Applied Statistics: Probability basics for Statistics in Data Science

Before we dive deep into Inferential statistics we need to know some basics of data visualization and probability.

Starting with which we will look into the concept of skewness in Normal distribution. Skewness is the asymmetry of the distribution.

Skewness refers to a distortion or asymmetry that deviates from the symmetrical bell curve, or normal distribution, in a set of data. If the curve is shifted to the left or to the right, it is said to be skewed.

 

 

The skewness for a normal distribution is zero, and any symmetric data should have a skewness near zero. Negative values for the skewness indicate data that are skewed left and positive values for the skewness indicate data that are skewed right. By skewed left, we mean that the left tail is long relative to the right tail. Similarly, skewed right means that the right tail is long relative to the left tail.

 

 

Outlier:

In statistics, an outlier is a data point that differs significantly from other observations. An outlier may be due to variability in the measurement or it may indicate the experimental error; the latter are sometimes excluded from the data set. An outlier can cause serious problems in statistical analyses.

 

Unimodal, Bimodal, and Multimodal distribution:

 

Unimodal distribution:

A distribution where one value or bin contains more data than the other values or bins

 

Bimodal distribution:

A distribution where there are two distinct values or bins that contain more data than the others, usually separated by a gap.

 

Multimodal distribution:

A distribution where there are many values or bins that contain more data than other nearby bins, usually separated by gaps.

 

 

There are various types of plots and plotting concepts that provides a visual view of the data. We will discuss it in upcoming modules. We will look into some software implementation using python where we need, box plots, scatter plots, histograms, pie charts, bar plots, line plots, Heat maps, area plots, etc. for analyzing certain data

These topics come under the concept of data visualization.

 

 For more details about the data visualization these some articles which give in-depth understanding about the plotting: 

https://towardsdatascience.com/data-visualization-which-graphs-should-i-use-55e214ee9cf1

https://towardsdatascience.com/top-16-types-of-chart-in-data-visualization-196a76b54b62

 

Why Data Visualization?

To gain insight into an information space by mapping data onto graphical primitives.

To provide a qualitative overview of large datasets and search for patterns, trends, structures, irregularities, relationships among data.

 

 

 

Probability and Distributions:

For Inferential statistics, the probability is a very important concept. Probability is the mathematical concept and Distributions being the way to organize probability and a reference to how data may occur.

• Probability: Meaning and Concept
• Probability refers to chance or likelihood of a particular event taking place.
• An event is an outcome of an experiment (collection of outcomes)
• An experiment is a process that is performed to understand and observe possible outcomes.
• Set of all outcomes of an experiment is called sample space.

 

Example: In a manufacturing unit, 3 parts from the assembly is selected. You are observing whether they are defective or non-defective. Determine a] Sample space b] Event of getting at least two defective parts

Solution:

 

S = Sample Space

D = Defective

G = Non-Defective

 

Sample space:

GGG GGD GDG DGG

GDD DGD DDG DDD

 

Let E denote the event of getting at least 2 defective parts. This implies that E will contain two and three defectives. Looking at the sample space above:

E = {GDD, DGD, DDG, DDD}

It is easy to see that E is a part of S and commonly called as a subset of S. Hence an event is always a subset of the sample space.

 

 

Definition of probability:

The probability of an event A is defined as the ratio of two numbers m and n. P(A) = m/n

Where m is a number of ways that are favorable to the occurrence of A, and n is the total number of outcomes of the experiment. P(A) is always >= 0 and always <= 1.

 

Extreme values of probability: The range within which the probability of an event lies, can be best understood by the following diagram. The glass shows three stages. Empty, Half-full, and full to explain the properties of probability.

 

 

0% chance, 50% chance or equally like, and 100% chance or certainty

 

 

Mutually exclusive events:

Two events A and B are said to be manually exclusive if the occurrence of A precludes the occurrence of B. For example, from a well-shuffled pack of cards, if you pick up one card at random and would like to know if it is a king or a queen, the selected card will either be a king or a queen. It cannot be both king and queen. If king occurs, t queen will not occur, and if queen occurs, a king will not occur

 

 

 

Independent events:

Two events A and B are said to be independent if the occurrence of A is in no way influenced by the occurrence of B. Likewise, the occurrence of B is in no way influenced by the occurrence of A. Mutually exclusive events are not independent.

 

 

Rules for computing Probability:

1] Addition rule - Mutually exclusive events

P(A U B) = P(A) + P(B)

This rule says that the probability of union A and B is determined by adding the probability of event A and B

Here symbol A U B is called A union B, meaning A occurs, or B occurs or both A and B simultaneously occur. When A and B mutually exclusive, A and B cannot simultaneously occur.

 

2] Addition rule - Non Mutually exclusive events

P(A U B) = P(A) + P(B) - P(A ∩ B)

This rule says that the probability of the union of A and B is determined by adding the probability of events A and B, and then subtracting the probability of the intersection of events A and B

 

What are Disjoint Events?

Disjoint events cannot happen at the same time. In other words, they are mutually exclusive.

Put in formal terms, events A and B are disjoint if their intersection is zero:

P(A ∩ B) = 0.

You’ll sometimes see this written as:

P(A and B) = 0.

 

 

 

Multiplication rule:

 

Independent events:

P(A ∩ B) = P(A) . P(B)

This rule says that when 2 events A and B are independent the probability of simultaneous occurrence of A and B equals the product of probability of A and probability of B.

 

Dependent events:

P(A ∩ B) = P(A) . P(B/A)

P(B/A) stands for B given A

This rule says that the probability of the intersection of the events A and B equals the product of probability of A and B, given that A has happened or known to you. This is symbolized in the second term of the above expression P(B/A).

P(B/A) is called conditional probability of B given fact that A has happened or

P(A ∩ B) = P(B) . P(A/B) if B has already happened

 

If independent P(B/A) = P(B) and P(A/B) = P(A)

 

P(B/A) = P(A ∩ B) / P(A) and P(A/B) = P(A ∩ B) / P(B)

 

 

Marginal Probability:

Contingency tables consist of rows and columns of two attributes at different levels with frequencies or numbers in each of the cells. It is a matrix of frequencies assigned to rows and columns.

The term marginal is used to indicate that the probabilities are calculated using a contingency table.

 

Upcoming next we will be going to know Bayes Theorem, Binomial Distribution, Poisson distribution, and normal distribution insights and a code in python for the same. Till then stay safe and enjoy life!