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Introduction to Statistics
Santosh Kumar
Director (iCISA)
Describe the Sample

Descriptive Statistics





Measures of Central Tendency
Measures of Variability
Other Descriptive Measures
Beware that poor samples may provide a distorted
view of the population
In general, larger samples a more representative of
the population.
2
Measures of Central Tendency


Measures the “center” of the data
Examples





Mean
Median
Mode
The choice of which to use, depends…
It is okay to report more than one.
3
Mean

The “average”.

If the data are made up of n observations x1, x2,…, xn, the
mean is given by the sum of the observations divided by the
number of observations. For example, if the data are x1=1,
x2=2, x3=3, then the mean is (1+2+3)/3=2.

We can calculate the sample mean from
the data. It is often denoted as
1
X 
n
n
X
i
i 1
4
Mean


The population mean is usually unknown
(although we try to make inferences
about this).
The sample mean is an unbiased
estimator of the population mean.
5
Median


The “middle observation” according to its
rank in the data.
The median is:


The observation with rank (n+1)/2 if n is odd
The average of observations with rank n/2 and
(n+2)/2 if n is even
6
Median


Example: If a very senior officer is present in the
room and we desire assess the income level of all the
officers present
The median is more robust than the mean to extreme
observations.



If data are skewed to the right, then the mean > median (in
general)
If data are skewed to the left, then mean < median (in
general)
If data are symmetric, then mean=median
7
Mode




The value that occurs most
Good for ordinal or nominal data in which
there are a limited number of categories
Not very useful for continuous data
Only measure of central tendency for
qualitative data.
8
Measures of Variability


Measure the “spread” in the data
Some important measures




Variance
Standard Deviation
Range
Interquartile Range
9
Variance

The sample variance (s2) may be calculated from the
data. It is the average of the square deviations of the
observations from the mean.


2
1  n
s 
 Xi  X 
n  1  i 1

2

The population variance is often denoted by
S2. This is usually unknown.
10
Variance



The deviations are squared because we are
only interested in the size of the deviation
rather than the direction (larger or smaller
than the mean).
Note
Why?  X  X    0
n
 i 1
i

11
Variance


The reason that we divide by n-1 instead of n has to
do with the number of “information units” in the
variance. After estimating the sample mean, there
are only n-1 observations that are a priori unknown
(degrees of freedom).
This also makes s2 an unbiased estimator of S2.
12
Standard Deviation

Square root of the variance

s = sqrt(s2) = sample SD


S = sqrt(S2) = population SD



Calculate from the data (see formula for s2)
Usually unknown
Expressed in the same units as the mean (instead of
squared units like the variance)
But, s is not an unbiased estimator of S.
13
Range


Maximum - Minimum
Very sensitive to extreme observations
(outliers)
14
Interquartile Range


Q3-Q1
More robust than the range to extreme
observations
15
Other Descriptive Measures

Minimum and Maximum



Very sensitive to extreme observations
Sample size (n)
Percentiles

Examples:


Median = 50th percentile
Q1, Q3
16
Small Samples

For very small samples (e.g., <5
observations), summary statistics are not
meaningful. Simply list the data.
17
Probability(P)

It measures the likelihood that an event will
occur
favourable number of cases
P
total number of cases
P lies between 0 and 1
probility of an impossible event  0
probabilit y of an certain event  1
18
Probability Distributions

A probability distribution describes the behavior of the
character of interest (called variable)


It identifies possible values of the variable and provides
information about the probability that these values (or ranges
of values) will occur.
Important probability distributions to be considered here
are Binomial , Poisson and Normal distribution
19
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