Download PPT 1 - Asian School of Business

Basic Statistics I
Biostatistics, MHA, CDC, Jul 09
Prof. K.G. Satheesh Kumar
Asian School of Business
What and why!
Statistics* is a discipline that deals with:
• collection of data;
• their classification and summarising;
• analysis
for drawing conclusions and making decisions
* Statistics also refers to the data obtained from sample (as against parameter for
population data)
Analysis provides us an understanding of the
variation# and its causes in a phenomenon
# World is full of variations that it is hard to tell real differences from natural variations
Descriptive & Inferential Statistics
• Descriptive Statistics is concerned with
organisation, summarisation and
presentation of data
• Inferential Statistics deals with drawing
conclusions about large groups of subjects
(population) on the basis of observations
obtained from some of them (sample)
Variables
• A variable is what is being observed or measured
• Gender, age, height, weight, colour of eye,
responsiveness to treatment, life expectancy,
preferences etc. are examples
• Dependent variable is an outcome of interest that
changes in response to some intervention
• Independent variable is the intervention, or what is being
manipulated (sometimes without manipulation e.g. age)
Types of Data
• Primary & Secondary
• Discrete & Continuous
– Discrete data can assume only specific values (e.g.
gender, number of children etc.)
– Continuous data may take any value within a defined
range (there is always an error) (e.g. height, weight
etc.)
• Quantitative (Numeric) & Qualitative
(Categorical)
– Quantitative: Amenable to Arithmetic operations
– Qualitative: Simply records a quality
Types of Data (Contd…)
Another way to classify data is Stevens’
Taxonomy (after S.S. Stevens) into four types:
•
•
•
•
Nominal (named categories)
Ordinal (Nominal plus ordered categories)
Interval (Ordinal plus meaningful intervals)
Ratio (Interval plus meaningful zero-reference)
(also called scales of measurement), nominal
being the weakest and ratio the strongest
Nominal Variable
• Named categories, with no implied order among them
• Qualitative data like classifications, dichotomous data,
existential variables etc
• E.g. Single/Married/Separated/Widowed; Male/Female;
Received/Not Received a treatment etc.
• Sometimes numbers are used as alternative names or
labels
• No category is better or worse than another
Ordinal Variable
• Ordered Categories, where we can say
one category is better or worse than
another, but not how much better or worse
• Excellent/Satisfactory/Unsatisfactory;
A/B/C/F grades; Ranks 1/2/3/4; Cancer
Stage I/II/III/IV; Emergent/Urgent/Elective;
Much improved/Somewhat
Improved/Same/Worse/Dead
Interval Variable
• Order and separation between variables become
meaningful, but zero point is arbitrary
• Ratio of measurements is not meaningful, while ratio of
intervals is
• E.g.: Time of day: We cannot say 10 AM is twice 5 AM.
But we can say the interval between 2 AM and 4 AM is
twice that between 4 AM and 5AM
• Temperature in degree Celsius or degree Fahrenheit;
height from an arbitrary reference; IQ (average IQ = 100)
are other examples
Ratio Variable
• If the zero point is meaningful, the ratio
between two measurements is also
meaningful – such variables are ratio
variables
• Lengths, weights, money, absolute
temperature, duration (not time of day),
volume, area etc. are examples
Describing Data
• In words: “54% of the students are boys and 46% girls”
• In the form of a Table*
• In charts (graphs) as in the next slide
* Source: Biostatistics; Geoffrey R. Norman, David L. Streiner, David L Streiner
Bar Charts
can be used
for all the
four types,
but moving
categories
around as in
Fig 2.1 is
possible
only for
nominal data
where the
order is not
meaningful
* Source: Biostatistics; Geoffrey R. Norman, David L. Streiner, David L Streiner
Raw Data
Look at the
raw data
here. It is
nearly
impossible
to make
sense of a
table like
this
* Source: Biostatistics; Geoffrey R. Norman, David L. Streiner, David L Streiner
Sorted Data
Putting
data in rank
order
creates
some
sense
though not
much
* Source: Biostatistics; Geoffrey R. Norman, David L. Streiner, David L Streiner
Grouped Data and
Loss of Information
Wider the
interval fewer
the number
of classes
and more the
loss of
information
* Source: Biostatistics; Geoffrey R. Norman, David L. Streiner, David L Streiner
Histogram
Bars touch each other unlike in bar charts
* Source: Biostatistics; Geoffrey R. Norman, David L. Streiner, David L Streiner
Frequency Polygon
Cumulative Frequency Polygon
(Less than Ogive or simply Ogive)
Describing Data using
Summary Measures
•
Describing data in tables and graphs may not
be possible/advisable in all situations; again
comparing tables and graphs of different
groups of data is not easy
•
Hence the need for describing data with a few
numbers, called summary measures
1.
2.
3.
4.
Cluster/Centre/Level/Central Tendency/Location
Scatter/Spread/Dispersion
Shape/Skewness
Tails/Kurtosis
Mean Median Mode
The Mean
(Arithmetic Mean)
is a measure of
location data and
is given by
1 n
X  1 X i
n
Data type
Central Tendency
The Median is a value
such that half of the data
points fall below it and
half above it, after the
data is arranged in
ascending order
The Mode is the most
frequently occurring
category
Nominal
Mode
Ordinal
Mode, Median
Interval & Ratio
All the three
Two identical
distributions, one
shifted with respect
to the other.
They have different
locations / means
Multi-modal Distributions
Illustration of Mean for
Ungrouped Data
Q. Find the mean equity holding of 20 Indian
Billionaires whose individual equity
holdings (in Millions of Rs.) are 2717,
2796, 3098, 3144, 3527, 3534, 3862,
4187, 4310, 4506, 4745, 4784, 4923,
5034, 5071, 5424, 5561, 6505, 6707,
6874
A. Mean = (2717+2796+…+6874)/20
= Rs.4565.40 Million
Another Example for Mean
Q. Find the mean of 34, 37, 45, 32, 50
A. Assume a mean, A = 40
Then the deviations are -6,-3,5,-8,10
which add up to -2. Hence average
deviation is -2/5 = -0.4
Mean = Assumed mean + Average deviation
from assumed mean = 40 – 0.4 = 39.6
Illustration of Mean for
Grouped Data
Assumed Mean, A = 45
X
f
d=X-A
fd
25
2
-20
-40
35
5
-10
-50
45
6
0
0
55
4
10
40
65
3
20
60
20
Average Deviation from Assumed mean = 10/20 = 0.5
Mean = 45 + 0.5 = 45.5
10
Illustration of Median
Q. For the data on equity holdings of 20
billionaires find the median value.
A. Arrange data in ascending order. Median
is the value at position (n+1)/2 = 10.5.
The 10th and 11th values are respectively
4506 and 4745. Hence the value at rank
10.5 is
Median = (4506+4745)/2 = 4625.5
Measures of Spread
This summary measure indicates the
spread or scatter or dispersion of data
around the measure of central tendency
Range, IQR and MAD
• Range is the difference between the highest and lowest
values
– Simple to compute and understand, but unstable (increases with
sample size), sensitive to extreme values and not amenable to
further processing
• Interquartile range or midspread or hingespread is the
difference between QL and QH and comprises the middle
50% of the data
– Also, relative midspread = midspread / median
• Mean Absolute Deviation or Mean Deviation (MD) is the
average of absolute deviations from the mean
Quartiles
• Quartiles are the three values that divide
ordered observations into four equal parts
• 25% of the observations lie below the First or
Lower Quartile QL
• 50% of the observations lie below the Second or
Middle Quartile M (Median)
• 75% of the observations lie below the Third of
Upper Quartile QH
Illustration
Q. For the equity holdings example, find the lower and
upper quartiles
A. Rank of QL = (n+1)*25% = 5.25
5th value = 3527 and 6th value = 3534
Hence the value at rank 5.25 = 3527 + 7/4 = 3528.75
Rank of QH = (n+1) * 75% = 15.75
15th value = 5071 and 16th value is 5424
Hence value at rank 15.75 = 5071 + 264.75 = 5335.75
Five Number Summary
Five numbers can comprehensively summarise the
features of a distribution without being unduly
affected by a small part of the data
Minimum (MN)
Lower Quartile (LQ)
Also called Lower Hinge
Median (MD)
Upper Quartile (UQ)
Maximum (MX)
Also called Upper Hinge
Five Number Summary is Comprehensive:
The Grand Summary of a Distribution
Lower Tail
Upper Tail
Variance and Standard Deviation
• Variance is the Mean Squared Deviation
(MSD) from the mean
• Square Root of Variance (RMSD) is called
Standard Deviation
• The ratio Standard Deviation / Mean is
called the Coefficient of Variation (CV)
Variance = Mean Squared Deviation from the Mean
1 n
2
(
X

X
)
 i
n 1
=
Serial
No.
1
2
3
4
5
Total
Average
Data
(X)
6
7
8
9
20
50
10
X–Mean
-4
-3
-2
-1
10
0
where
1 n
X  1 X i
n
(X–Mean)2
16
9
4
1
100
130
26 = Variance
Covariance
• Variance(X) =
•
•
•
•
1 n
2
(
X

X
)

i
1
n
Hence n Variance(X) = SSxx
Similarly nVariance(Y) = SSYY
Further n Covariance(X,Y) = SSXY
Cov(X,Y) is the expected value of the product
of the deviation of X from its mean and the
deviation of Y from its mean
Shape or Skewness
•
•
•
Skewness refers to the symmetry of the distribution
Curve A has positive or right skew; Curve B has negative of left skew
Direction of the skew refers to the direction of the longer tail and not to
where the bulk of data are located
Kurtosis
• Curve A: Mesokurtic,
normal curve, kurtosis = 3
• Curve B: Leptokurtic,
peaked than normal,
kurtosis > 3
Kurtosis refers to how flat or
peaked the distribution is
• Curve C: Platy kurtic,
flatter than normal,
kurtosis < 3
Continued…