introduction to medical statistics - chiang mai university · 2020. 10. 26. · rs introduction to...
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1
Introduction to
Medical Statistics For Medical Students
Phichayut PhinyoMD, Dip(ClinEpidemio) Dip(ClinStat)
Center for Clinical Epidemiology and Clinical Statistics
Department of Family Medicine
Faculty of Medicine, Chiang Mai University
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2RS Introduction To Medical Statistics P Phinyo
Class objectives▪ After the end of this session, (hopefully) the
students would be able to understand and
correctly describe the concept of the following
topics:
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3RS Introduction To Medical Statistics P Phinyo
Topics outline▪ Medical statistics I
▪ Statistical concepts in medical practice
▪ Quantitative and Qualitative data
▪ Types of variables
▪ Frequency distributions
▪ Measures of central tendency
▪ Measures of dispersion
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4RS Introduction To Medical Statistics P Phinyo
Topics outline▪ Medical statistics II
▪ Statistical inference
▪ Hypothesis generation and testing
▪ The alpha level and p value
▪ Type I error and type II error
▪ Test of statistical significance
▪ Test of clinical importance
▪ Confidence intervals
▪ Impact (size of effect)
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5RS Introduction To Medical Statistics P Phinyo
Topics outline▪ Medical research methodology
▪ Statistical tests
▪ Internal and external validity
▪ Statistical significance
▪ Statistical power
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6RS Introduction To Medical Statistics P Phinyo
Nature of data▪ Learning objectives
▪ Variables and Data
▪ Nominal data
▪ Ordinal data
▪ Discrete metric data
▪ Continuous metric data
▪ Scale of measurements
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7RS Introduction To Medical Statistics P Phinyo
Variables and data▪ Variables: something whose value can vary
▪ Data: value that explains the variable
Variables▪ Age▪ Gender▪ Zodiac signs▪ Glasses▪ Etc.
We get the data when we determine the value of a variable for each unit of observation
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8RS Introduction To Medical Statistics P Phinyo
Types of variables (data)
Categorical variables (Qualitative data)
Numerical variables(Quantitative data)
Nominal
Ordinal
Continuous (non-discrete)
Discrete (count)
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9RS Introduction To Medical Statistics P Phinyo
Types of variables (data)▪ Nominal data
▪ Related to named things
▪ Particularly not numeric, but categories
▪ Can be more than 2 categories
▪ The ordering is arbitrary (or does not make sense)
▪ Example
▪ Gender (male or female)
▪ Ethnics (Thai, Burmese, Laos, etc.)
▪ Zodiac signs (Cancer, Virgo, Sagittarius, etc.)
▪ Insurance (UC, SSSS, CSS, etc.)
▪ Religion (Buddhism, Islam, Christian, etc.)
▪ Marital status (Single, Married, Divorce, etc.)
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10RS Introduction To Medical Statistics P Phinyo
Types of variables (data)▪ Nominal data
▪ Related to named things
▪ Particularly not numeric, but categories
▪ Can be more than 2 categories
▪ The ordering is arbitrary (or does not make sense)
▪ Clinical example
▪ Location of headache (diffuse, temporal, etc.)
▪ Mechanism of injury (traffic, falling, etc.)
▪ Site of metastasis (brain, bone, liver, etc.)
▪ …
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11RS Introduction To Medical Statistics P Phinyo
Types of variables (data)▪ Ordinal data
▪ Usually are assessed measurements
▪ Not numerical data but categorical
▪ Can be more than 2 categories (usually ≥2)
▪ The ordering is not arbitrary (make sense to order)
▪ Example
▪ Cancer staging (stage I, II, III, IV)
▪ Disease severity (mild, moderate, severe)
▪ Risk evaluation (low, intermediate, high)
▪ Glasgow Coma Scale (GCS)
▪ Motor power (grade 1,2,3,4,5)
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12RS Introduction To Medical Statistics P Phinyo
Types of variables (data)▪ Ordinal data
▪ Special consideration about ordinal data
▪ The ordinal scales are not real number but only
numeric labels that were attached to each category.
▪ The difference between each value might not be
proportional. In other words, we cannot quantify the
interval difference of ordinal data.
▪ Not appropriate to apply any rules of basic arithmetic
to this type of data. You should not add, subtract,
multiply, divide, or even find the average value of
ordinal data.
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13RS Introduction To Medical Statistics P Phinyo
Types of variables (data)▪ Discrete metric data (discrete data or count data)
▪ Comes from counting process
▪ Unit of measurements: number of things
▪ Real numbers (integers)
▪ Decimals do not make sense
▪ Equal interval difference, unlike ordinal
▪ Clinical example
▪ Number of students in the classroom
▪ Number of patients
▪ Number of deaths
▪ Number of angina attacks
▪ Number of hospital visits
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14RS Introduction To Medical Statistics P Phinyo
Types of variables (data)▪ Continuous metric data (continuous data)
▪ Comes from measurements
▪ Have units of measurement
▪ The data are real numbers
▪ Decimals do have meaning
▪ Equal interval difference
▪ Clinical example
▪ Body weight, height, body mass index
▪ Blood sugar, Serum cholesterol level
▪ Hemoglobin level
▪ Tumor marker level
▪ Blood pressure (continuous or discrete?)
▪ Heart rate (continuous or discrete?)
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15RS Introduction To Medical Statistics P Phinyo
Types of variables (data)▪ What is the main difference between discrete
and continuous metric data?
(a) List the possible number of eggs that the carton contain(counting)
(b) List the number of possible value for the weight of each egg in the carton(measuring)
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16RS Introduction To Medical Statistics P Phinyo
Types of variables (data)▪ What is the main difference between discrete
and continuous metric data?
(a) List the possible number of eggs that the carton contain(counting 0,1,2,3,4,5,6)limited range of possible data
(b) List the number of possible value for the weight of each egg in the carton(measuring 70,70.1,70.001,70.000001,70.000000000000001 etc.)Infinite range of possible data
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17RS Introduction To Medical Statistics P Phinyo
Categorical variables (Qualitative data)
Numerical variables(Quantitative data)
Nominal
Ordinal
Continuous (non-discrete)
Discrete (count)
Arithmetically operable
Have unit
Arithmetically inoperable
Does not Have unit
Has the data got units?
Can the data be put in meaningful order?
Do the data come from counting or measuring?
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18RS Introduction To Medical Statistics P Phinyo
Quizzes▪ Numeric rating scale (NRS)
▪ What types of data is it?
▪ Nominal
▪ Ordinal
▪ Discrete metric
▪ Continuous metric
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19RS Introduction To Medical Statistics P Phinyo
Quizzes▪ Visual Analogue Scale
▪ What types of data is it?
▪ Nominal
▪ Ordinal
▪ Discrete metric
▪ Continuous metric
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20RS Introduction To Medical Statistics P Phinyo
Scale of measurement
Nominal
Ordinal
Interval
Ratio
Named variable
Named variableOrdered variable
Named variableOrdered variableProportionate interval
Named variableOrdered variableProportionate intervalAccommodate absolute zero
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21RS Introduction To Medical Statistics P Phinyo
Describing data▪ Learning objectives
▪ Frequency distribution
▪ Frequency table
▪ Relative frequency
▪ Cumulative frequency
▪ Relative cumulative frequency
▪ Grouped frequency tables
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22RS Introduction To Medical Statistics P Phinyo
Raw data: GENDER (nominal)M F F F M F F M M F
F F M M F F F M F F
F M F F M M M M M F
M M M F M M M M F F
M F M M F F M M F M
F F F M M M M F F F
F F M M F F F M M M
M M F F M M M M M F
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23RS Introduction To Medical Statistics P Phinyo
Raw data: GENDER (nominal)M F F F M F F M M F
F F M M F F F M F F
F M F F M M M M M F
M M M F M M M M F F
M F M M F F M M F M
F F F M M M M F F F
F F M M F F F M M M
M M F F M M M M M F
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24RS Introduction To Medical Statistics P Phinyo
Frequency tables (nominal)
GenderCount
(Frequency)Percent
Cumulative percentage
Female 38 47.5 47.5
Male 42 52.5 100
Total 80 100
To describe nominal data, frequency (percentage) tables are used. The order in the table is not important.
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25RS Introduction To Medical Statistics P Phinyo
Frequency distributions
InsuranceCount
(Frequency)Percent
Cumulative percentage
UC 50 50.0 50.0
SSSS 30 30.0 80.0
CSS 15 15.0 95.0
Others 5 5.0 100
Total 100 100
Even though ordering or nominal categories is arbitrary, arranging the category from highest frequency to lowest frequency could help the readers understand the data more quickly.
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26RS Introduction To Medical Statistics P Phinyo
Frequency tables (nominal)
InsuranceCount
(Frequency)Percent
Cumulative percentage
UC 50 50.0 50.0
SSSS 30 30.0 80.0
CSS 15 15.0 95.0
Others 5 5.0 100
Total 100 100
Even though ordering or nominal categories is arbitrary, arranging the category from highest frequency to lowest frequency could help the readers understand the data more quickly.
Number (frequency)
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27RS Introduction To Medical Statistics P Phinyo
Frequency distributions
InsuranceCount
(Frequency)Percent
Cumulative percentage
UC 50 50.0 50.0
SSSS 30 30.0 80.0
CSS 15 15.0 95.0
Others 5 5.0 100
Total 100 100
Even though ordering or nominal categories is arbitrary, arranging the category from highest frequency to lowest frequency could help the readers understand the data more quickly.
Percentage (relative frequency)
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28RS Introduction To Medical Statistics P Phinyo
Frequency distributions
InsuranceCount
(Frequency)Percent
Cumulative percentage
UC 50 50.0 50.0
SSSS 30 30.0 80.0
CSS 15 15.0 95.0
Others 5 5.0 100
Total 100 100
Even though ordering or nominal categories is arbitrary, arranging the category from highest frequency to lowest frequency could help the readers understand the data more quickly.
Does this make sense?
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29RS Introduction To Medical Statistics P Phinyo
Frequency tables (ordinal)
Cancer stageCount
(Frequency)Percent
Cumulative percentage
Stage I 30 12.0 12.0
Stage II 50 20.0 32.0
Stage III 100 40.0 72.0
Stage IV 70 28.0 100.0
Total 250 100
Ordering of categories is not arbitrary and is meaningful for ordinal data, such as this case. The frequency distribution shown here would reflect the distribution of cancer stages in the sample.
PercentageCumulativeFrequencyOrRelativeCumulativeFrequency
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30RS Introduction To Medical Statistics P Phinyo
Frequency tables (discrete)Parity
Count (Frequency)
PercentCumulative percentage
0 49 49.0 49.0
1 18 18.0 67.0
2 17 17.0 84.0
3 11 11.0 95.0
4 2 2.0 97.0
5 1 1.0 98.0
≥6 2 2.0 100
Total 100
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31RS Introduction To Medical Statistics P Phinyo
Frequency tables (continuous)Hemoglobin
Count (Frequency)
PercentCumulative percentage
8.80 1 1.0 1.0
8.90 1 1.0 2.0
8.91 1 1.0 3.0
8.92 1 1.0 4.0
8.93 1 1.0 5.0
9.01 1 1.0 6.0
9.02 1 1.0 7.0
… … … 100
Total 100
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32RS Introduction To Medical Statistics P Phinyo
Frequency tables (continuous)Hemoglobin
Count (Frequency)
PercentCumulative percentage
8.80 1 1.0 1.0
8.90 1 1.0 2.0
8.91 1 1.0 3.0
8.92 1 1.0 4.0
8.93 1 1.0 5.0
9.01 1 1.0 6.0
9.02 1 1.0 7.0
… … … 100
Total 100
Need to be grouped
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33RS Introduction To Medical Statistics P Phinyo
Grouped frequency tableHemoglobin
Count (Frequency)
PercentCumulative percentage
7.0-8.0 10 6.5 6.5
8.0-9.0 20 12.9 19.4
9.0-10.0 30 19.4 38.8
11.0-12.0 35 22.4 61.2
12.0-13.0 30 19.4 80.6
13.0-14.0 20 12.9 93.5
14.0-15.0 10 6.5 100
Total 155
General rule of thumb: no lesser than five and no more than 10 groups
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34RS Introduction To Medical Statistics P Phinyo
Grouped frequency table
HemoglobinCount
(Frequency)Percent
7.0-8.0 10 6.5
8.0-9.0 20 12.9
9.0-10.0 30 19.4
11.0-12.0 35 22.4
12.0-13.0 30 19.4
13.0-14.0 20 12.9
14.0-15.0 10 6.5
Total 155
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35RS Introduction To Medical Statistics P Phinyo
Grouped frequency table
HemoglobinCount
(Frequency)Percent
7.0-8.0 10 6.5
8.0-9.0 20 12.9
9.0-10.0 30 19.4
11.0-12.0 35 22.4
12.0-13.0 30 19.4
13.0-14.0 20 12.9
14.0-15.0 10 6.5
Total 155
Around 60% are among these categories(from 9.0-13.0)
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36RS Introduction To Medical Statistics P Phinyo
Frequency distribution▪ Commonly illustrated via histogram
▪ Measure of shape
▪ Skewness
▪ Kurtosis
▪ Common distribution
▪ Normal distribution
▪ Skewed distribution
▪ Uniform distribution
▪ Bimodal or multimodal distribution
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37RS Introduction To Medical Statistics P Phinyo
Frequency distribution▪ Measure of shape
▪ Skewness
▪ Measure of the symmetry
▪ Measured by skewness coefficient
▪ Range from -1 and +1
▪ When the distribution is symmetry, the
skewness coefficient will be near 0
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38RS Introduction To Medical Statistics P Phinyo
HistogramFr
eque
ncy
Resting Heart rate (beat per minute)55 60 65 70 75 80 85 90 95
Symmetrical distributionNormal distributionSkewness near 0Mean = Median = Mode
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39RS Introduction To Medical Statistics P Phinyo
HistogramFr
eque
ncy
Resting Heart rate (beat per minute)55 60 65 70 75 80 85 90 95
Symmetrical distributionNormal distributionSkewness near 0Mean = Median = Mode
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40RS Introduction To Medical Statistics P Phinyo
HistogramFr
eque
ncy
Asymmetrical distributionLeft skewedNegatively skewedSkewness coefficient close to -1Mean < Median < ModeWith some outliers
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41RS Introduction To Medical Statistics P Phinyo
HistogramFr
eque
ncy
Asymmetrical distributionRight skewedPositively skewedSkewness coefficient close to +1Mode < Median < Mean
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42RS Introduction To Medical Statistics P Phinyo
HistogramFr
eque
ncy
Uniform distribution
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43RS Introduction To Medical Statistics P Phinyo
HistogramFr
eque
ncy
Bimodal distribution
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44RS Introduction To Medical Statistics P Phinyo
Frequency distribution
What shape is the distribution?
a) Normal distribution
b) Positively skewed distribution
c) Negatively skewed distribution
d) Uniform distribution
e) Multimodal distribution
Age groups n (%)
<15 0.2%
15-29 5%
30-39 7%
40-49 11%
50-59 16%
60-69 30%
70-79 35%
>80 14%
This table shows the age distribution of 2454 patients with acute pulmonary embolism. Source: modified from Goldhaber et al. (1999)
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45RS Introduction To Medical Statistics P Phinyo
Frequency distribution▪ Measure of shape
▪ Kurtosis
▪ Apply to symmetrical distribution
▪ Whether the distribution has, to its left
and right, short tails or long tails.
▪ High kurtosis value = long tails (tails
are longer and fatter, central peak is
higher and more pointy)
▪ Low kurtosis value = short tails (tails are
shorter and thinner, central peak is
lower and wider)
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46RS Introduction To Medical Statistics P Phinyo
Frequency distribution
https://analystprep.com/cfa-level-1-exam/quantitative-methods/kurtosis-and-skewness-types-of-distributions/
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47RS Introduction To Medical Statistics P Phinyo
Summarizing data▪ Learning objectives
▪ Measure of central tendency
▪ Mode
▪ Median
▪ Mean
▪ Minimum
▪ Maximum
▪ Percentile
▪ Interquartile range
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48RS Introduction To Medical Statistics P Phinyo
Mode▪ Mode, or modal values, is the value in the data
with the highest frequency.
▪ Not actually a measure of central-ness.
▪ May be useful for nominal and ordinal data
▪ It is not useful for metric data types
▪ There may be more than one mode.
▪ Not very common in clinical research
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49RS Introduction To Medical Statistics P Phinyo
Median▪ A measure of central-ness of the data.
▪ If we arrange the data in ascending order, the
median is the middle value.
▪ Median is meaningless for nominal data.
▪ Example I
30 31 32 33 35
▪ Example II
30 31 32 33 35 50
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50RS Introduction To Medical Statistics P Phinyo
Median▪ Property of the median
▪ Not affected much by skewness of the
distribution (good representative)
▪ Not affected by outliers
▪ Consider a very stable measure of centerness
▪ Disadvantages
▪ Discards lots of information
▪ Not easy to determine by hand
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51RS Introduction To Medical Statistics P Phinyo
Mean▪ Mean or arithmetic mean or the average
▪ It uses all the information in the data to estimate
▪ It is affected by skewness and by outliers
▪ Mean might not be a good representative in
some occasions.
▪ It should not be used with ordinal data
(although very common in practice)
▪ Example: 30 31 32 33 32 Mean = 31.6
▪ Example: 30 31 32 33 32 70 Mean = 38.0
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52RS Introduction To Medical Statistics P Phinyo
Percentiles▪ Are values which divide an ordered set of data
into 100 equal-sized groups.
2170 2220 2460 2640 2660 2780 … 3000 … 3410 3460 3490 3500 3540 3740
N=100
Middle position = 50Median or Percentile 50
n=50 below this value n=50 above this value
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53RS Introduction To Medical Statistics P Phinyo
Percentiles▪ Are values which divide an ordered set of data
into 100 equal-sized groups.
2170 2220 2460 2640 2660 2780 … 3000 … 3410 3460 3490 3500 3540 3740
N=100
n≤5 below this value n≥95 above this value
5th percentile
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54RS Introduction To Medical Statistics P Phinyo
What is appropriate?▪ Depends on data type and distribution.
Type of data Mode Median Mean
Nominal Yes No No
Ordinal Yes Yes Should not
Discrete Yes Yes if skewed Yes if Normal
Continuous No Yes if skewed Yes if Normal
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55RS Introduction To Medical Statistics P Phinyo
Summarizing data▪ Learning objectives
▪ Measure of dispersion (spread)
▪ Range
▪ Interquartile range
▪ Standard deviation
▪ Transformation of data
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56RS Introduction To Medical Statistics P Phinyo
Range▪ Distance from the smallest value to the largest.
▪ Minimum to maximum
▪ Not affected by skewness
▪ Very sensitive to outliers in both ends.
▪ Example: 10 40 41 42 43 43 44 45 46 47 50 90 100
▪ Range (10 to 100)
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57RS Introduction To Medical Statistics P Phinyo
Interquartile range (IQR)▪ A solution to the problem of the sensitivity of the
range to outliers in both ends.
▪ Slitted the data into 4 equal-sized quartiles
▪ 25% 25% 25% 25%
▪ Q1 Q2 Q3 Q4
▪ Then, chop a quarter of the values off both ends.
▪ IQR is written as Q1-Q3
▪ Not affected by outliers but can be affected by
skewness. It does not use all the information in
the data.
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58RS Introduction To Medical Statistics P Phinyo
Box and whisker plot
https://towardsdatascience.com/understanding-boxplots-5e2df7bcbd51
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59RS Introduction To Medical Statistics P Phinyo
https://towardsdatascience.com/understanding-boxplots-5e2df7bcbd51
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60RS Introduction To Medical Statistics P Phinyo
Standard deviation▪ To correct the limitation of IQR (not using all
information in the data), the standard deviation is
used instead.
▪ Can only be meaningfully used for metric data.
▪ Measure of the average distance/deviation of the
data values from their collective mean.
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61RS Introduction To Medical Statistics P Phinyo
Standard deviationFr
eque
ncy
Resting Heart rate (beat per minute)55 60 65 70 75 80 85 90 95
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62RS Introduction To Medical Statistics P Phinyo
Standard deviationFr
eque
ncy
Resting Heart rate (beat per minute)55 60 65 70 75 80 85 90 95
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63RS Introduction To Medical Statistics P Phinyo
Standard deviationFr
eque
ncy
Resting Heart rate (beat per minute)55 60 65 70 75 80 85 90 95
Distance = 83-75 = +8
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64RS Introduction To Medical Statistics P Phinyo
Standard deviationFr
eque
ncy
Resting Heart rate (beat per minute)55 60 65 70 75 80 85 90 95
Distance = 83-75 = +8Distance = 65-75 = -10
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65RS Introduction To Medical Statistics P Phinyo
Standard deviationFr
eque
ncy
Resting Heart rate (beat per minute)55 60 65 70 75 80 85 90 95
Distance = 83-75 = +8Distance = 67-75 = -8
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66RS Introduction To Medical Statistics P Phinyo
Standard deviationFr
eque
ncy
Resting Heart rate (beat per minute)55 60 65 70 75 80 85 90 95
Distance = 83-75 = +8Distance = 67-75 = -8
Total distance=0 ???
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67RS Introduction To Medical Statistics P Phinyo
Standard deviationFr
eque
ncy
Resting Heart rate (beat per minute)55 60 65 70 75 80 85 90 95
(Distance)2 = (83-75) 2 = 64
(Distance)2 = (67-75) 2 = 64
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68RS Introduction To Medical Statistics P Phinyo
Standard deviationFr
eque
ncy
Resting Heart rate (beat per minute)55 60 65 70 75 80 85 90 95
(Distance)2 = (83-75) 2 = 64
(Distance)2 = (67-75) 2 = 64
Total distance 2 =128Variance = total distance 2 /n
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69RS Introduction To Medical Statistics P Phinyo
Variance and SD▪ Variance =
(𝑇𝑜𝑡𝑎𝑙 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑓𝑟𝑜𝑚𝑚𝑒𝑎𝑛)2
𝑛
The unit is also squared; beat per min 2
▪ Standard Deviation = 𝑉𝑎𝑟𝑖𝑎𝑛𝑐𝑒
The unit is back to beat per min
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70RS Introduction To Medical Statistics P Phinyo
Standard deviation▪ Data spread => Distance from mean => SD
▪ If the data are widely spread, the average distance
of the values from their mean will be large, and
thus the standard deviation.
▪ If the data are narrowly spread, the average
distance of the values from their mean will be
small, and thus the standard deviation.
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71RS Introduction To Medical Statistics P Phinyo
Standard deviation
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72RS Introduction To Medical Statistics P Phinyo
Standard deviation▪ Consideration
▪ Only is meaningful if the data is
normally distributed, hence the use of mean.
Normal distribution?
Mean and SD
Median and IQR
Ordinal data Median and IQR
Metric data
Yes
No
Not at all correct, but very common in practice
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73RS Introduction To Medical Statistics P Phinyo
Standard deviation▪ Interpretation?
Atthakomol et al. (2020)
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74RS Introduction To Medical Statistics P Phinyo
Standard deviation▪ Interpretation?
Atthakomol et al. (2020)
The average distance/deviation of the age values is about 8.5 years
from the sample mean of 81.6 years
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75RS Introduction To Medical Statistics P Phinyo
Standard deviation▪ Area properties of the Normal distribution
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76RS Introduction To Medical Statistics P Phinyo
Standard deviation▪ Area properties of the Normal distribution
68% within ±1SD
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77RS Introduction To Medical Statistics P Phinyo
Standard deviation▪ Area properties of the Normal distribution
68% within ±1SD
95% within ±2SD
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78RS Introduction To Medical Statistics P Phinyo
Standard deviation▪ Area properties of the Normal distribution
68% within ±1SD
95% within ±2SD
99% within ±3SD
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79RS Introduction To Medical Statistics P Phinyo
Standard deviation▪ Area properties of the Normal distribution
68% within ±1SD
95% within ±2SD
99% within ±3SD
Assumption: The data distribution must be Normal!!
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80RS Introduction To Medical Statistics P Phinyo
Standard deviation▪ Interpretation?
Atthakomol et al. (2020)
68% of the age values lie between 73.11 to 90.15 years
95% of the age values lie between 64.6 To 98.67 years
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81RS Introduction To Medical Statistics P Phinyo
Normal distribution▪ Symmetrical bell-shaped distribution
▪ Special place in the heart of statisticians!
▪ What statisticians really want to know is whether or not a
distribution is Normal or Normal-ish!
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82RS Introduction To Medical Statistics P Phinyo
Normal distribution▪ Three key characteristics!
▪ Visualization
▪ Histogram
▪ Box plot
▪ Mean and median approximation
▪ Large standard deviation
▪ Statistical tests for normality
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83RS Introduction To Medical Statistics P Phinyo
Normal distribution
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84RS Introduction To Medical Statistics P Phinyo
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85RS Introduction To Medical Statistics P Phinyo
Normal distribution
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86RS Introduction To Medical Statistics P Phinyo
Normal distribution ?????
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87RS Introduction To Medical Statistics P Phinyo
Positively skewed distribution
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88RS Introduction To Medical Statistics P Phinyo
Positively skewed distribution
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89RS Introduction To Medical Statistics P Phinyo
Transformation of data
Right skewness (positively skewed)Can be transformed to normal or near-normal distribution by logarithmic transformation.
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90RS Introduction To Medical Statistics P Phinyo
Transformation of data