1.1 Preliminary Concepts

1.1.1 Statistics: What & How

1.1.2 Topics

  • What is statistics?
  • How Statistics works?
  • Probability and Statistics
  • Application of Statistics
  • Example Problem

1.1.3 What is statistics?

Three Meanings

  • Plural of statistic
  • Table of data
  • Methodology

1.1.4 How Statistics works?

Takes a sample from a population.

drawing

There are many sampling techniques.

1.1.5 Probability and Statistics

1.1.6 Application of Statistics

1.1.7 Examples

  • Identification of unwanted spam messages in e-mail

  • Segmentation of customer behavior for targeted advertising

  • Forecasts of weather behavior and long-term climate changes

  • Prediction of popular election outcomes

  • Development of algorithms for auto-piloting drones and self-driving cars

  • Optimization of energy use in homes and office buildings

  • Projection of areas where criminal activity is most likely

  • Discovery of genetic sequences linked to diseases

1.1.8 Chapter Overview

  • Definition
  • Population and sample
  • Variable and its types
  • Scale of measurement
  • Use of summation sign
  • Main Discussion

1.1.9 Definition

Coxton and Crowden

Statistics may be defined as the science of collection, presentation, analysis and interpretation of numerical data.

1.1.10 Mechanism

  • Data Collection
  • Organization
  • Analysis
  • Interpretation
  • Presentation

1.1.11 Population and Sample

Population: A set of similar items or events which is of interest Sample: Any subset of population

drawing

  • Finite
  • Infinite

1.1.12 Variable and Constant

  • Variable
  • Random Variable
  • Constant

Examples

  • Income of a regular employee
  • Income of a freelancer
  • Any unchanging number, e.g, \(\pi\)
  • Result of a die throw
  • Father’s name
  • Mark of a subject
  • GPA of a student

1.1.13 Types of Variable

  • Qualitative
  • Quantitative
  • Discrete: Limited and pre-specified
  • Continuous: Can take on any values between any two given number

1.1.14 Univariate, Multivariate

1.1.15 Scale of Measurement

Describes nature of information within the values.

  • Nominal: Name of Insignificant number, e.g., color, Street no.,
  • Ordinal: Order matters, e.g., rating
  • Interval: Zero may not be zero, like temperature
  • Ratio: Zero is 0; most variables fall in this category

1.1.16 Examples

  • Gender
  • Religion
  • Temperature
  • Income group (Lower class, Low, Middle, High)
  • Income
  • Distance of stars
  • Radius of screws
  • Diameter of trees
  • Room no.

1.1.17 Another Example

Match as per suitable scale

Movie Rating Scale
Poor, bad, good, excellent ratio
In a scale of -10 to 10: -10, -2, 0, 5, 10 interval
Awesome, Amazing, Mind-blowing, Stunning nominal
In a scale of 0 to 10: 0, 5, 8, 10 ordinal

1.1.18 Operation with scales

drawing

1.1.19 Shifting origin and scale

Say we have values, \(x_1, x_2, \cdot \cdot \cdot , x_n\)

  • Origin shift: Adding/Subtracting
  • \(y_1 = x_1-a \space or \space x_1+a\)
  • Scale shift: Multiplying/Division
  • \(y_1 = b \cdot x_1 \space or \space x_1/b\)
  • both: \(y_i = \frac{x_i-a}{b}\)

1.1.20 Use of Summation sign

\[x_1 + x_2 + x_3 + x_4 = \sum_{i=1}^4 x_i\]

\[x_1 + x_2 + ... x_n = \sum_{i=1}^n x_i\]

\[x_1 + x_2 + ... x_{10} = ?\]

1.1.21 Theorem 01

\[\sum_{i=1}^n bx_i=b \sum_{i=1}^n x_i\]

1.1.22 Theorem 02

\[\sum_{i=1}^n (ax_i-b)=a \sum_{i=1}^n x_i-nb\]

1.1.23 Quick tips

  • \(\sum_{i=1}^n a = na\)
  • Can you prove it?

1.1.24 Theorem 03

\[\sum_{i=1}^n (ax_i^2-bx_i+c)=a\sum_{i=1}^n x_i^2-b\sum_{i=1}^n x_i + nc\]

1.1.25 Theorem 04

\[\sum_{i=1}^n (ax_i-by_i)=a\sum_{i=1}^n x_i - b \sum_{i=1}^n y_i\]

1.1.26 Theorem 05

\[\sum_{i=1}^n (ax_i-b)^2=a^2 \sum_{i=1}^n x_i^2 - 2ab \sum_{i=1}^n x_i + nb^2\]

1.1.27 Theorem 06

\[(\sum_{i=1}^n x_i)^2=\sum_{i=1}^n x_i^2 + \sum_{i \ne j}^n\sum x_ix_j\]

1.1.28 Quick Tip

\[\prod_{i=1}^k x_i = x_1 \times x_2 \times \cdot \cdot \cdot \times x_n\]

1.1.29 Theorem 07

\[\prod_{i=1}^k x_iy_i = (\prod_{i=1}^k x_i)(\prod_{i=1}^k y_i)\]

1.1.30 Theorem 08

\[\sum_{i=1}^m \sum_{i=1}^n (x_i+y_j)=n\sum_{i=1}^m x_i + m \sum_{i=1}^n y_j\]

1.1.31 Theorem 09

m\(\sum_{i=1}^m \sum_{i=1}^n (x_iy_j)=(\sum_{i=1}^n x_i) (\sum_{i=1}^n y_j)\)

1.1.32 Example

Given

\(f_1=2, f_2 = 4, f_3 = 6\)

\(x_1 = -3, x_2 =7, x_3 = 4\)

Find the values of

  1. \(\sum f_ix_i\)
  2. \(\sum f_ix_i^2\)
  3. \(\sum f_i(x_i-5)^2\)

1.1.33 Textbook Exercise -01

  1. Find the value of \(\sum_{i=1}^{10} (x_i-4)\)

where \(\sum_{i=1}^{10} x_i = 20\)

1.1.34 Exercise-01

    1. Discrete vs continuous variable
    1. Prove \[\sum_{i=1}^k abx_i = ab \sum_{i=1}^k x_i\]

1.1.35 Exercise -01

Prove \[\prod_{i=1}^n c =c^n\]

1.1.36 Exercise -01

  1. Find the value of \[\sum_{i=1}^{10} (x_i-4)\] where \[\sum_{i=1}^{10}=20\]

1.1.37 Creative Question

Given below are the daily income and expense of ten workers.

Income (x) 120 130
Expense (y) 80 120

From above data, prove

  • \[\sum_{i=1}^{2}x_iy_i \ne (\sum_{i=1}^{2}x_i)(\sum_{i=1}^{2}y_j)\]
  • \[\sum_{i=1}^{2} \sum_{j=1}^{2}x_iy_j=(\sum_{i=1}^{2}x_i)(\sum_{j=1}^{2}y_j)\]
  • \[\sum_{i=1}^{2} \sum_{j=1}^{2}(x_i-y_j)=2 \times \sum_{i=1}^{2}x_i- 2 \times \sum_{j=1}^{2}y_j\]

1.1.38 Creative Question -07

Given below are the daily income and expense of ten workers.

Income (x) 120 130 88 150 175 144 180 200 160 155
Expense (y) 80 120 70 100 160 114 170 195 140 131
  1. What do you mean by bivariate data?
  2. From above data, prove

\[\sum_{i=1}^{10} \sum_{j=1}^{10}x_iy_j=(\sum_{i=1}^{10}x_i)(\sum_{j=1}^{10}y_j)\] c. \[\sum_{i=1}^{10} \sum_{j=1}^{10}(x_i-y_j)=10 \times \sum_{i=1}^{10}x_i- 10 \times \sum_{j=1}^{10}y_j\] d. Prove \[\sum_{i=1}^{10}x_iy_i \ne (\sum_{i=1}^{10}x_i)(\sum_{i=1}^{10}y_j)\]