWN AS THE
VARIANCE
2
SD 
Sum(Value  Mean)
Number of values
SD decreases as a function of:
• smaller spread of values
about the mean
• larger number of values
IN A NORMAL DISTRIBUTION,
95% OF THE VALUES WILL LIE
WITHIN 2 SDs OF THE MEAN
Standard Deviation (σ)
99%
95%
9/14/2010
27
STANDARD DEVIATION AND SAMPLE
SIZE
As sample size
increases, so
SD decreases
n=150
n=50
n=10
How do we ESTIMATE
Experimental Uncertainty due to
unavoidable random errors?
Uncertainty in Multiple Measurements
Since random errors are by nature, erratic, they are
subject to the laws of probability or chance. A good way to
estimate uncertainty is to take multiple measurements and
use statistical methods to calculate the standard deviation.
The uncertainty of a measurement can be reduced
by repeating a measurement many times and
taking the average. The individual
measurements will be scattered around the
average.
The amount of spread of individual measurements
around the average value is a measure of the
uncertainty.
Avr = 65.36 cm
62
63
64
65
66
67
68
Larger spread or
uncertainty
Same average
values
Smaller spread or
uncertainty
The spread of the multiple measurements around an
average value represents the uncertainty and is called the
standard deviation, STD.
2/3 (68%) of all the
measurements fall within
1 STD
95% of all the
measurements fall within
2 STDs
99% of all the
measurements fall within
3 STDs
The spread of the multiple measurements around an
average value represents the uncertainty and is called the
standard deviation, STD.
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
avr
7
33

STD 
( x  xavr ) 2
n 1
33
STD 
 1.9 (27%)
9
average  7  1.9 ( STD)
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
avr
7
33

STD 
( x  xavr ) 2
n 1
average  7  1.9 ( STD)
68% confidence that another
measurement would be within one
STD of the average value.
(between 5.1-8.9)
95% confidence that another
measurement would be within two
STDs of the average value.
(between 3.2-10.8)
99% confidence that another
measurement would be within three
STDs of the average value.
(between 1.3-12.7)
SKEWED DISTRIBUTION
MEAN
MEDIAN – 50% OF
VALUES WILL LIE ON
EITHER SIDE OF THE
MEDIAN
DOES A VARIABLE FOLLOW 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)
DISTRIBUTIONS: EXAMPLES
NORMAL
DISTRIBUTION
SKEWED
DISTRIBUTION
•
•
•
•
•
Height
Weight
Haemoglobin
Bankers’ bonuses
Number of
marriages
DISTRIBUTIONS AND
STATISTICAL TESTS
• Many common statistical tests rely on the variables
being tested having a normal distribution
• These are known as parametric tests
• Where parametric tests cannot be used, other, nonparametric tests are applied which do not require
normally distributed variables
• Sometimes, a skewed distribution can be made
sufficiently normal to apply parametric statistics by
transforming the variable (by taking its square root,
squaring it, taking its log, etc)
EXAMPLE: IQ
Say that you have tested a sample of people on a
validated IQ test
The IQ test has been
carefully standardized
on a large sample to
have a mean of 100
and an SD of 15
94
97
100
SD 
103
106
Sum of (Individua l Value - Mean Value)2
Number of values
EXAMPLE: IQ
Say you now administer the test to
repeated samples of 25 people
Expected random variation of
these means equals the Standard
Error
SE 
SD
Sample Size
 15
94
97
100
103
106
25
 3.0
STANDARD DEVIATION vs STADARD
ERROR
• Standard Deviation is a measure of variability
of scores in a particular sample
• Standard Error of the Mean is an estimate of
the variability of estimated population means
taken from repeated samples of that
population (in other words, it gives an
estimate of the precision of the sample mean)
See Douglas G. Altman and J. Martin Bland. Standard
deviations and standard errors. BMJ 331 (7521):903, 2005.
EXAMPLE: IQ
One sample of 25 people yields a mean IQ
score of 107.5
What are the chances of
obtaining an IQ of 107.5
or more in a sample of 25
people from the same
population as that on
which the test was
standardized?
94
97
100
103
106
EXAMPLE: IQ
How far out the sample IQ is in the population
distribution is calculated as the area under the
curve to the right of the sample mean:
Sample Mean - Population Mean
Standard Error

94
97
100
103
106
107.5 - 100
3.0
 2 .5
This ratio tells us how far out
on the standard distribution
we are – the higher the
number, the further we are
from the population mean
EXAMPLE: IQ
Look up this figure (2.5) in a table of
values of the normal distribution
From the table, the area in the tail
to the right of our sample mean is
0.006 (approximately 1 in 160)
94
97
100
103
106
This means that there is a
1 in 160 chance that our
sample mean came from
the same population as
the IQ test was
standardized on
EXAMPLE: IQ
This is commonly referred to as p=0.006
By convention, we accept as
significantly different a sample
mean which has a 1 in 20 chance
(or less) of coming from the
population in which the test was
standardized (commonly referred
to as p=0.05)
94
97
100
103
106
Thus our sample had a
significantly greater IQ
than the reference
population (p<0.05)
EXAMPLE: IQ
If we move the sample
mean (green) closer to
the population mean
(red), the area of the
distribution to the right
of the sample mean
increases
94
97
100
103
106
Even by inspection, the
sample is more likely
than our previous one to
come from the original
population
UNPAIRED OR INDEPENDENT-SAMPLE
t-TEST: PRINCIPLE
The two distributions are
widely separated so their
means clearly different
The distributions overlap, so
it is unclear whether the
samples come from the
same population
Difference between means
t
SE of the difference
In essence, the t-test
gives a measure of the
difference between the
sample means in relation
to the overall spread
UNPAIRED OF INDEPENDENT-SAMPLE
t-TEST: PRINCIPLE
SE 
Difference between means
t
SE of the difference
SD
Sample Size
With smaller sample sizes,
SE increases, as does the
overlap between the two
curves, so value of t
decreases
THE PREVIOUS IQ EXAMPLE
• In the previous IQ example, we were assessing
whether a particular sample was likely to have
come from a particular population
• If we had two samples (rather than sample
plus population), we would compare these
two samples using an independent-sample ttest
SUMMARY THUS FAR …
ONE-SAMPLE
(INDEPENDENT
SAMPLE) t-TEST
Used to compare means of
two independent samples
PAIRED (MATCHED
PAIR) t-TEST
Used to compare two
(repeated) measures from
the same subjects
COMPARISONS BETWEEN THREE OR
MORE SAMPLES
• Cannot use t-test (only for 2 samples)
• Use analysis of variance (ANOVA)
• Essentially, ANOVA involves dividing the
variance in the results into:
– Between groups variance
– Within groups variance
Measure of Between Groups variance
F
Measure of Within Groups variance
The greater F, the more significant the
result (values of F in standard tables)
ANOVA - AN EXAMPLE
Between-Group
Variance
Within-Group
Variance
Here, the between-group variance is large relative to
the within-group variance, so F will be large
ANOVA - AN EXAMPLE
Between-Group
Variance
Within-Group
Variance
Here, the within-group variance is larger, and the
between-group variance smaller, so F will be smaller
(reflecting the likeli-hood of no significant differences
between these three sample means
ANOVA – AN EXAMPLE
• Data from SPSS sample data file
‘dvdplayer.sav’
• Focus group where 68
participants were asked to rate
DVD players
• Results from running ‘One Way
ANOVA’ (found under ‘Compare
Means’)
• Table shows scores for ‘Total DVD
assessment’ by different age
groups
Age
Group
N
Mean
SD
18-24
13
31.9
5.0
25-31
12
31.1
5.7
32-38
10
35.8
5.3
39-45
10
38.0
6.6
46-52
12
29.3
6.0
53-59
11
28.5
5.3
Total
68
32.2
6.4
ANOVA – SPSS PRINT-OUT
Data from SPSS print-out shown below
Sum of
Squares
df
Mean Square
F
Sig.
Between Groups
733.27
5
146.65
4.60
0.0012
Within Groups
1976.42
62
31.88
Total
2709.69
67
•
•
‘Between Groups’ Sum of Squares concerns the variance (or
variability) between the groups
‘Within Groups’ Sum of Squares concerns the variance within the
groups
ANOVA – MAKING SENSE OF THE SPSS
PRINT-OUT
•
•
•
•
Sum of
Squares
df
Mean Square
F
Sig.
Between Groups
733.27
5
146.65
4.60
0.0012
Within Groups
1976.42
62
31.88
Total
2709.69
67
The degrees of freedom (df) represent the number of independent data points
required to define each value calculated.
If we know the overall mean, once we know the ratings of 67 respondents, we can
work out the rating given by the 68th (hence Total df = N-1 = 67).
Similarly, if we know the overall mean plus means of 5 of the 6 groups, we can
calculate the mean of the 6th group (hence Between Groups df = 5).
Within Groups df = Total df – Between Groups df
ANOVA – MAKING SENSE OF THE SPSS
PRINT-OUT
•
Sum of
Squares
df
Mean Square
F
Sig.
Between Groups
733.27
5
146.65
4.60
0.0012
Within Groups
1976.42
62
31.88
Total
2709.69
67
This would be reported as follows:
Mean scores of total DVD assessment varied significantly between age groups
(F(5,62)=4.60, p=0.0012)
•
Have to include the Between Groups and Within Groups degrees of freedom
because these determine the significance of F