The document provides an introduction to statistics concepts including central tendency, dispersion, probability, and random variables. It discusses different measures of central tendency like mean, median and mode. It also covers dispersion concepts like variance and standard deviation. The document introduces key probability concepts such as experiments, sample spaces, events, and conditional probability. It defines random variables and discusses discrete and continuous random variables.

1. MONDAY, DECEMBER 23
2013

2. INTRODUCTION TO THEORY OF
STATISTICS
BY
SRIRAM C
I sem M.Tech Geoinformatics
KSRSAC

3. Today:
Central Tendency , Dispersion & Probability
 From frequency tables to distributions
 Types of Distributions: Normal, Skewed
 Level of Measurement:
Nominal, Ordinal, Interval
 Central Tendency: Mode, Median, Mean
 Dispersion: Variance, Standard Deviation

4. Descriptive statistics are concerned with
describing the characteristics of frequency
distributions
 Where is the center?
 What is the range?
 What is the shape [of the
distribution]?

5. Frequency Distributions OR HISTOGRAMS
 Simple depiction of all the data
 Graphic — easy to understand
 Problems
 Not always precisely measured
 Not summarized in one number or datum
 Simple depiction of all the data
 Graphic — easy to understand
 Problems
 Not always precisely measured
 Not summarized in one number or datum

6. Frequency Table
Test Scores
Observation Frequency
65 1
70 2
75 3
80 4
85 3
90 2
95 1

7. Frequency Distributions
Test Score
Frequency
4
3
2
1
65 70 75 80 85 90 95

8. Normally Distributed Curve

9. Skewed Distributions

10. Summarizing Distributions
Two key characteristics of a frequency distribution
are especially important when summarizing
data or when making a prediction from one set
of results to another:
 Central Tendency
 What is in the “Middle”?
 What is most common?
 What would we use to predict?
 Dispersion
 How Spread out is the distribution?
 What Shape is it?

11. Three measures of central tendency are commonly
used in statistical analysis - the mode, the median,
and the mean
Each measure is designed to represent a typical score
The choice of which measure to use depends on:
• the shape of the distribution (whether normal or
skewed), and
• the variable’s “level of measurement” (data are
nominal, ordinal or interval).

12. Appropriate Measures of
Central Tendency
• Nominal variables Mode
• Ordinal variables Median
• Interval level variables Mean
- If the distribution is normal (median is better
with skewed distribution)
• Nominal variables Mode
• Ordinal variables Median
• Interval level variables Mean
- If the distribution is normal (median is better
with skewed distribution)

13. Mode
Most Common Outcome
Find the Mode
4 5 6 6 7 8 9 10 12
Ans:6

14. Median
Middle-most Value
50% of observations are above the Median, 50% are
below it
The difference in magnitude between the
observations does not matter
Therefore, it is not sensitive to outliers
Formula Median = n + 1 / 2

15. To compute the median
• first you rank order the values of X from low to
high:  85, 94, 94, 96, 96, 96, 96, 97, 97, 98
• then count number of observations = 10.
• add 1 = 11.
• divide by 2 to get the middle score  the 5 ½
score
here 96 is the middle score score

16. Mean - Average
 Most common measure of central tendency
 Best for making predictions
 Applicable under two conditions:
1. scores are measured at the interval level, and
2. distribution is more or less normal [symmetrical].
 Symbolized as:
 for the mean of a sample
 μ for the mean of a population
X

17. Finding the MeanFinding the Mean
• X = (Σ X / N)
• If X = {3, 5, 10, 4, 3}
X = (3 + 5 + 10 + 4 + 3) / 5
= 25 / 5
= 5
• X = (Σ X / N)
• If X = {3, 5, 10, 4, 3}
X = (3 + 5 + 10 + 4 + 3) / 5
= 25 / 5
= 5

18. Find the Mean
Q: 4, 5, 8, 7
A: 6
Median: 6.5
Q: 4, 5, 8, 1000
A: 254.25
Median: 6.5

19. Why can’t the mean tell us everything?
Mean describes Central Tendency, what the
average outcome is.
We also want to know something about how
accurate the mean is when making predictions.
The question becomes how good a representation
of the distribution is the mean? How good is the
mean as a description of central tendency -- or
how good is the mean as a predictor?
Answer -- it depends on the shape of the
distribution. Is the distribution normal or
skewed?

20. Measures of Variability
Central Tendency doesn’t tell us everything
Dispersion/Deviation/Spread tells us a lot about how a
variable is distributed.
We are most interested in Standard Deviations (σ) and
Variance (σ2
)

21. Dispersion
Once you determine that the variable of interest is
normally distributed, ideally by producing a
histogram of the scores, the next question to be
asked about the Normally Distributed Curve is its
dispersion: how spread out are the scores
around the mean.
Dispersion is a key concept in statistical thinking.
The basic question being asked is how much do the
scores deviate around the Mean? The more
“bunched up” around the mean the better your
ability to make accurate predictions.

22. How well does the mean represent the scores in a
distribution? The logic here is to determine
how much spread is in the scores. How much
do the scores "deviate" from the mean? Think
of the mean as the true score or as your best
guess. If every X were very close to the Mean,
the mean would be a very good predictor.
If the distribution is very sharply peaked then the
mean is a good measure of central tendency
and if you were to use the mean to make
predictions you would be right or close much of
the time.

23. Mean Deviation
The key concept for describing normal distributions
and making predictions from them is called
deviation from the mean.
We could just calculate the average distance between
each observation and the mean.
• We must take the absolute value of the distance,
otherwise they would just cancel out to zero!
Formula:
| |iX X
n
−
∑

24. Mean Deviation: An ExampleMean Deviation: An Example
X – Xi Abs. Dev.
7 – 6 1
7 – 10 3
7 – 5 2
7 – 4 3
7 – 9 2
7 – 8 1
1. Compute X (Average)
2. Compute X – X and take
the Absolute Value to get
Absolute Deviations
3. Sum the Absolute
Deviations
4. Divide the sum of the
absolute deviations by N
Data: X = {6, 10, 5, 4, 9, 8} X = 42 / 6 = 7
Total: 12 12 / 6 = 2

25. What Does it Mean?
On Average, each observation is two units
away from the mean.
Is it Really that Easy?
• No!
• Absolute values are difficult to manipulate algebraically
• Absolute values cause enormous problems for calculus
(Discontinuity)
• We need something else…

26. Variance and Standard Deviation
Instead of taking the absolute value, we square
the deviations from the mean. This yields a
positive value.
This will result in measures we call the Variance
and the Standard Deviation
Sample- Population-
s: Standard Deviation σ: Standard Deviation
s2
: Variance σ2
: Variance

27. Example:
-1 1
3 9
-2 4
-3 9
2 4
1 1
Data: X = {6, 10, 5, 4, 9, 8}; N = 6
Total: 42 Total: 28
Standard Deviation:
7
6
42
===
∑
N
X
X
Mean:
Variance:
2
2
( ) 28
4.67
6
X X
s
N
−
= = =
∑
16.267.42
=== ss
XX − 2
)( XX −X
6
10
5
4
9
8

28. Introduction to Probability
Experiments, Counting Rules,Experiments, Counting Rules,
and Assigning Probabilitiesand Assigning Probabilities
Events and Their ProbabilityEvents and Their Probability
Some Basic RelationshipsSome Basic Relationships
of Probabilityof Probability
Conditional ProbabilityConditional Probability

29. Probability as a Numerical MeasureProbability as a Numerical Measure
of the Likelihood of Occurrenceof the Likelihood of Occurrence
00 11.5.5
Increasing Likelihood of OccurrenceIncreasing Likelihood of Occurrence
Probability:Probability:
The eventThe event
is veryis very
unlikelyunlikely
to occur.to occur.
The occurrenceThe occurrence
of the event isof the event is
just as likely asjust as likely as
it is unlikely.it is unlikely.
The eventThe event
is almostis almost
certaincertain
to occur.to occur.

30. An Experiment and Its Sample SpaceAn Experiment and Its Sample Space
AnAn experimentexperiment is any process that generatesis any process that generates
well-defined outcomes.well-defined outcomes.
AnAn experimentexperiment is any process that generatesis any process that generates
well-defined outcomes.well-defined outcomes.
TheThe sample spacesample space for an experiment is the set offor an experiment is the set of
all experimental outcomes.all experimental outcomes.
TheThe sample spacesample space for an experiment is the set offor an experiment is the set of
all experimental outcomes.all experimental outcomes.
An experimental outcome is also called aAn experimental outcome is also called a samplesample
pointpoint..
An experimental outcome is also called aAn experimental outcome is also called a samplesample
pointpoint..

31. Events & Probabilities…
An individual outcome of a sample space is called a simple
event [cannot break it down into several other events],
An event is a collection or set of one or more simple events
in a sample space.
Roll of a die: S = {1, 2, 3, 4, 5, 6}
Simple event: the number “3” will be rolled
Event: an even number (one of 2, 4, or 6) will be rolled

32. Events & Probabilities…
The probability of an event is the sum of the probabilities of
the simple events that constitute the event.
E.g. (assuming a fair die) S = {1, 2, 3, 4, 5, 6} and
P(1) = P(2) = P(3) = P(4) = P(5) = P(6) = 1/6
Then:
P(EVEN) = P(2) + P(4) + P(6) = 1/6 + 1/6 + 1/6 = 3/6 = 1/2

33. Probability Rules:
Mathematical Notation

34. Random Variables
 A random variable is a variable whose value
is a numerical outcome of a random
phenomenon
 often denoted with capital alphabetic symbols
(X, Y, etc.)
 a normal random variable may be denoted as
X ~ N(µ, σ)
 The probability distribution of a random
variable X tells us what values X can take and
how to assign probabilities to those values

35. Random Variables
 Random variables that have a finite
(countable) list of possible outcomes, with
probabilities assigned to each of these
outcomes, are called discrete
 Random variables that can take on any
value in an interval, with probabilities
given as areas under a density curve, are
called continuous

36. Random Variables
 Discrete random variables
 number of pets owned (0, 1, 2, … )
 numerical day of the month (1, 2, …, 31)
 how many days of class missed
 Continuous random variables
 weight
 temperature
 time it takes to travel to work

37. Conditional Probability…
Conditional probability is used to determine how two events
are related; that is, we can determine the probability of one
event given the occurrence of another related event.
Experiment: random select one student in class.
P(randomly selected student is male) =
P(randomly selected student is male/student is on 3rd
row) =
Conditional probabilities are written as P(A | B) and read as
“the probability of A given B” and is calculated as:

38. Conditional Probability…
Again, the probability of an event given that another event
has occurred is called a conditional probability…
P( A and B) = P(A)*P(B/A) = P(B)*P(A/B) both are true
Keep this in mind!

39. Data ExplorationSUMMARY
Descriptive statistics help describe your data’s distribution
A measure of central tendency and dispersion are needed to
describe your data’s distribution statistically
Ideally your data fits the descriptions of a normal distribution
with data distributed evenly on either side of the measure of
central tendency.
The following are measures of central tendency: mean, median
and mode
The following are measure of dispersion: range, variance, and
standard deviation
Histograms and box plots can help you illustrate your data’s
distribution
Your descriptive statistics, histograms and/or box plots together
help you describe the nature of your data
After exploring your data using descriptive statistics it’s good to
reflect on your question and modify or refine it as needed.

41. Thanks
MONDAY, DECEMBER 23
2013