SlideShare a Scribd company logo
© 2006
1
Biostatistics Basics
An introduction to an expansive
and complex field
© 2006
Evidence-based Chiropractic 2
Common statistical terms
• Data
– Measurements or observations of a variable
• Variable
– A characteristic that is observed or
manipulated
– Can take on different values
© 2006
Evidence-based Chiropractic 3
Statistical terms (cont.)
• Independent variables
– Precede dependent variables in time
– Are often manipulated by the researcher
– The treatment or intervention that is used in a
study
• Dependent variables
– What is measured as an outcome in a study
– Values depend on the independent variable
© 2006
Evidence-based Chiropractic 4
Statistical terms (cont.)
• Parameters
– Summary data from a population
• Statistics
– Summary data from a sample
© 2006
Evidence-based Chiropractic 5
Populations
• A population is the group from which a
sample is drawn
– e.g., headache patients in a chiropractic
office; automobile crash victims in an
emergency room
• In research, it is not practical to include all
members of a population
• Thus, a sample (a subset of a population)
is taken
© 2006
Evidence-based Chiropractic 6
Random samples
• Subjects are selected from a population so
that each individual has an equal chance
of being selected
• Random samples are representative of the
source population
• Non-random samples are not
representative
– May be biased regarding age, severity of the
condition, socioeconomic status etc.
© 2006
Evidence-based Chiropractic 7
Random samples (cont.)
• Random samples are rarely utilized in
health care research
• Instead, patients are randomly assigned to
treatment and control groups
– Each person has an equal chance of being
assigned to either of the groups
• Random assignment is also known as
randomization
© 2006
Evidence-based Chiropractic 8
Descriptive statistics (DSs)
• A way to summarize data from a sample or
a population
• DSs illustrate the shape, central tendency,
and variability of a set of data
– The shape of data has to do with the
frequencies of the values of observations
© 2006
Evidence-based Chiropractic 9
DSs (cont.)
– Central tendency describes the location of the
middle of the data
– Variability is the extent values are spread
above and below the middle values
• a.k.a., Dispersion
• DSs can be distinguished from inferential
statistics
– DSs are not capable of testing hypotheses
© 2006
Evidence-based Chiropractic 10
Hypothetical study data
(partial from book)
Case # Visits
1 7
2 2
3 2
4 3
5 4
6 3
7 5
8 3
9 4
10 6
11 2
12 3
13 7
14 4
• Distribution provides a summary of:
– Frequencies of each of the values
• 2 – 3
• 3 – 4
• 4 – 3
• 5 – 1
• 6 – 1
• 7 – 2
– Ranges of values
• Lowest = 2
• Highest = 7
etc.
© 2006
Evidence-based Chiropractic 11
Frequency distribution table
Frequency Percent Cumulative %
• 2 3 21.4 21.4
• 3 4 28.6 50.0
• 4 3 21.4 71.4
• 5 1 7.1 78.5
• 6 1 7.1 85.6
• 7 2 14.3 100.0
© 2006
Evidence-based Chiropractic 12
Frequency distributions are
often depicted by a histogram
© 2006
Evidence-based Chiropractic 13
Histograms (cont.)
• A histogram is a type of bar chart, but
there are no spaces between the bars
• Histograms are used to visually depict
frequency distributions of continuous data
• Bar charts are used to depict categorical
information
– e.g., Male–Female, Mild–Moderate–Severe,
etc.
© 2006
Evidence-based Chiropractic 14
Measures of central tendency
• Mean (a.k.a., average)
– The most commonly used DS
• To calculate the mean
– Add all values of a series of numbers and
then divided by the total number of elements
© 2006
Evidence-based Chiropractic 15
Formula to calculate the mean
• Mean of a sample
• Mean of a population
 (X bar) refers to the mean of a sample and refers to the
mean of a population
 EX is a command that adds all of the X values
 n is the total number of values in the series of a sample and
N is the same for a population
X μ
N
X



n
X
X


© 2006
Evidence-based Chiropractic 16
Measures of central
tendency (cont.)
• Mode
– The most frequently
occurring value in a
series
– The modal value is
the highest bar in a
histogram
Mode
© 2006
Evidence-based Chiropractic 17
Measures of central
tendency (cont.)
• Median
– The value that divides a series of values in
half when they are all listed in order
– When there are an odd number of values
• The median is the middle value
– When there are an even number of values
• Count from each end of the series toward the
middle and then average the 2 middle values
© 2006
Evidence-based Chiropractic 18
Measures of central
tendency (cont.)
• Each of the three methods of measuring
central tendency has certain advantages
and disadvantages
• Which method should be used?
– It depends on the type of data that is being
analyzed
– e.g., categorical, continuous, and the level of
measurement that is involved
© 2006
Evidence-based Chiropractic 19
Levels of measurement
• There are 4 levels of measurement
– Nominal, ordinal, interval, and ratio
1. Nominal
– Data are coded by a number, name, or letter
that is assigned to a category or group
– Examples
• Gender (e.g., male, female)
• Treatment preference (e.g., manipulation,
mobilization, massage)
© 2006
Evidence-based Chiropractic 20
Levels of measurement (cont.)
2. Ordinal
– Is similar to nominal because the
measurements involve categories
– However, the categories are ordered by rank
– Examples
• Pain level (e.g., mild, moderate, severe)
• Military rank (e.g., lieutenant, captain, major,
colonel, general)
© 2006
Evidence-based Chiropractic 21
Levels of measurement (cont.)
• Ordinal values only describe order, not
quantity
– Thus, severe pain is not the same as 2 times
mild pain
• The only mathematical operations allowed
for nominal and ordinal data are counting
of categories
– e.g., 25 males and 30 females
© 2006
Evidence-based Chiropractic 22
Levels of measurement (cont.)
3. Interval
– Measurements are ordered (like ordinal
data)
– Have equal intervals
– Does not have a true zero
– Examples
• The Fahrenheit scale, where 0° does not
correspond to an absence of heat (no true zero)
• In contrast to Kelvin, which does have a true zero
© 2006
Evidence-based Chiropractic 23
Levels of measurement (cont.)
4. Ratio
– Measurements have equal intervals
– There is a true zero
– Ratio is the most advanced level of
measurement, which can handle most types
of mathematical operations
© 2006
Evidence-based Chiropractic 24
Levels of measurement (cont.)
• Ratio examples
– Range of motion
• No movement corresponds to zero degrees
• The interval between 10 and 20 degrees is the
same as between 40 and 50 degrees
– Lifting capacity
• A person who is unable to lift scores zero
• A person who lifts 30 kg can lift twice as much as
one who lifts 15 kg
© 2006
Evidence-based Chiropractic 25
Levels of measurement (cont.)
• NOIR is a mnemonic to help remember
the names and order of the levels of
measurement
– Nominal
Ordinal
Interval
Ratio
© 2006
Evidence-based Chiropractic 26
Levels of measurement (cont.)
Measurement scale
Permissible mathematic
operations
Best measure of
central tendency
Nominal Counting Mode
Ordinal
Greater or less than
operations
Median
Interval Addition and subtraction
Symmetrical – Mean
Skewed – Median
Ratio
Addition, subtraction,
multiplication and division
Symmetrical – Mean
Skewed – Median
© 2006
Evidence-based Chiropractic 27
The shape of data
• Histograms of frequency distributions have
shape
• Distributions are often symmetrical with
most scores falling in the middle and fewer
toward the extremes
• Most biological data are symmetrically
distributed and form a normal curve (a.k.a,
bell-shaped curve)
© 2006
Evidence-based Chiropractic 28
The shape of data (cont.)
Line depicting
the shape of
the data
© 2006
Evidence-based Chiropractic 29
The normal distribution
• The area under a normal curve has a
normal distribution (a.k.a., Gaussian
distribution)
• Properties of a normal distribution
– It is symmetric about its mean
– The highest point is at its mean
– The height of the curve decreases as one
moves away from the mean in either direction,
approaching, but never reaching zero
© 2006
Evidence-based Chiropractic 30
The normal distribution (cont.)
Mean
A normal distribution is symmetric about its mean
As one moves away from
the mean in either direction
the height of the curve
decreases, approaching,
but never reaching zero
The highest point of
the overlying
normal curve is at
the mean
© 2006
Evidence-based Chiropractic 31
The normal distribution (cont.)
Mean = Median = Mode
© 2006
Evidence-based Chiropractic 32
Skewed distributions
• The data are not distributed symmetrically
in skewed distributions
– Consequently, the mean, median, and mode
are not equal and are in different positions
– Scores are clustered at one end of the
distribution
– A small number of extreme values are located
in the limits of the opposite end
© 2006
Evidence-based Chiropractic 33
Skewed distributions (cont.)
• Skew is always toward the direction of the
longer tail
– Positive if skewed to the right
– Negative if to the left
The mean is shifted
the most
© 2006
Evidence-based Chiropractic 34
Skewed distributions (cont.)
• Because the mean is shifted so much, it is
not the best estimate of the average score
for skewed distributions
• The median is a better estimate of the
center of skewed distributions
– It will be the central point of any distribution
– 50% of the values are above and 50% below
the median
© 2006
Evidence-based Chiropractic 35
More properties
of normal curves
• About 68.3% of the area under a normal
curve is within one standard deviation
(SD) of the mean
• About 95.5% is within two SDs
• About 99.7% is within three SDs
© 2006
Evidence-based Chiropractic 36
More properties
of normal curves (cont.)
© 2006
Evidence-based Chiropractic 37
Standard deviation (SD)
• SD is a measure of the variability of a set
of data
• The mean represents the average of a
group of scores, with some of the scores
being above the mean and some below
– This range of scores is referred to as
variability or spread
• Variance (S2) is another measure of
spread
© 2006
Evidence-based Chiropractic 38
SD (cont.)
• In effect, SD is the average amount of
spread in a distribution of scores
• The next slide is a group of 10 patients
whose mean age is 40 years
– Some are older than 40 and some younger
© 2006
Evidence-based Chiropractic 39
SD (cont.)
Ages are spread
out along an X axis
The amount ages are
spread out is known as
dispersion or spread
© 2006
Evidence-based Chiropractic 40
Distances ages deviate above
and below the mean
Adding deviations
always equals zero
Etc.
© 2006
Evidence-based Chiropractic 41
Calculating S2
• To find the average, one would normally
total the scores above and below the
mean, add them together, and then divide
by the number of values
• However, the total always equals zero
– Values must first be squared, which cancels
the negative signs
© 2006
Evidence-based Chiropractic 42
Calculating S2 cont.
Symbol for SD of a sample
 for a population
S2 is not in the
same units (age),
but SD is
© 2006
Evidence-based Chiropractic 43
Calculating SD with Excel
Enter values in a column
© 2006
Evidence-based Chiropractic 44
SD with Excel (cont.)
Click Data Analysis
on the Tools menu
© 2006
Evidence-based Chiropractic 45
SD with Excel (cont.)
Select Descriptive
Statistics and click OK
© 2006
Evidence-based Chiropractic 46
SD with Excel (cont.)
Click Input Range icon
© 2006
Evidence-based Chiropractic 47
SD with Excel (cont.)
Highlight all the
values in the column
© 2006
Evidence-based Chiropractic 48
SD with Excel (cont.)
Check if labels are
in the first row
Check Summary
Statistics
Click OK
© 2006
Evidence-based Chiropractic 49
SD with Excel (cont.)
SD is calculated precisely
Plus several other DSs
© 2006
Evidence-based Chiropractic 50
Wide spread results in higher SDs
narrow spread in lower SDs
© 2006
Evidence-based Chiropractic 51
Spread is important when
comparing 2 or more group means
It is more difficult to
see a clear distinction
between groups
in the upper example
because the spread is
wider, even though the
means are the same
© 2006
Evidence-based Chiropractic 52
z-scores
• The number of SDs that a specific score is
above or below the mean in a distribution
• Raw scores can be converted to z-scores
by subtracting the mean from the raw
score then dividing the difference by the
SD




X
z
© 2006
Evidence-based Chiropractic 53
z-scores (cont.)
• Standardization
– The process of converting raw to z-scores
– The resulting distribution of z-scores will
always have a mean of zero, a SD of one,
and an area under the curve equal to one
• The proportion of scores that are higher or
lower than a specific z-score can be
determined by referring to a z-table
© 2006
Evidence-based Chiropractic 54
z-scores (cont.)
Refer to a z-table
to find proportion
under the curve
© 2006
Evidence-based Chiropractic 55
z-scores (cont.)
Partial z-table (to z = 1.5) showing proportions of the
area under a normal curve for different values of z.
Z 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09
0.0 0.5000 0.5040 0.5080 0.5120 0.5160 0.5199 0.5239 0.5279 0.5319 0.5359
0.1 0.5398 0.5438 0.5478 0.5517 0.5557 0.5596 0.5636 0.5675 0.5714 0.5753
0.2 0.5793 0.5832 0.5871 0.5910 0.5948 0.5987 0.6026 0.6064 0.6103 0.6141
0.3 0.6179 0.6217 0.6255 0.6293 0.6331 0.6368 0.6406 0.6443 0.6480 0.6517
0.4 0.6554 0.6591 0.6628 0.6664 0.6700 0.6736 0.6772 0.6808 0.6844 0.6879
0.5 0.6915 0.6950 0.6985 0.7019 0.7054 0.7088 0.7123 0.7157 0.7190 0.7224
0.6 0.7257 0.7291 0.7324 0.7357 0.7389 0.7422 0.7454 0.7486 0.7517 0.7549
0.7 0.7580 0.7611 0.7642 0.7673 0.7704 0.7734 0.7764 0.7794 0.7823 0.7852
0.8 0.7881 0.7910 0.7939 0.7967 0.7995 0.8023 0.8051 0.8078 0.8106 0.8133
0.9 0.8159 0.8186 0.8212 0.8238 0.8264 0.8289 0.8315 0.8340 0.8365 0.8389
1.0 0.8413 0.8438 0.8461 0.8485 0.8508 0.8531 0.8554 0.8577 0.8599 0.8621
1.1 0.8643 0.8665 0.8686 0.8708 0.8729 0.8749 0.8770 0.8790 0.8810 0.8830
1.2 0.8849 0.8869 0.8888 0.8907 0.8925 0.8944 0.8962 0.8980 0.8997 0.9015
1.3 0.9032 0.9049 0.9066 0.9082 0.9099 0.9115 0.9131 0.9147 0.9162 0.9177
1.4 0.9192 0.9207 0.9222 0.9236 0.9251 0.9265 0.9279 0.9292 0.9306 0.9319
1.5 0.9332 0.9345 0.9357 0.9370 0.9382 0.9394 0.9406 0.9418 0.9429 0.9441
0.9332
Corresponds to the area
under the curve in black

More Related Content

Similar to dokumen.tips_biostatistics-basics-biostatistics.ppt

PARAMETRIC TESTS.pptx
PARAMETRIC TESTS.pptxPARAMETRIC TESTS.pptx
PARAMETRIC TESTS.pptx
DrLasya
 
Data Display and Summary
Data Display and SummaryData Display and Summary
Data Display and Summary
DrZahid Khan
 
Univariate Analysis
 Univariate Analysis Univariate Analysis
Univariate Analysis
Soumya Sahoo
 
Scales of measurement
Scales of measurementScales of measurement
Scales of measurement
Regent University
 
3 measures of central dendency
3  measures of central dendency3  measures of central dendency
3 measures of central dendency
Dr. Nazar Jaf
 
Introduction to statistics RSS6 2014
Introduction to statistics RSS6 2014Introduction to statistics RSS6 2014
Introduction to statistics RSS6 2014
RSS6
 
Biostatistics
Biostatistics Biostatistics
Biostatistics
Vaibhav Ambashikar
 
Research Course - RCT.pdf
Research Course - RCT.pdfResearch Course - RCT.pdf
Research Course - RCT.pdf
MarioKopljar1
 
Research Course - RCT.pptx
Research Course - RCT.pptxResearch Course - RCT.pptx
Research Course - RCT.pptx
MarioKopljar1
 
Research Course - RCT.pdf
Research Course - RCT.pdfResearch Course - RCT.pdf
Research Course - RCT.pdf
MarioKopljar1
 
Business statistics
Business statisticsBusiness statistics
Business statistics
Ravi Prakash
 
Biostatistics mean median mode unit 1.pptx
Biostatistics mean median mode unit 1.pptxBiostatistics mean median mode unit 1.pptx
Biostatistics mean median mode unit 1.pptx
SailajaReddyGunnam
 
Intro statistics
Intro statisticsIntro statistics
Intro statistics
Arash Kamrani
 
satatistics Presentation.pptx
satatistics Presentation.pptxsatatistics Presentation.pptx
satatistics Presentation.pptx
MdMatiurRahman25
 
Clinical research ( Medical stat. concepts)
Clinical research ( Medical stat. concepts)Clinical research ( Medical stat. concepts)
Clinical research ( Medical stat. concepts)
Mohamed Fahmy Dehim
 
3. Statistical Analysis.pptx
3. Statistical Analysis.pptx3. Statistical Analysis.pptx
3. Statistical Analysis.pptx
jeyanthisivakumar
 
Biostatistics ppt
Biostatistics  pptBiostatistics  ppt
Biostatistics ppt
santhoshikayithi
 
Data Display and Summary
Data Display and SummaryData Display and Summary
Data Display and Summary
DrZahid Khan
 
Biostatistics
BiostatisticsBiostatistics
Introduction to Statistics23122223.ppt
Introduction to Statistics23122223.pptIntroduction to Statistics23122223.ppt
Introduction to Statistics23122223.ppt
pathianithanaidu
 

Similar to dokumen.tips_biostatistics-basics-biostatistics.ppt (20)

PARAMETRIC TESTS.pptx
PARAMETRIC TESTS.pptxPARAMETRIC TESTS.pptx
PARAMETRIC TESTS.pptx
 
Data Display and Summary
Data Display and SummaryData Display and Summary
Data Display and Summary
 
Univariate Analysis
 Univariate Analysis Univariate Analysis
Univariate Analysis
 
Scales of measurement
Scales of measurementScales of measurement
Scales of measurement
 
3 measures of central dendency
3  measures of central dendency3  measures of central dendency
3 measures of central dendency
 
Introduction to statistics RSS6 2014
Introduction to statistics RSS6 2014Introduction to statistics RSS6 2014
Introduction to statistics RSS6 2014
 
Biostatistics
Biostatistics Biostatistics
Biostatistics
 
Research Course - RCT.pdf
Research Course - RCT.pdfResearch Course - RCT.pdf
Research Course - RCT.pdf
 
Research Course - RCT.pptx
Research Course - RCT.pptxResearch Course - RCT.pptx
Research Course - RCT.pptx
 
Research Course - RCT.pdf
Research Course - RCT.pdfResearch Course - RCT.pdf
Research Course - RCT.pdf
 
Business statistics
Business statisticsBusiness statistics
Business statistics
 
Biostatistics mean median mode unit 1.pptx
Biostatistics mean median mode unit 1.pptxBiostatistics mean median mode unit 1.pptx
Biostatistics mean median mode unit 1.pptx
 
Intro statistics
Intro statisticsIntro statistics
Intro statistics
 
satatistics Presentation.pptx
satatistics Presentation.pptxsatatistics Presentation.pptx
satatistics Presentation.pptx
 
Clinical research ( Medical stat. concepts)
Clinical research ( Medical stat. concepts)Clinical research ( Medical stat. concepts)
Clinical research ( Medical stat. concepts)
 
3. Statistical Analysis.pptx
3. Statistical Analysis.pptx3. Statistical Analysis.pptx
3. Statistical Analysis.pptx
 
Biostatistics ppt
Biostatistics  pptBiostatistics  ppt
Biostatistics ppt
 
Data Display and Summary
Data Display and SummaryData Display and Summary
Data Display and Summary
 
Biostatistics
BiostatisticsBiostatistics
Biostatistics
 
Introduction to Statistics23122223.ppt
Introduction to Statistics23122223.pptIntroduction to Statistics23122223.ppt
Introduction to Statistics23122223.ppt
 

More from Abebe334138

Advanced Biostatistics presentation pptx
Advanced Biostatistics presentation  pptxAdvanced Biostatistics presentation  pptx
Advanced Biostatistics presentation pptx
Abebe334138
 
Regression Analysis.ppt
Regression Analysis.pptRegression Analysis.ppt
Regression Analysis.ppt
Abebe334138
 
Lecture_5Conditional_Probability_Bayes_T.pptx
Lecture_5Conditional_Probability_Bayes_T.pptxLecture_5Conditional_Probability_Bayes_T.pptx
Lecture_5Conditional_Probability_Bayes_T.pptx
Abebe334138
 
3. Statistical inference_anesthesia.pptx
3.  Statistical inference_anesthesia.pptx3.  Statistical inference_anesthesia.pptx
3. Statistical inference_anesthesia.pptx
Abebe334138
 
chapter-7b.pptx
chapter-7b.pptxchapter-7b.pptx
chapter-7b.pptx
Abebe334138
 
chapter -7.pptx
chapter -7.pptxchapter -7.pptx
chapter -7.pptx
Abebe334138
 
7 Chi-square and F (1).ppt
7 Chi-square and F (1).ppt7 Chi-square and F (1).ppt
7 Chi-square and F (1).ppt
Abebe334138
 
Presentation1.pptx
Presentation1.pptxPresentation1.pptx
Presentation1.pptx
Abebe334138
 
RCT CH0.ppt
RCT CH0.pptRCT CH0.ppt
RCT CH0.ppt
Abebe334138
 
1. intro_biostatistics.pptx
1. intro_biostatistics.pptx1. intro_biostatistics.pptx
1. intro_biostatistics.pptx
Abebe334138
 
Lecture_R.ppt
Lecture_R.pptLecture_R.ppt
Lecture_R.ppt
Abebe334138
 
ppt1221[1][1].pptx
ppt1221[1][1].pptxppt1221[1][1].pptx
ppt1221[1][1].pptx
Abebe334138
 

More from Abebe334138 (12)

Advanced Biostatistics presentation pptx
Advanced Biostatistics presentation  pptxAdvanced Biostatistics presentation  pptx
Advanced Biostatistics presentation pptx
 
Regression Analysis.ppt
Regression Analysis.pptRegression Analysis.ppt
Regression Analysis.ppt
 
Lecture_5Conditional_Probability_Bayes_T.pptx
Lecture_5Conditional_Probability_Bayes_T.pptxLecture_5Conditional_Probability_Bayes_T.pptx
Lecture_5Conditional_Probability_Bayes_T.pptx
 
3. Statistical inference_anesthesia.pptx
3.  Statistical inference_anesthesia.pptx3.  Statistical inference_anesthesia.pptx
3. Statistical inference_anesthesia.pptx
 
chapter-7b.pptx
chapter-7b.pptxchapter-7b.pptx
chapter-7b.pptx
 
chapter -7.pptx
chapter -7.pptxchapter -7.pptx
chapter -7.pptx
 
7 Chi-square and F (1).ppt
7 Chi-square and F (1).ppt7 Chi-square and F (1).ppt
7 Chi-square and F (1).ppt
 
Presentation1.pptx
Presentation1.pptxPresentation1.pptx
Presentation1.pptx
 
RCT CH0.ppt
RCT CH0.pptRCT CH0.ppt
RCT CH0.ppt
 
1. intro_biostatistics.pptx
1. intro_biostatistics.pptx1. intro_biostatistics.pptx
1. intro_biostatistics.pptx
 
Lecture_R.ppt
Lecture_R.pptLecture_R.ppt
Lecture_R.ppt
 
ppt1221[1][1].pptx
ppt1221[1][1].pptxppt1221[1][1].pptx
ppt1221[1][1].pptx
 

Recently uploaded

The Ipsos - AI - Monitor 2024 Report.pdf
The  Ipsos - AI - Monitor 2024 Report.pdfThe  Ipsos - AI - Monitor 2024 Report.pdf
The Ipsos - AI - Monitor 2024 Report.pdf
Social Samosa
 
原版一比一利兹贝克特大学毕业证(LeedsBeckett毕业证书)如何办理
原版一比一利兹贝克特大学毕业证(LeedsBeckett毕业证书)如何办理原版一比一利兹贝克特大学毕业证(LeedsBeckett毕业证书)如何办理
原版一比一利兹贝克特大学毕业证(LeedsBeckett毕业证书)如何办理
wyddcwye1
 
一比一原版(UCSB文凭证书)圣芭芭拉分校毕业证如何办理
一比一原版(UCSB文凭证书)圣芭芭拉分校毕业证如何办理一比一原版(UCSB文凭证书)圣芭芭拉分校毕业证如何办理
一比一原版(UCSB文凭证书)圣芭芭拉分校毕业证如何办理
nuttdpt
 
原版制作(unimelb毕业证书)墨尔本大学毕业证Offer一模一样
原版制作(unimelb毕业证书)墨尔本大学毕业证Offer一模一样原版制作(unimelb毕业证书)墨尔本大学毕业证Offer一模一样
原版制作(unimelb毕业证书)墨尔本大学毕业证Offer一模一样
ihavuls
 
University of New South Wales degree offer diploma Transcript
University of New South Wales degree offer diploma TranscriptUniversity of New South Wales degree offer diploma Transcript
University of New South Wales degree offer diploma Transcript
soxrziqu
 
原版一比一弗林德斯大学毕业证(Flinders毕业证书)如何办理
原版一比一弗林德斯大学毕业证(Flinders毕业证书)如何办理原版一比一弗林德斯大学毕业证(Flinders毕业证书)如何办理
原版一比一弗林德斯大学毕业证(Flinders毕业证书)如何办理
a9qfiubqu
 
一比一原版巴斯大学毕业证(Bath毕业证书)学历如何办理
一比一原版巴斯大学毕业证(Bath毕业证书)学历如何办理一比一原版巴斯大学毕业证(Bath毕业证书)学历如何办理
一比一原版巴斯大学毕业证(Bath毕业证书)学历如何办理
y3i0qsdzb
 
在线办理(英国UCA毕业证书)创意艺术大学毕业证在读证明一模一样
在线办理(英国UCA毕业证书)创意艺术大学毕业证在读证明一模一样在线办理(英国UCA毕业证书)创意艺术大学毕业证在读证明一模一样
在线办理(英国UCA毕业证书)创意艺术大学毕业证在读证明一模一样
v7oacc3l
 
A presentation that explain the Power BI Licensing
A presentation that explain the Power BI LicensingA presentation that explain the Power BI Licensing
A presentation that explain the Power BI Licensing
AlessioFois2
 
Orchestrating the Future: Navigating Today's Data Workflow Challenges with Ai...
Orchestrating the Future: Navigating Today's Data Workflow Challenges with Ai...Orchestrating the Future: Navigating Today's Data Workflow Challenges with Ai...
Orchestrating the Future: Navigating Today's Data Workflow Challenges with Ai...
Kaxil Naik
 
STATATHON: Unleashing the Power of Statistics in a 48-Hour Knowledge Extravag...
STATATHON: Unleashing the Power of Statistics in a 48-Hour Knowledge Extravag...STATATHON: Unleashing the Power of Statistics in a 48-Hour Knowledge Extravag...
STATATHON: Unleashing the Power of Statistics in a 48-Hour Knowledge Extravag...
sameer shah
 
一比一原版(UO毕业证)渥太华大学毕业证如何办理
一比一原版(UO毕业证)渥太华大学毕业证如何办理一比一原版(UO毕业证)渥太华大学毕业证如何办理
一比一原版(UO毕业证)渥太华大学毕业证如何办理
aqzctr7x
 
一比一原版(GWU,GW文凭证书)乔治·华盛顿大学毕业证如何办理
一比一原版(GWU,GW文凭证书)乔治·华盛顿大学毕业证如何办理一比一原版(GWU,GW文凭证书)乔治·华盛顿大学毕业证如何办理
一比一原版(GWU,GW文凭证书)乔治·华盛顿大学毕业证如何办理
bopyb
 
办(uts毕业证书)悉尼科技大学毕业证学历证书原版一模一样
办(uts毕业证书)悉尼科技大学毕业证学历证书原版一模一样办(uts毕业证书)悉尼科技大学毕业证学历证书原版一模一样
办(uts毕业证书)悉尼科技大学毕业证学历证书原版一模一样
apvysm8
 
Analysis insight about a Flyball dog competition team's performance
Analysis insight about a Flyball dog competition team's performanceAnalysis insight about a Flyball dog competition team's performance
Analysis insight about a Flyball dog competition team's performance
roli9797
 
Intelligence supported media monitoring in veterinary medicine
Intelligence supported media monitoring in veterinary medicineIntelligence supported media monitoring in veterinary medicine
Intelligence supported media monitoring in veterinary medicine
AndrzejJarynowski
 
End-to-end pipeline agility - Berlin Buzzwords 2024
End-to-end pipeline agility - Berlin Buzzwords 2024End-to-end pipeline agility - Berlin Buzzwords 2024
End-to-end pipeline agility - Berlin Buzzwords 2024
Lars Albertsson
 
Predictably Improve Your B2B Tech Company's Performance by Leveraging Data
Predictably Improve Your B2B Tech Company's Performance by Leveraging DataPredictably Improve Your B2B Tech Company's Performance by Leveraging Data
Predictably Improve Your B2B Tech Company's Performance by Leveraging Data
Kiwi Creative
 
Udemy_2024_Global_Learning_Skills_Trends_Report (1).pdf
Udemy_2024_Global_Learning_Skills_Trends_Report (1).pdfUdemy_2024_Global_Learning_Skills_Trends_Report (1).pdf
Udemy_2024_Global_Learning_Skills_Trends_Report (1).pdf
Fernanda Palhano
 
4th Modern Marketing Reckoner by MMA Global India & Group M: 60+ experts on W...
4th Modern Marketing Reckoner by MMA Global India & Group M: 60+ experts on W...4th Modern Marketing Reckoner by MMA Global India & Group M: 60+ experts on W...
4th Modern Marketing Reckoner by MMA Global India & Group M: 60+ experts on W...
Social Samosa
 

Recently uploaded (20)

The Ipsos - AI - Monitor 2024 Report.pdf
The  Ipsos - AI - Monitor 2024 Report.pdfThe  Ipsos - AI - Monitor 2024 Report.pdf
The Ipsos - AI - Monitor 2024 Report.pdf
 
原版一比一利兹贝克特大学毕业证(LeedsBeckett毕业证书)如何办理
原版一比一利兹贝克特大学毕业证(LeedsBeckett毕业证书)如何办理原版一比一利兹贝克特大学毕业证(LeedsBeckett毕业证书)如何办理
原版一比一利兹贝克特大学毕业证(LeedsBeckett毕业证书)如何办理
 
一比一原版(UCSB文凭证书)圣芭芭拉分校毕业证如何办理
一比一原版(UCSB文凭证书)圣芭芭拉分校毕业证如何办理一比一原版(UCSB文凭证书)圣芭芭拉分校毕业证如何办理
一比一原版(UCSB文凭证书)圣芭芭拉分校毕业证如何办理
 
原版制作(unimelb毕业证书)墨尔本大学毕业证Offer一模一样
原版制作(unimelb毕业证书)墨尔本大学毕业证Offer一模一样原版制作(unimelb毕业证书)墨尔本大学毕业证Offer一模一样
原版制作(unimelb毕业证书)墨尔本大学毕业证Offer一模一样
 
University of New South Wales degree offer diploma Transcript
University of New South Wales degree offer diploma TranscriptUniversity of New South Wales degree offer diploma Transcript
University of New South Wales degree offer diploma Transcript
 
原版一比一弗林德斯大学毕业证(Flinders毕业证书)如何办理
原版一比一弗林德斯大学毕业证(Flinders毕业证书)如何办理原版一比一弗林德斯大学毕业证(Flinders毕业证书)如何办理
原版一比一弗林德斯大学毕业证(Flinders毕业证书)如何办理
 
一比一原版巴斯大学毕业证(Bath毕业证书)学历如何办理
一比一原版巴斯大学毕业证(Bath毕业证书)学历如何办理一比一原版巴斯大学毕业证(Bath毕业证书)学历如何办理
一比一原版巴斯大学毕业证(Bath毕业证书)学历如何办理
 
在线办理(英国UCA毕业证书)创意艺术大学毕业证在读证明一模一样
在线办理(英国UCA毕业证书)创意艺术大学毕业证在读证明一模一样在线办理(英国UCA毕业证书)创意艺术大学毕业证在读证明一模一样
在线办理(英国UCA毕业证书)创意艺术大学毕业证在读证明一模一样
 
A presentation that explain the Power BI Licensing
A presentation that explain the Power BI LicensingA presentation that explain the Power BI Licensing
A presentation that explain the Power BI Licensing
 
Orchestrating the Future: Navigating Today's Data Workflow Challenges with Ai...
Orchestrating the Future: Navigating Today's Data Workflow Challenges with Ai...Orchestrating the Future: Navigating Today's Data Workflow Challenges with Ai...
Orchestrating the Future: Navigating Today's Data Workflow Challenges with Ai...
 
STATATHON: Unleashing the Power of Statistics in a 48-Hour Knowledge Extravag...
STATATHON: Unleashing the Power of Statistics in a 48-Hour Knowledge Extravag...STATATHON: Unleashing the Power of Statistics in a 48-Hour Knowledge Extravag...
STATATHON: Unleashing the Power of Statistics in a 48-Hour Knowledge Extravag...
 
一比一原版(UO毕业证)渥太华大学毕业证如何办理
一比一原版(UO毕业证)渥太华大学毕业证如何办理一比一原版(UO毕业证)渥太华大学毕业证如何办理
一比一原版(UO毕业证)渥太华大学毕业证如何办理
 
一比一原版(GWU,GW文凭证书)乔治·华盛顿大学毕业证如何办理
一比一原版(GWU,GW文凭证书)乔治·华盛顿大学毕业证如何办理一比一原版(GWU,GW文凭证书)乔治·华盛顿大学毕业证如何办理
一比一原版(GWU,GW文凭证书)乔治·华盛顿大学毕业证如何办理
 
办(uts毕业证书)悉尼科技大学毕业证学历证书原版一模一样
办(uts毕业证书)悉尼科技大学毕业证学历证书原版一模一样办(uts毕业证书)悉尼科技大学毕业证学历证书原版一模一样
办(uts毕业证书)悉尼科技大学毕业证学历证书原版一模一样
 
Analysis insight about a Flyball dog competition team's performance
Analysis insight about a Flyball dog competition team's performanceAnalysis insight about a Flyball dog competition team's performance
Analysis insight about a Flyball dog competition team's performance
 
Intelligence supported media monitoring in veterinary medicine
Intelligence supported media monitoring in veterinary medicineIntelligence supported media monitoring in veterinary medicine
Intelligence supported media monitoring in veterinary medicine
 
End-to-end pipeline agility - Berlin Buzzwords 2024
End-to-end pipeline agility - Berlin Buzzwords 2024End-to-end pipeline agility - Berlin Buzzwords 2024
End-to-end pipeline agility - Berlin Buzzwords 2024
 
Predictably Improve Your B2B Tech Company's Performance by Leveraging Data
Predictably Improve Your B2B Tech Company's Performance by Leveraging DataPredictably Improve Your B2B Tech Company's Performance by Leveraging Data
Predictably Improve Your B2B Tech Company's Performance by Leveraging Data
 
Udemy_2024_Global_Learning_Skills_Trends_Report (1).pdf
Udemy_2024_Global_Learning_Skills_Trends_Report (1).pdfUdemy_2024_Global_Learning_Skills_Trends_Report (1).pdf
Udemy_2024_Global_Learning_Skills_Trends_Report (1).pdf
 
4th Modern Marketing Reckoner by MMA Global India & Group M: 60+ experts on W...
4th Modern Marketing Reckoner by MMA Global India & Group M: 60+ experts on W...4th Modern Marketing Reckoner by MMA Global India & Group M: 60+ experts on W...
4th Modern Marketing Reckoner by MMA Global India & Group M: 60+ experts on W...
 

dokumen.tips_biostatistics-basics-biostatistics.ppt

  • 1. © 2006 1 Biostatistics Basics An introduction to an expansive and complex field
  • 2. © 2006 Evidence-based Chiropractic 2 Common statistical terms • Data – Measurements or observations of a variable • Variable – A characteristic that is observed or manipulated – Can take on different values
  • 3. © 2006 Evidence-based Chiropractic 3 Statistical terms (cont.) • Independent variables – Precede dependent variables in time – Are often manipulated by the researcher – The treatment or intervention that is used in a study • Dependent variables – What is measured as an outcome in a study – Values depend on the independent variable
  • 4. © 2006 Evidence-based Chiropractic 4 Statistical terms (cont.) • Parameters – Summary data from a population • Statistics – Summary data from a sample
  • 5. © 2006 Evidence-based Chiropractic 5 Populations • A population is the group from which a sample is drawn – e.g., headache patients in a chiropractic office; automobile crash victims in an emergency room • In research, it is not practical to include all members of a population • Thus, a sample (a subset of a population) is taken
  • 6. © 2006 Evidence-based Chiropractic 6 Random samples • Subjects are selected from a population so that each individual has an equal chance of being selected • Random samples are representative of the source population • Non-random samples are not representative – May be biased regarding age, severity of the condition, socioeconomic status etc.
  • 7. © 2006 Evidence-based Chiropractic 7 Random samples (cont.) • Random samples are rarely utilized in health care research • Instead, patients are randomly assigned to treatment and control groups – Each person has an equal chance of being assigned to either of the groups • Random assignment is also known as randomization
  • 8. © 2006 Evidence-based Chiropractic 8 Descriptive statistics (DSs) • A way to summarize data from a sample or a population • DSs illustrate the shape, central tendency, and variability of a set of data – The shape of data has to do with the frequencies of the values of observations
  • 9. © 2006 Evidence-based Chiropractic 9 DSs (cont.) – Central tendency describes the location of the middle of the data – Variability is the extent values are spread above and below the middle values • a.k.a., Dispersion • DSs can be distinguished from inferential statistics – DSs are not capable of testing hypotheses
  • 10. © 2006 Evidence-based Chiropractic 10 Hypothetical study data (partial from book) Case # Visits 1 7 2 2 3 2 4 3 5 4 6 3 7 5 8 3 9 4 10 6 11 2 12 3 13 7 14 4 • Distribution provides a summary of: – Frequencies of each of the values • 2 – 3 • 3 – 4 • 4 – 3 • 5 – 1 • 6 – 1 • 7 – 2 – Ranges of values • Lowest = 2 • Highest = 7 etc.
  • 11. © 2006 Evidence-based Chiropractic 11 Frequency distribution table Frequency Percent Cumulative % • 2 3 21.4 21.4 • 3 4 28.6 50.0 • 4 3 21.4 71.4 • 5 1 7.1 78.5 • 6 1 7.1 85.6 • 7 2 14.3 100.0
  • 12. © 2006 Evidence-based Chiropractic 12 Frequency distributions are often depicted by a histogram
  • 13. © 2006 Evidence-based Chiropractic 13 Histograms (cont.) • A histogram is a type of bar chart, but there are no spaces between the bars • Histograms are used to visually depict frequency distributions of continuous data • Bar charts are used to depict categorical information – e.g., Male–Female, Mild–Moderate–Severe, etc.
  • 14. © 2006 Evidence-based Chiropractic 14 Measures of central tendency • Mean (a.k.a., average) – The most commonly used DS • To calculate the mean – Add all values of a series of numbers and then divided by the total number of elements
  • 15. © 2006 Evidence-based Chiropractic 15 Formula to calculate the mean • Mean of a sample • Mean of a population  (X bar) refers to the mean of a sample and refers to the mean of a population  EX is a command that adds all of the X values  n is the total number of values in the series of a sample and N is the same for a population X μ N X    n X X  
  • 16. © 2006 Evidence-based Chiropractic 16 Measures of central tendency (cont.) • Mode – The most frequently occurring value in a series – The modal value is the highest bar in a histogram Mode
  • 17. © 2006 Evidence-based Chiropractic 17 Measures of central tendency (cont.) • Median – The value that divides a series of values in half when they are all listed in order – When there are an odd number of values • The median is the middle value – When there are an even number of values • Count from each end of the series toward the middle and then average the 2 middle values
  • 18. © 2006 Evidence-based Chiropractic 18 Measures of central tendency (cont.) • Each of the three methods of measuring central tendency has certain advantages and disadvantages • Which method should be used? – It depends on the type of data that is being analyzed – e.g., categorical, continuous, and the level of measurement that is involved
  • 19. © 2006 Evidence-based Chiropractic 19 Levels of measurement • There are 4 levels of measurement – Nominal, ordinal, interval, and ratio 1. Nominal – Data are coded by a number, name, or letter that is assigned to a category or group – Examples • Gender (e.g., male, female) • Treatment preference (e.g., manipulation, mobilization, massage)
  • 20. © 2006 Evidence-based Chiropractic 20 Levels of measurement (cont.) 2. Ordinal – Is similar to nominal because the measurements involve categories – However, the categories are ordered by rank – Examples • Pain level (e.g., mild, moderate, severe) • Military rank (e.g., lieutenant, captain, major, colonel, general)
  • 21. © 2006 Evidence-based Chiropractic 21 Levels of measurement (cont.) • Ordinal values only describe order, not quantity – Thus, severe pain is not the same as 2 times mild pain • The only mathematical operations allowed for nominal and ordinal data are counting of categories – e.g., 25 males and 30 females
  • 22. © 2006 Evidence-based Chiropractic 22 Levels of measurement (cont.) 3. Interval – Measurements are ordered (like ordinal data) – Have equal intervals – Does not have a true zero – Examples • The Fahrenheit scale, where 0° does not correspond to an absence of heat (no true zero) • In contrast to Kelvin, which does have a true zero
  • 23. © 2006 Evidence-based Chiropractic 23 Levels of measurement (cont.) 4. Ratio – Measurements have equal intervals – There is a true zero – Ratio is the most advanced level of measurement, which can handle most types of mathematical operations
  • 24. © 2006 Evidence-based Chiropractic 24 Levels of measurement (cont.) • Ratio examples – Range of motion • No movement corresponds to zero degrees • The interval between 10 and 20 degrees is the same as between 40 and 50 degrees – Lifting capacity • A person who is unable to lift scores zero • A person who lifts 30 kg can lift twice as much as one who lifts 15 kg
  • 25. © 2006 Evidence-based Chiropractic 25 Levels of measurement (cont.) • NOIR is a mnemonic to help remember the names and order of the levels of measurement – Nominal Ordinal Interval Ratio
  • 26. © 2006 Evidence-based Chiropractic 26 Levels of measurement (cont.) Measurement scale Permissible mathematic operations Best measure of central tendency Nominal Counting Mode Ordinal Greater or less than operations Median Interval Addition and subtraction Symmetrical – Mean Skewed – Median Ratio Addition, subtraction, multiplication and division Symmetrical – Mean Skewed – Median
  • 27. © 2006 Evidence-based Chiropractic 27 The shape of data • Histograms of frequency distributions have shape • Distributions are often symmetrical with most scores falling in the middle and fewer toward the extremes • Most biological data are symmetrically distributed and form a normal curve (a.k.a, bell-shaped curve)
  • 28. © 2006 Evidence-based Chiropractic 28 The shape of data (cont.) Line depicting the shape of the data
  • 29. © 2006 Evidence-based Chiropractic 29 The normal distribution • The area under a normal curve has a normal distribution (a.k.a., Gaussian distribution) • Properties of a normal distribution – It is symmetric about its mean – The highest point is at its mean – The height of the curve decreases as one moves away from the mean in either direction, approaching, but never reaching zero
  • 30. © 2006 Evidence-based Chiropractic 30 The normal distribution (cont.) Mean A normal distribution is symmetric about its mean As one moves away from the mean in either direction the height of the curve decreases, approaching, but never reaching zero The highest point of the overlying normal curve is at the mean
  • 31. © 2006 Evidence-based Chiropractic 31 The normal distribution (cont.) Mean = Median = Mode
  • 32. © 2006 Evidence-based Chiropractic 32 Skewed distributions • The data are not distributed symmetrically in skewed distributions – Consequently, the mean, median, and mode are not equal and are in different positions – Scores are clustered at one end of the distribution – A small number of extreme values are located in the limits of the opposite end
  • 33. © 2006 Evidence-based Chiropractic 33 Skewed distributions (cont.) • Skew is always toward the direction of the longer tail – Positive if skewed to the right – Negative if to the left The mean is shifted the most
  • 34. © 2006 Evidence-based Chiropractic 34 Skewed distributions (cont.) • Because the mean is shifted so much, it is not the best estimate of the average score for skewed distributions • The median is a better estimate of the center of skewed distributions – It will be the central point of any distribution – 50% of the values are above and 50% below the median
  • 35. © 2006 Evidence-based Chiropractic 35 More properties of normal curves • About 68.3% of the area under a normal curve is within one standard deviation (SD) of the mean • About 95.5% is within two SDs • About 99.7% is within three SDs
  • 36. © 2006 Evidence-based Chiropractic 36 More properties of normal curves (cont.)
  • 37. © 2006 Evidence-based Chiropractic 37 Standard deviation (SD) • SD is a measure of the variability of a set of data • The mean represents the average of a group of scores, with some of the scores being above the mean and some below – This range of scores is referred to as variability or spread • Variance (S2) is another measure of spread
  • 38. © 2006 Evidence-based Chiropractic 38 SD (cont.) • In effect, SD is the average amount of spread in a distribution of scores • The next slide is a group of 10 patients whose mean age is 40 years – Some are older than 40 and some younger
  • 39. © 2006 Evidence-based Chiropractic 39 SD (cont.) Ages are spread out along an X axis The amount ages are spread out is known as dispersion or spread
  • 40. © 2006 Evidence-based Chiropractic 40 Distances ages deviate above and below the mean Adding deviations always equals zero Etc.
  • 41. © 2006 Evidence-based Chiropractic 41 Calculating S2 • To find the average, one would normally total the scores above and below the mean, add them together, and then divide by the number of values • However, the total always equals zero – Values must first be squared, which cancels the negative signs
  • 42. © 2006 Evidence-based Chiropractic 42 Calculating S2 cont. Symbol for SD of a sample  for a population S2 is not in the same units (age), but SD is
  • 43. © 2006 Evidence-based Chiropractic 43 Calculating SD with Excel Enter values in a column
  • 44. © 2006 Evidence-based Chiropractic 44 SD with Excel (cont.) Click Data Analysis on the Tools menu
  • 45. © 2006 Evidence-based Chiropractic 45 SD with Excel (cont.) Select Descriptive Statistics and click OK
  • 46. © 2006 Evidence-based Chiropractic 46 SD with Excel (cont.) Click Input Range icon
  • 47. © 2006 Evidence-based Chiropractic 47 SD with Excel (cont.) Highlight all the values in the column
  • 48. © 2006 Evidence-based Chiropractic 48 SD with Excel (cont.) Check if labels are in the first row Check Summary Statistics Click OK
  • 49. © 2006 Evidence-based Chiropractic 49 SD with Excel (cont.) SD is calculated precisely Plus several other DSs
  • 50. © 2006 Evidence-based Chiropractic 50 Wide spread results in higher SDs narrow spread in lower SDs
  • 51. © 2006 Evidence-based Chiropractic 51 Spread is important when comparing 2 or more group means It is more difficult to see a clear distinction between groups in the upper example because the spread is wider, even though the means are the same
  • 52. © 2006 Evidence-based Chiropractic 52 z-scores • The number of SDs that a specific score is above or below the mean in a distribution • Raw scores can be converted to z-scores by subtracting the mean from the raw score then dividing the difference by the SD     X z
  • 53. © 2006 Evidence-based Chiropractic 53 z-scores (cont.) • Standardization – The process of converting raw to z-scores – The resulting distribution of z-scores will always have a mean of zero, a SD of one, and an area under the curve equal to one • The proportion of scores that are higher or lower than a specific z-score can be determined by referring to a z-table
  • 54. © 2006 Evidence-based Chiropractic 54 z-scores (cont.) Refer to a z-table to find proportion under the curve
  • 55. © 2006 Evidence-based Chiropractic 55 z-scores (cont.) Partial z-table (to z = 1.5) showing proportions of the area under a normal curve for different values of z. Z 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.0 0.5000 0.5040 0.5080 0.5120 0.5160 0.5199 0.5239 0.5279 0.5319 0.5359 0.1 0.5398 0.5438 0.5478 0.5517 0.5557 0.5596 0.5636 0.5675 0.5714 0.5753 0.2 0.5793 0.5832 0.5871 0.5910 0.5948 0.5987 0.6026 0.6064 0.6103 0.6141 0.3 0.6179 0.6217 0.6255 0.6293 0.6331 0.6368 0.6406 0.6443 0.6480 0.6517 0.4 0.6554 0.6591 0.6628 0.6664 0.6700 0.6736 0.6772 0.6808 0.6844 0.6879 0.5 0.6915 0.6950 0.6985 0.7019 0.7054 0.7088 0.7123 0.7157 0.7190 0.7224 0.6 0.7257 0.7291 0.7324 0.7357 0.7389 0.7422 0.7454 0.7486 0.7517 0.7549 0.7 0.7580 0.7611 0.7642 0.7673 0.7704 0.7734 0.7764 0.7794 0.7823 0.7852 0.8 0.7881 0.7910 0.7939 0.7967 0.7995 0.8023 0.8051 0.8078 0.8106 0.8133 0.9 0.8159 0.8186 0.8212 0.8238 0.8264 0.8289 0.8315 0.8340 0.8365 0.8389 1.0 0.8413 0.8438 0.8461 0.8485 0.8508 0.8531 0.8554 0.8577 0.8599 0.8621 1.1 0.8643 0.8665 0.8686 0.8708 0.8729 0.8749 0.8770 0.8790 0.8810 0.8830 1.2 0.8849 0.8869 0.8888 0.8907 0.8925 0.8944 0.8962 0.8980 0.8997 0.9015 1.3 0.9032 0.9049 0.9066 0.9082 0.9099 0.9115 0.9131 0.9147 0.9162 0.9177 1.4 0.9192 0.9207 0.9222 0.9236 0.9251 0.9265 0.9279 0.9292 0.9306 0.9319 1.5 0.9332 0.9345 0.9357 0.9370 0.9382 0.9394 0.9406 0.9418 0.9429 0.9441 0.9332 Corresponds to the area under the curve in black