1. STATISTICS IN METEOROLOGY

2. COVERAGE Introduction Statistics and its applications in Meteorology
Raw data, array, grouped data
Cumulative frequency
Ogives
Summary

3. STATISTICS An imposing form of Mathematics
Suggests tables, charts and figures
Numbers play essential role:-
Provide raw material
Must be processed further, to be useful

4. STATISTICS Statistics is concerned with scientific methods for:-
Collection of Data
Organisation of Data
Summarising and Presentation of Data
Analysis of Data
Drawing valid Conclusions and making reasonable Decisions based on analysis

5. STATISTICS
Involves methods of refining numerical and non-numerical information into useful forms
When numbers are collected and compiled, they become Statistics
Synonymous with ways and means of presenting and handling Data, making inferences logically and drawing relevant conclusions

6. CHARACTERISTICS
Numerical data must possess the following characteristics to be called as statistics.
Aggregate of Facts: Single and isolated figures are not statistics – they are unrelated and cannot be compared.Ex - Monthly Income of Mr X is Rs 50000/-. It would not constitute statistics although it is a numerical statement of fact

7. CHARACTERISTICS Affected by multiplicity of causes
Facts and figures are affected by number of forces operating together.Ex- Statistics of production of rice are affected by the rainfall, quality of soil,seeds and manure ,methods of cultivation,etc It is difficult to study separately the effect of each of these forces separately on the production of rice.
Numerically expressed
All stats are numerical statements of facts i.e. expressed in numbers.Qualitative statements such as the population of india is rapidly increasing is not statistics

8. CHARACTERISTICS Accuracy
Facts and figures about any phenomenon can be derived in two ways:
By actual counting and measurement.
By estimate.
Collected in systematic manner
Before collecting data a suitable plan should be made and worked out in systematic manner.
Data collected in haphazard manner would very likely lead to fallacious conclusions.

9. CHARACTERISTICS Collected for pre-determined purpose
Purpose must be decided in advance
It should be specific and well defined
Should be placed in relation to each other
They should be comparable

10. FUNCTIONS Presents facts in definite form. Simplifies mass of data.
Facilitates comparison.
Helps in formulating and testing hypothesis.
Helps in prediction.
Helps in formulation of suitable policies.

11. LIMITATIONS Does not deal with individual measurements.
Deals only with quantitative characteristics.
Only one of the methods to study a problem.

12. LIMITATIONS Does not always yield best results. Can be misused.
Anybody (without knowledge) can not deal with it.
Requires skills and experience.
No qualitative inferences.

13. DISTRUST OF STATISTICS
Statistics can prove almost anything : figures are convincing; hence, people are easily led to believe them.
Data can be manipulated : to establish foregone conclusions.
Even with correct figures, misled presentation can be made.

14. STATISTICS
Descriptive Statistics. Collection, Presentation and Description of numerical data (e.g., Means, Medians, Counts, Variance, Deviations, etc.).
Inferential Statistics. Process of interpreting what the values of your statistical tests mean and making decisions from those.

15. BASIC DEFINITIONS
It is often impossible or impractical to observe the entire group of data collected, especially if it is large.
Population. A collection, or set of individuals, objects, or measurements whose properties are to be analysed.
A population can be finite or infinite .

16. BASIC DEFINITIONS
Instead of examining the entire group, called population or universe, one examine a small part of group called sample.
Sample (a subset of the population). It consists of the individuals, objects or measurements selected by the sample collector from the Population.

17. BASIC DEFINITIONS
Variables. A variable is a symbol X ,Y A etc which can assume any of a prescribed set of values called the domain of the variable.
Constant. If the variable can assume only one value it is called a constant
Data. The set of values collected for the variable from each of the elements belonging to the sample.

18. VARIABLES Discrete : Result of counting (counts), usually Integers.
Counting give rise to discrete data.
Example : No. of children in each house of a village.
Continuous : A measurement of quantity, can assume any value between two given values.
Measurement give rise to continous data.
Example : Rainfall amount, Temperature, etc.

19. Variable X is independent variable Variable Y is dependent variable
BASIC DEFINITIONS
FUNCTION
If to each value which a variable X can assume there corresponds one or more values of a variable Y, we say that Y is a function of X.
Y = F (X)
Variable X is independent variable
Variable Y is dependent variable
Single valued function
Multiple valued function

20. Many types of graphs are used in stats depending on
BASIC DEFINITIONS
GRAPHS
A graph is a pictorial presentation of the relationship between variables.
Many types of graphs are used in stats depending on
Nature of data involved.
Purpose for which graph is intended.
Ex – Bar graphs,Pie graph, Picto-graphs etc.
Graphs also referred as charts / diagrams

21. TABULATION OF DATA : FREQUENCY DISTRIBUTION

22. BASIC DEFINITIONS
Raw Data : Collected data, which has not been numerically organized.
Arrays : An arrangement of raw data in ascending / descending order of magnitude.
Frequency Distribution : A tabular arrangement of data by classes, together with the corresponding Class Frequencies.

23. CLASSIFICATION OF DATA
After collection and editing of data the next step is classification
Classification is grouping of related facts into classes
Sorting of facts
Ex:
Post Office
School

24. CLASSIFICATION OF DATA
Objectives
To condense the mass of data in such a manner that similarities and dissimilarities can be readily apprehended
To Facilitate comparison
To pin point the most significant features of data at a glance
To give prominence to the important information gathered and dropping out the unnecessary elements
To enable statistical treatment of data

25. TYPES OF CLASSIFICATION
Geographical – Area wise ,cities, districts etc.(State wise production of food grains)
Chronological – on basis of time( Population of india from )
Qualitative - According to attributes or qualities (On basis of literacy, religion etc)
Dichotomous or two fold classification
Manifold classification
Quantitative – In terms of magnitudes(Students of college on the basis of weight)

26. DISCRETE FREQUENCY DISTRIBUTION
FORMATION OF
DISCRETE FREQUENCY DISTRIBUTION
Very simple method.
Count the number of times, a particular value is repeated
Table represents frequency of that particular class
Example

27. RAW DATA
No. of Children per family at Air Force Station, XYZ is:- 2 1 4 3

28. FREQUENCY DISTRIBUTION
NO. OF CHILDREN
TALLIES
FREQUENCY II 2 1
IIIII III 8
IIIII IIIII I 11 3
IIIII I 6 4
III
TOTAL 30

29. FORMATION OF FREQUENCY DISTRIBUTION
Determine the largest and smallest numbers in the raw data and thus find the range
Divide the range into a convenient number of class intervals having the same size
Determine the number of observations falling into each class interval i.e find the class frequencies. This is best done by using a tally or score sheet

30. FORMATION OF CONTINUOUS FREQUENCY DISTRIBUTION
More popular and widely used
Class Limits : LCL & UCL
Class Interval : UCL – LCL
Class Frequency
Class Mid-points / Class Marks :-
Mid-point = (UCL + LCL) / 2

31. CLASSIFICATION OF DATA ACCORDING TO CLASS INTERVALS
Exclusive Method - Upper limit of one class is the lower limit of the next class
It ensures continuity of data
It is always assumed that the upper limit is exclusive.i.e the item of that value is not included in that class.( Ex ,20-30,30-40)
Inclusive Method – The upper limit of one class is included in that class itself.(10-19,20-29)

32. RAW DATA
Marks obtained by students in Maths are:- 35 36 28 41 40 29 37 33 26 21 31 38 30 43 39

33. FORMATION OF CONTINUOUS FREQUENCY DISTRIBUTION
MARKS
TALLIES
FREQ.
CUM. FREQ.
0 – 10
10 – 20
20 – 30
IIIII I 6
30 – 40
IIIII IIIII I 11 17
40 – 50
III 3 20
TOTAL
EXCLUSIVE & INCLUSIVE METHODS

34. TABULATION OF DATA
A Table is a systematic arrangement of statistical data in columns and rows
Rows are horizontal arrangements
Columns are vertical arrangements
Purpose: The purpose of a table is to simplify the presentation and to facilitate comparisons

35. TABULATION OF DATA
ONE OF THE SIMPLEST AND MOST REVEALING DEVICES FOR SUMMARISING DATA.
ROLE OF TABULATION
IT SIMPLIFIES COMPLEX DATA
IT FACILITATES COMPARISON
IT GIVES IDENTITY TO THE DATA
IT REVEALS PATTERNS
PARTS OF TABLE
TABLE NUMBER,TITLE OF TABLE,CAPTION, STUB,BODY OF THE TABLE,HEADNOTE,FOOTNOTE.

36. DIAGRAMMATIC & GRAPHIC PRESENTATION
One of the most convincing and appealing ways in which statistical results may be presented
Significance of Diagrams and Graph
They give bird’s eye view of the entire data ,information presented is easily understood. “A picture is worth 10,000 words”.
They are attractive to the eye
They facilitate comparison

37. DIAGRAMMATIC & GRAPHIC PRESENTATION
Comparison of Tabular and Diagrammatic Presentation
Tables contain precise figures whereas diagrams give only an appx data
More information can be presented in one table than either in one graph
Table more difficult to interpret than diagrams
Graphs and Diagrams have a visual appeal thus more impressive

38. DIAGRAMMATIC & GRAPHIC PRESENTATION
Types of Diagrams
ONE-D Diagrams e.g., Bar diagrams
TWO-D Diagrams e.g., Rectangular, Squares ,Circles
Three-D Diagrams e.g., Cubes, Cylinders and spheres
Pictograms & Cartograms
Types of Graphs
Graphs of time series
Graphs of frequency distributions

39. DIAGRAMMATIC & GRAPHIC PRESENTATION
Difference between Diagrams & Graphs
Graphs are generally constructed on a graph paper whereas diagram is constructed on a plain paper. A graph represents mathematical relationship between variables whereas diagram does not
Diagrams are more attractive to eyes thus better suited for publicity and propaganda. They do not add anything to the meaning of the data thus not helpful to statisticians and researchers
For representing frequency distribution and time series, graphs are more appropriate than diagrams. In fact for presenting frequency distribution diagrams are rarely used

40. GRAPHICAL PRESENTATION

41. GRAPHS OF FREQUENCY DISTRIBUTION
Histogram.
Frequency Polygon
Smoothed Frequency Curve
Cumulative Frequency Curve, i.e., ‘Ogive’

42. HISTOGRAM Most popular and widely used
Set of Vertical Bars, whose areas are proportional to frequency represented
Variables always on X-axis and frequency on Y-axis
Bases on horizontal axis (X-axis) with centers at the class marks and lengths equal to class interval sizes
Areas proportional to class frequencies

43. CONSTRUCTION OF HISTOGRAM
For Distributions having Equal Class-Intervals
Take frequency on Y-axis
Variable on X-axis
Construct adjacent rectangles
Height of the rectangles will be proportional to the frequencies

44. CONSTRUCTION OF HISTOGRAM
For Distributions having Unequal Class-Intervals.
A correction for unequal class intervals must be made
Finding frequency density or relative frequency density
The frequency density is the frequency for that class divided by the width of that class
Areas proportional to class frequencies

45. FORMATION OF CONTINUOUS FREQUENCY DISTRIBUTION
MARKS
TALLIES
FREQ.
CUM. FREQ.
0 – 10
10 – 20
20 – 30
IIIII I 6
30 – 40
IIIII IIIII I 11 17
40 – 50
III 3 20
TOTAL
EXCLUSIVE & INCLUSIVE METHODS

46. HISTOGRAM

47. FREQUENCY POLYGON Graph for Frequency Distribution
Draw Histogram, join mid-points of upper side of each bar with straight line

48. FREQUENCY POLYGON
By constructing a frequency polygon the value of mode can be easily ascertained
It facilitate comparison of two or more frequency distribution on the same graph

49. ADVANTAGES OF FREQUENCY POLYGON OVER HISTOGRAM
Several distributions can be plotted on same axis
Much simpler
Sketches outline of data pattern more clearly
Becomes increasingly smooth as observations increase

50. SMOOTHED FREQUENCY CURVE
Instead of straight line, curved line is used for smoothening.

51. CUMULATIVE FREQUENCY CURVE
(OGIVE)
The curve obtained by plotting cumulative frequencies is called cumulative curve or Ogive
Two methods of constructing Ogive:
Less than method : We start with upper limits of the classes and go on adding the frequencies. When these frequencies are plotted we get a rising curve
More than method : We start with lower limits of the classes and go on subtracting the frequencies of each class. When these frequencies are plotted we get a declining curve

52. CUMULATIVE FREQUENCY CURVE (OGIVE)
To determine and portray number or population of cases above or below a given value
To compare two or more Frequency Distributions
To determine certain values graphically
Not so simple to interpret (be careful to use)

53. CUMULATIVE FREQUENCY CURVE (OGIVE)

54. TYPES OF FREQUENCY CURVES
Symmetrical or bell-shaped (Normal curve)
Positive skewness (Skewed to the right)
Negative skewness(Skewed to the left)
J – Shaped ( Max occurs at one end )
Reversed J – Shaped.(Max occurs at one end )
U – Shaped ( Maxima at both ends)
Bimodal ( Two Maxima)
Multimodal ( More than two Maxima )

55. LIMITATIONS OF DIAGRAMS AND GRAPHS
Can present only approximate values
Can approximately represent only limited information
Intended mostly to explain quantitative facts to general public
Can be misinterpreted

56. FREQUENCY DISTRIBUTION
SALIENT FEATURES OF
FREQUENCY DISTRIBUTION

57. SALIENT FEATURES OF FREQUENCY DISTRIBUTION
Measures of Central Tendency
Measures of Dispersion
Measures of Relative Position
Skewness
Kurtosis

58. SUMMARY Importance and Applications of Statistics in Meteorology
Frequency Distribution and Tabulation of Data
Graphical Presentation of Data
Salient Features

59. ANY QUESTION ?

60. MEASURES OF CENTRAL TENDENCY

61. COVERAGE Introduction. Requisites of a Good Average
Types, Merits and Limitations:-
Mean
Median
Mode
Quartiles and Percentiles
Relation between Mean, Median and Mode
Summary

62. INTRODUCTION
One of the most important objectives of statistical analysis is to obtain one single value that describes characteristics of the entire class / group
Known as ‘Central Value’ or ‘Average’
Facilitates comparison between two different classes / groups

63. AVERAGES & MEASURES OF CENTRAL TENDENCY
An average is a value which is typical or representative of a set of data
Why are they called measures of central tendency ?
As average tend to lie centrally within a set of data arranged according to magnitude ,they are also termed as measures of central tendency
Several types of averages can be defined , the most common being the Arithmetic mean, median , mode , the geometric mean, and the harmonic mean.
Each has advantages and disadvantages depending on: -
The data
The intended purpose

64. REQUISITES OF A GOOD AVERAGE
Easy to understand and simple to compute
Based on all items in the group
Not to be affected by extreme values
Rigidly defined and capable of further algebraic treatment
Sampling Stability

65. TYPES Mean:- Arithmetic Mean Geometric Mean Harmonic Mean Median Mode
Quartiles and Percentiles

66. ARITHMETIC MEAN
Individual Observations. or

67. ARITHMETIC MEAN
Discrete Series. or
Short cut method
A is assumed mean

68. Practice time
Illustration 3, page 184
Ilustartion 4, page 184

69. MERITS OF ARITHMETIC MEAN
Simplest and Easiest measure for Central Tendency.
Affected by each and every value in the group.
Rigid formula, can be mathematically treated later.
Relatively reliable (some sampling stability).
It is centre of gravity
Calculated value and not based upon position.

70. LIMITATIONS OF ARITHMETIC MEAN
Affected by extreme values.
Not always a good measure (good in case of normal distribution, U shaped curve???).
0˚C
50˚C
MEAN = 25˚C ?

71. GEOMETRIC MEAN
The geometric mean G of a set of numbers X1,X2,-----xN is the nth root of the product of the numbers. or

72. USES OF GEOMETRIC MEAN
To find average percentage increase in sales, production, population or other economic / business series.
Theoretically considered as the best measure in construction of Index numbers.
Most suitable, when large weights given to small items, or small weights to large items.

73. Individual Observations.
HARMONIC MEAN
The harmonic mean H of a set of N numbers X1,X2,------Xn is the reciprocal of the arithmetic mean of the reciprocals of the numbers.
Individual Observations.
Discrete & Continuous Series.

74. HARMONIC MEAN Example Find the harmonic mean of the numbers : 2, 4 , 8

75. USES OF HARMONIC MEAN Based on reciprocals of numbers.
Restricted usage.
Tedious calculation, when no. of items is large.
Useful for computation of average rate of increase in profit / loss, average speed during journey, average selling price of items, etc.

76. RELATION BETWEEN ARITHMETIC, GEOMETRIC AND HARMONIC MEAN
H < G < A
Calculate Arithmetic , Geometric and Harmonic mean for the following set of data
2,4 and 8
A =
G = 4
H =

77. MEDIAN Refers to the middle (i.e., Central) value of distribution.
Splits the data set into two halves, 50% items to each side.
Unlike mean, it is Positional Average.
To compute, first arrange the series into ascending or descending order, then find the frequency of central item.

78. COMPUTATION OF MEDIAN Continuous Series.
L : Lower Class Limit for Median class.
c.f. : Cum. Freq. of class preceding Median class.
N : Total No. of items (i.e., Size).
f : Freq. of Median class.
i : Class Interval of Median class.

79. Practice time
Illustration 14, page 200
Ilustartion 16, page 201

80. MERITS OF MEDIAN Useful for open-ended classes (Positional Average).
Not affected by extreme values.
useful in markedly skewed distribution.
Most appropriate in dealing with qualitative data.
Value can be found from graph.
Indicates value of middle item.

81. LIMITATIONS OF MEDIAN Necessary to arrange data series.
Not determined by all values, as it is positional average.
Value affected by sampling fluctuations.
Not further mathematical treatment can be applied.
Erratic, if sample is small.
Less reliable than arithmetic mean.

82. OTHER POSITIONAL MEASURES
Quartile
Q1= Size of (N+1)/4 th term or N/4 th term

83. MODE
Mode (or Modal Value) of the distribution is that value, for which the frequency is maximum.
Value, around which, items tend to be most heavily concentrated. Y X
MODE

84. COMPUTATION OF MODE Continuous Series.
L : Lower Class Limit for Median class.
f1 : Freq. of Modal class.
f0 : Freq. of class preceding Modal class.
f2 : Freq. of class succeeding Modal class.
i : Class Interval of Modal class.

85. Practice time
Illustration 25 and 26, page 216

86. MERITS OF MODE Most frequently occurring value.
Not affected by extreme values.
Can be used to describe qualitative phenomena.
Value can also be found from graph.

87. LIMITATIONS OF MODE
Can not be always determined (sometimes, Bi-modal also).
Not based on each and every value.
Not further mathematical treatment can be applied.
Not rigidly defined.
With quantitative data, disadvantages outweigh its merits.

88. RELATIONSHIP BETWEEN MEAN, MEDIAN AND MODE
MODE = 3 x MEDIAN – 2 x MEAN Y X
MODE
MEDIAN
MEAN
CENTRE OF GRAVITY
DIVIDES AREA IN TWO HALVES
MAX. FREQ. (PEAK)

89. COMPARISON OF VARIOUS AVERAGES
Arithmetic Mean. Influenced by each & every value, but affected by extreme values.
Median. Not affected by extreme values (better for groups with more extreme values).
Mode. Very unstable and has limited use.

90. QUARTILES, DECILES AND PERCENTILES
Divide a series into equal parts.
Quartiles : divide total frequency into FOUR (04) equal parts.
Deciles : divide total frequency into TEN (10) equal parts.
Percentiles : divide total frequency into HUNDRED (100) equal parts.

91. QUARTILES, DECILES AND PERCENTILES
Quartiles are used widely in Economics and Business statistics.
Deciles and Percentiles are important in Psychological and Educational statistics concerning Rank, Grade, Rate, etc.
These are not Averages.

92. SUMMARY
Importance and Applications of various measures of Central Tendency.
Merits and Limitations of Mean, Median and Mode.
Calculation of each measure of Central Tendency.
Relation between various measures.

93. ANY QUESTION ?