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Lecture Objectives
? Overall: To give a basic understanding
of descriptive statistics
? Specific:
– understand the branches of statistics
– understand the different types of data that
can be collected

Statistics
? The science of collecting, monitoring,
analyzing, summarizing, and
interpreting data.
– This includes design issues as well.

Branches of Statistics
4
? Descriptive statistics
– Gives numerical and graphic procedures to
summarize a collection of data in a clear and
understandable way.
– Provide summary indices for a given data, e.g.
arithmetic mean, median, standard deviation,
coefficient of variation, etc.
? Inductive (inferential) statistics
– Provides procedures to draw inferences about a
population from a sample
Population
sample
Estimating population values from sample values

Why need biostatistics?
? Main reason: handling variations
–Biological variation
? Attribute differ not only among individuals
but also within same individual over time
? Example: height, weight, blood pressure,
eye color ...
–Sample variation
? Biomedical research projects are usually
carried out on small numbers of study
subjects 5

Role of biostatistics in Epidemiology
6
? Epidemiology is the study of the distribution and
determinants of health-related states or events
(including disease), and the application of this study to
the control of diseases and other health problems.
? Essential for scientific method of investigation
– Formulate hypothesis
– Design study to objectively test hypothesis
– Collect reliable and unbiased data
– Process and evaluate data rigorously
– Interpret and draw appropriate conclusions
? Essential for understanding, appraisal and
critique of scientific literature.

Variable
? Any measurable characteristic that
assumes different values for different
subjects, e.g., age, height, hair colour,
gender
? Observation of variables on different
subjects gives rise to data

Types of data
? Qualitative (Categorical data)
– Gender, disease severity
? Quantitative (Measurement) data
– Age, BP,Weight

Categorical Data
? The variable being studied are grouped
into categories based on some
qualitative trait.
? The resulting data are merely labels or
categories.

Examples: Categorical Data
? Hair color
– blonde, brown, red, black, etc.
? Opinion of students about riots
– ticked off, neutral, happy
? Smoking status
– smoker, non-smoker

Categorical data classified as Nominal,
Ordinal, and/or Binary
Categorical data
Not binary
Binary
Ordinal
data
Nominal
data
Binary Not binary

Nominal Data
? A type of categorical data in which objects fall
into unordered categories. E.g.
? Hair color
– blonde, brown, red, black, etc.
? Race
– Caucasian, African-American, Asian, etc.
? Smoking status

Ordinal Data
? A type of categorical data in which
order is important. Examples
? Education level
– None, Primary, Post primary
? Degree of illness
– none, mild, moderate, severe

Binary Data
? A type of categorical data in which there
are only two categories.
? Binary data can either be nominal or
ordinal. Examples
? Smoking status
? Education
– Primary, Post primary

Measurement Data
? The variables being studied are
“measured” based on some
quantitative trait.
? The resulting data are set of numbers.

Measurement data classified as
Discrete or Continuous
Measurement
data
Continuous
Discrete

Discrete Measurement Data
Only certain values are possible (there
are gaps between the possible values).
Continuous Measurement Data
Theoretically, any value within an interval is possible
with a fine enough measuring device.

Discrete data -- Gaps between possible values
0 1 2 3 4 5 6 7
Continuous data -- Theoretically,
no gaps between possible values
0 1000

Discrete Measurement Data
Examples
? Number of pregnancies
? Number of students late for class
? Number of crimes reported
? Number of huts in a sampled rural home
? CD4 counts
Generally, discrete data are counts.

Continuous Measurement Data
Examples
? Cholesterol level
? Height
? Body weight
? BP
Generally, continuous data come from
measurements.

Descriptive Statistics
A first step to summarizing
or describing raw data

What to describe?
? What is the “location” or “center” of the
data? (“measures of location”)
? How do the data vary? (“measures of
variability”)

Measures of Location
Measures of location indicate where on the
number line the data are to be found.
Common measures of location are:
? Mean
? Median
? Mode

Mean
? Another name for average.
? Let X1,X2,X3,…,Xn be the realised
values of a variable X, from a sample of
size n. Then the mean is
Formula:
n
i
X
X
?
?
That is, add up all of the data points and divide
by the number of data points.

Median
? Another name for 50th percentile
? ( Middle value).
? Appropriate for describing measurement
data.
? “Robust to outliers,” that is, not
affected much by unusual values.

Example
? ?
14
.
137
7
113
124
146
170
132
124
151
?
?
?
?
?
?
?
?
X
The systolic blood pressure of seven
middle aged men were as follows:
151, 124, 132, 170, 146, 124 and 113.
The mean is

Median
? Also known as the 50th percentile or
simply the middle value
? If the sample data are arranged in
increasing order, the median is
(i) the middle value if n is an odd number, or
(ii) midway between the two middle values if
n is an even number

Example 1. Median- n is odd
The reordered systolic blood pressure data
seen earlier are:
113, 124, 124, 132, 146, 151, and 170.
Median=132

Example 2. Median if– n is even
Six men with high cholesterol participated in a study to
investigate the effects of diet on cholesterol level. At the
beginning of the study, their cholesterol levels (mg/dL)
were as follows:
366, 327, 274, 292, 274 and 230.
Rearrange the data in numerical order as follows:
230, 274, 274, 292, 327 and 366.
The Median is half way between the middle two readings,
i.e. (274+292) ? 2 = 283.

Quartiles
31
? Quantiles: dividing the distribution of
ordered values into 4 equal-sized parts
First 25% Second 25% Third 25% Fourth 25%
Q1 Q2 Q3
Q1: first quartile
Q2 : second quartile = median
Q3: third quartile

Mode
? The value that occurs most frequently.
? One data set can have many modes.
? Appropriate for all types of data, but
most useful for categorical data or
discrete data with only a few number of
possible values.

The most appropriate measure
of location depends on …
the shape of the data’s
distribution.

Most appropriate measure of
location
? Depends on whether or not data are
“symmetric” or “skewed”.
? Depends on whether or not data have
one (“unimodal”) or more
(“multimodal”) modes.

Choosing Appropriate Measure of
Location
? If data are symmetric, the mean,
median, and mode will be approximately
the same.
? If data are multimodal, report the mean,
median and/or mode for each
subgroup.
? If data are skewed, report the median.

Mean versus Median
? Large sample values tend to inflate the
mean. This will happen if the
histogram of the data is right-skewed.
? The median is not influenced by large
sample values and is a better measure
of centrality if the distribution is
skewed.

Mean versus Median
37
? Median is less sensitive to extreme
values
x1 87 87
x2 95 95
x3 98 98
x4 101 101
x5 105.0 1050
Median is unchanged

Measures of Variation
38
? Summarize the dispersion of individual
values from some central value like the
mean
? Measures of dispersion characterise how
spread out the distribution is, i.e., how variable
the data are.
mean
x
x
x
x
x
x

Indices of Variation
? Commonly used measures of
dispersion include:
– Range
– Variance & standard deviation
– Inter-quartile range (IQR)
– Coefficient of Variation (or
relative standard deviation)

Range
?R= largest obs. - smallest obs.
or, equivalently
R = xmax - xmin
or, at times present
R = (xmin ,xmax )

Inter-quartile Range
? IQR = third quartile - first quartile
or, equivalently
IQR = Q3 - Q1
Q1 =lower quartile (has 25% of data
below and 75% above)
Q3=upper quartile (has 75% of data
below and 25% above)

IQR:-Example
? Consider the ages of 8 patients
18,21,23,24,24,32,42,59
Q1 =22 , Q3= 37
IQR=37-22=15

Variance
43
? Variance of a population : average of
squares of deviation from the mean
? Variance of a sample: usually subtract 1
from n in the denominator
n
X
X
n
i
i
2
1
)
( ?
?
?
1
)
( 2
1
?
?
?
?
n
X
X
n
i
i
effective sample
size, also called
degree of freedom

Standard deviation
44
? Problem with variance: its awkward unit
of measurement as value are squared
? Solution: taking square root of variance
=> standard deviation
? Sample standard deviation ( s or sd)
? ?
s s
x x
n
i
i
n
? ?
?
?
?
?
2
2
1
1

? it is the typical (standard) difference
(deviation) of an observation from the mean
? think of it as the average distance a data
point is from the mean, although this is not
strictly true
What is a standard deviation?

Example
Data Deviation Deviation2
151 13.86 192.02
124 -13.14 172.73
132 -5.14 26.45
170 32.86 1079.59
146 8.86 78.45
124 -13.14 172.73
113 -24.14 582.88
Sum = 960.0 Sum = 0.00 Sum = 2304.86
14
.
137
?
x

Example (contd.)
Therefore,
? ? 86
.
2304
7
1
2
?
?
?
?
i
i x
x
6
.
19
1
7
86
.
2304
?
?
?
s

Standard deviation
48
? Caution must be exercised when using
standard deviation as a comparative index of
dispersion
Weights of
newborn elephants
(kg)
929 853
878 939
895 972
937 841
801 826
Weights of
newborn mice (kg)
0.72 0.42
0.63 0.31
0.59 0.38
0.79 0.96
1.06 0.89
n=10
=887.1
sd =56.50
X
n=10
= 0.68
sd = 0.255
X
Incorrect to say that elephants show greater
variation for birth-weights than mice because of
higher standard deviation

Coefficient of variance
49
? Coefficient of variance expresses
standard deviation relative to its mean
X
s
cv ?
Mice show greater birth-weight variation
0637
.
0
?
elephants
cv
375
.
0
?
mice
cv

Measures of Variation -
Some Comments
50
? When comparison groups have very
different means (CV is suitable as it
expresses the standard deviation
relative to its corresponding mean)
? When different units of measurements
are involved, e.g. group 1 unit is mm,
and group 2 unit is gm (CV is suitable
for comparison as it is unit-free)
? In such cases, standard deviation
should not be used for comparison.

Measures of Variation -
Some Comments
? Range is the simplest, but is very
sensitive to outliers
? Variance units are the square of the
original units
? Interquartile range is mainly used with
skewed data (or data with outliers)
? standard deviation is the most
commonly used measure of variation.

.
? An outlier is an observation which does not
appear to belong with the other data
? Outliers can arise because of a
measurement or recording error or because
of equipment failure during an experiment,
etc.
? An outlier might be indicative of a sub-
population, e.g. an abnormally low or high
value in a medical test could indicate
presence of an illness in the patient.
Outliers

Q & A
53
? Thank you for your attention!

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