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Lecture No. 1

Dated: 04-11-2024

What is Statistics

It is the science of drawing conclusions from real life data and is a tool for data based research.
It is also called quatitative analysis.
It has applications in various disciplines such as:

  • Agriculture
  • Anthropology
  • Astronomy
  • Biology
  • Economic
  • Engineering
  • Geology
  • Genetics
  • Medicine
  • Physics
  • Psychology
  • Zoology

Nature of This Discipline

graph LR
A["Descriptive Statistics"] --> Probability
Probability --> C["Inferential Statistics"]

Meaning of Statistics

It comes from the latin word status which means political state.
It originally meant information useful to the state.

  • It refers to "numerical facts systematically arranged". In this case, we always use the plural form of statistics. This is what the the word statistics normal is referred to but the word data is used instead.
  • It is a discipline that involves procedures and techniques to collect, process and analyze numerical data to make inferences and research decisions against uncertainty which is incompleteness and instability of available data. In this case, the word is used as singular.
  • It can also refer to numerical quantities calculated from sample observations. A single quantity that is collected, is called a statistic.

Characteristics of the Science of Statistics

It deals with

  • Behavior of aggregates or large bodies of data of same kind, not isolated figures, ignoring what is happening to the individual or object of that aggregate.
  • Variability which obscures the underlying patterns.
  • Uncertainties as every process of gathering information involves deficiencies and variations.
  • Characteristics of things which can be expressed numerically either by count or measurements.
  • Aggregates which are subject to a number of random causes. e.g. the height of a person is subject to his race, age, diet, habits, climate etc.
  • Statistical laws are valid for the average or in long run. Therefore, statistical inference is therefore made against uncertainty.
  • If the data collection and processing is not handled properly, the results could be misleading.

They way in Which Statistics Works

It assists in

  • Summarizing the larger set of data in a form that is easily understandable.
  • Efficient design of laboratories, field experiments, surveys.
  • Effective planning in any field of inquiry.
  • Making general conclusions that how much of a certain thing will happen in certain conditions.

Importance of Statistics in Various Fields

  • A modern administrator uses statistical data to provide a factual basis for decision.
  • A politician uses statistics to support his arguments.
  • A businessman, industrial or research worker all use statistics in their work.
Quote

“a social scientist without an adequate understanding of statistics, is often like the blind man groping in a dark room for a black cat that is not there”

The Meaning of Data

It comes from the latin word datum(singular) and data means "those which are given".
Data may therefore be thought as the results of observations.

Observations

An observation means any sort of numerical recording of information whether it is physical such as measuring height, width, a classification such as heads or tails, or an answer to a question such as yes or no.

Variables

A characteristic which varies with an object or an individual such as age.
The set of all possible values from which a variable can take, is called domain.
If the domain consists of only one value then the variable is called a constant.

Quantitative and Qualitative Variables

Quantitative

The variables which are numerical such as age, height etc are called quantitative variables.

Qualitative

The variables which are not numerical such as eye color, poverty, education etc are called qualitative variables.

Discrete and Continuous Variables

Discrete

These variables take a discrete set of integers or whole numbers, as jumps or breaks.

Examples

  • Number of rooms in a house
  • Number of family members

Continuous

These variables can take any fractional or integer values given within an interval without gaps.

Examples

  • Age
  • Height
  • Temperature

Measurement Scales

Measurement

Measurement means assigning a number to observations or objects.

Scaling

It is the processing of measuring.
Following are the scales discussed.

Nominal Scale

Classification of observations into mutually exclusive qualitative categories.

Examples
  • Students in a class are males or females, hence we can assign 1 and 2 respectively.
  • Rainfall can be heavy, moderate or light, hence we can assign 1, 2 and 3 respectively.

Ordinal or Ranking Scale

Similar to nominal scale but with addition of the property of ordering or ranking.

Examples
  • good, fair, poor can be ranked 1, 2 and 3.

Interval Scale

A measurement scale possessing a constant interval size but not a true zero point, is called interval scale.

Examples
  • \(20^\circ C\) (\(68^\circ F\)) and \(30^\circ C\) (\(86^\circ F\))
  • \(5^\circ C\) (\(41^\circ F\)) and \(15^\circ C\) (\(59^\circ F\))

Both of the scales have equal intervals.
10 units for celsius scale and 18 units for farenheit.

Ratio Scale

Same as interval scale but it has a true meaningful zero point.

Errors of Measurement

The true value can never be measured because of certain habits, practices, methods and instruments used for measurements.
This departure of recorded value from the true value is called error of measurement.

If the observed value is \(x\) but the true value is \(x + \epsilon\) then the absolute error is \((x + \epsilon) - \epsilon\)
The ratio of absolute error to true value \(\frac{(x + \epsilon) - \epsilon}{x + \epsilon}\) is called the relative error.
Multiplying relative error by 100 gives us percentage error.

Example

If a student's weight is measured as \(60KG\) then we know for certain that his true weight is within \(60.5KG\) to \(59.5KG\) range.

Biased and Random Errors

Biased

An error is called biased if the observed value is constantly or consistently higher or lower than the true value.
This happens due to to limitations of measuring instrument and is cumulative.
Being cumulative means that the more measurements we do, the error adds up and its magnitude in overall result increases.
They are also called cumulative or systematic errors and are not revealed by repeating measurements.

Random or Unbiased

An error is said to be unbiased or random when the deviations from the true value, either excesses or defects tend to occur equally often.
When measurements happen often, they tend to cancel out in the long run.
Therefore, these errors are compensating and are also called accidental errors.