1. Survey Feedback

Information is collected through survey method. This is the most popular and widely used method of data collection. The managers use this information collected through survey for making decisions. The wide range of data is collected regarding working conditions, quality of work, working hours, wages and salaries, attitude of employees relating to above.

These data are then analyzed by the team of managers. They find out the problem, evaluate the results and find out solutions. Information is collected from all the members of the organization. Managers conduct meetings with their subordinates and discuss the information, allow subordinates to interpret the data. After this plans are prepared for making necessary changes. This procedure is followed at all levels of management involving all the employees of the organization.

2. Team Building

Team Building is another method of organization development. This method is specifically designed to make improvement in the ability of employees and motivating them to work together. It is the organization development technique which emphasizes on team building or forming work groups in order to improve organizational effectiveness.

These teams consist of employees of the same rank and a supervisor. This technique is an application of sensitivity training to the teams of different departments. The teams or work groups are pretty small consisting of 10 to 15 persons. They undergo group discussion under the supervision of an expert trainer usually a supervisor. The trainer only guides but does not participate in the group discussion.

This method of team building is used because people in general do not open up their mind and not honest to their fellows. As they does not mix up openly and fail to express their views to the peers and superiors. This technique helps them express their views and see how others interpret their views. It increases the sensitivity to others’ behaviour.

They become aware of group functioning. They get exposed to the creative thinking of others and socio-psychological behaviour at the workplace. They learn many aspects of interpersonal behaviour and interactions.

3. Sensitivity Training

It is quite popular OD intervention. It is also known as laboratory training. Under this technique the employees in groups are asked to interact. The aim of sensitivity training is to help people understand each other and gain insight so that they feel free and become fearless.

Abraham Korman has rightly observed that, “the assumptions of sensitivity training procedure are that, if these goals are achieved, one will become defensive about himself, less fearful of the intentions of others, more responsive to others and their needs, and less likely to misinterpret others’ behaviours in a negative fashion.

“Under this technique the different groups of employees are allowed to mix up with each other and communicate freely and build up interpersonal relationship. They learn the reflection of their behaviour and try to improve it. In the words of Chris Argyris, “sensitivity training is a group experience designed to provide maximum possible opportunity for the individuals to expose their behaviour, give and receive feedback, experiment with new behaviour and develop awareness of self and of others.”

The employees through this technique know others feelings and behaviour and the impact of their behaviour on others. It builds up openness, improves listening skills, tolerate individual differences and the art of resolving conflicts. It helps in reducing interpersonal conflicts in the organization.

It is up to the executives at the top level of management in the organization to take decision regarding appropriateness of this technique but they must see that the objectives of organizational development are achieved with the help of this method.

However there is every likelihood that some culprits will exploit the opportunity to fulfill their vested goals at the cost of organization’s interests. There is one more serious drawback of the method that it may give rise to groupism in the organization which will defeat the purpose of OD. To make this technique effective and fulfill the purpose of OD, the selection of trainer must be cautiously made. He must be a man of integrity and responsibility and must command respect from the participating groups.

He plays a crucial role in making the OD programme successful. He should maintain cordial atmosphere throughout the training programme. He must see that each member of the groups learn the behaviour of others and to be creative and get more exposure to group life.

4. Managerial Grid

This technique is developed by industrial psychologists duo Robert Blake and Jane Mouton. The concept of managerial grid identifies two major dimensions of management behaviour. They are people oriented and production oriented behaviours. Attempts are made to pay increased attention to both the variables.

5. Management by Objectives (MBO)

MBO is a technique of management development which was put forward for the first time by Peter Drucker in 1954. It is a method of achieving organizational objectives and a technique of evaluation and review of performance. Under this method objectives of the organization are fixed and responsibility to achieve them lie on the managers and results are expected from them.

Achievement of organizational objectives is considered as the joint and individual responsibility of all managers. It also provides a perfect appraisal system. Performance of the managers is measured against the specific objectives. It is result oriented technique.

George Odiorne observed that MBO is, “a system wherein superior and subordinate managers of an organization jointly identify its common aims, define each individual’s major areas of responsibility in terms of the results expected of him and use these measures as guides for operating the unit, assessing the contribution benefits of its members.”

According to D D White and D A Bednar, “MBO is a technique designed to:-

• Increase the precision of the planning process at the organizational level.
• Reduce the gap between employee and organizational goals.”

6. Brain Storming

It is a technique where a group of five to eight managers come together and find a solution to a problem. As the name suggests it involves storming of the brain to develop creativity in thinking. It gives rise to new ideas. The principle involves in it is that any idea, thought or plan put forward in a meeting must be critically evaluated. The participants are asked to come forward with novel ideas generated in their mind. It works on a premise that everyone has a creative mind and capability to generate new ideas.

Participants are closely observed at the discussion and no expert is provided to conduct the meeting. The participants sit across the table for close communication. The brainstorming technique can generate an atmosphere where people can express freely. This encourages group interaction and creative thinking. The only limitation this method has is that it is time consuming and hence expensive.

7. Process Consultation

The technique of process consultation is an improvement over the method of sensitivity training or T Group in the sense that both are based on the similar premise of improving organizational effectiveness through dealing with interpersonal problems but process consultation is more tasks oriented than sensitivity training.

In process consultation the consultant or expert provides the trainee feedback and tell him what is going around him as pointed out by E H Schein that the consultant, “gives the client ‘insight’ into what is going on around him, within him, and between him and other people.”

Under this technique the consultant or expert provides necessary guidance or advice as to how the participant can solve his own problem. Here the consultant makes correct diagnosis of the problem and then guides the participants.

The consultant according to E H Schein, “helping the client to perceive, understand and act upon process events which occur in the clients’ environment.” Process consultation technique is developed to find solutions to the important problems faced by the organization such as decision making and problem solving, communication, functional role of group members, leadership qualities. Consultant is an expert outside the organization.

8. Quality Circles

Under this system a group of 5 to 12 come together at their own free will during working hours once in a week and discuss out the problems and suggests solution to the management for implementation. The supervisors remain present during the meeting. Quality Circles have their origin in Japan in nineteen sixties which improved the quality, reduced cost and heightened the morale of the workers. The success was due to workers’ participation. Total quality management or TQM is the recent development. This concept was adopted by the USA in 1980.

9. Transactional Analysis

Transactional analysis helps people to understand each other better. It is a useful tool for organizational development but it has diverse applications in training, counselling, interpersonal communication and making analysis of group dynamics. Nowadays, it is widely used as OD technique. It helps in developing more adult ego states among people of the organization. It is also used in process consultation and team building.

Stress is a general term applied to various psychologic (mental) and physiologic (bodily) pressures experienced or felt by people throughout their lives.

Stress is defined as “a state of psychological and physiological imbalance resulting from the disparity between situational demand and the individual’s ability and motivation to meet those needs.”

Dr. Hans Selye, one of the leading authorities on the concept of stress, described stress as “the rate of all wear and tear caused by life.”

Stress can be positive or negative:

• Stress is good when the situation offers an opportunity to a person to gain something. It acts as a motivator for peak performance.
• Stress is negative when a person faces social, physical, organizational and emotional problems.

Factors that are responsible for causing stress are called stressors.

Types of Stress

Many people feel stress after dealing with the loss of a loved one, while going through a divorce or even after a tough day at work, but most people don’t realize is that there are different kinds of stress. The types of stress relate to how the stress comes on or what symptoms are associated with the stress, but psychologists typically differentiate between the different types based on how long the periods last.

1. Physical Stress

A common type of stress is physical stress, which refers to actual physical activities and events that wreak havoc on the human body. One good example is travel. Traveling frequently can send you to different time zones, which makes sleeping and waking difficult. Physical stress also includes stress brought on by sleeping too much, not getting enough sleep, spending too many hours on your feet or working long hours. If you ever spent a day chasing your kids around an amusement park or stuck in an airport and dealing with flight delays, you have likely experienced physical stress.

2. Emotional Stress

Out of all the different kinds of stress, emotional stress is the most common. This can occur after you go through an intense break up or divorce, lose a loved one, have a fight with your spouse or experience any other problem that causes you to feel depressed or anxious. Emotional stress often manifests in the same way that depression does. You may experience weight changes, changes in how you fall asleep or how long you sleep, feelings of isolation and mood swings. Emotional stress can also occur when you feel overwhelmed at home or at work.

3. Traumatic Stress

When thinking about the types of stress, many people don’t think about traumatic stress. Traumatic stress is a type of stress that occurs because of some type of trauma to the human body and may lead to intense pain, coma or even death. It often relates to some kind of physical change that occurs. If you went through an operation, your body may experience stress until you recover from that surgery. A car accident, second or third degree burns or even a case of pneumonia may all cause traumatic stress.

4. Acute vs. Chronic Stress

In addition to the different types of stress, psychologists differentiate between acute stress and chronic stress. According to the American Psychological Association, chronic stress refers to an extended type of stress that impacts people every day of the year and can last for years or even decades. Acute stress is a type of stress that only occurs for a set period of time or only because of certain factors in the environment. This may include the stress you feel after a fight with your kids, a meeting at work or an encounter with another driver on the road. The APA also identifies something called episodic acute stress, which refers to intense periods of stress.

Stress can wreak havoc on the human body. It causes people to feel sick and tired and may lead some to contemplate suicide. Psychologists can help you learn more about the different kinds of stress, show you how to combat that stress and learning coping mechanisms for use at home and work.

Causes of Stress

1. Career Concern

If an employee feels that he is very much behind in the corporate ladder, then he may experience stress. If he seems that there are no opportunities for self-growth, he may experience stress. Hence, unfulfilled career expectations are the significant source of stress.

2. Role Ambiguity

It occurs when the person doesn’t know what he is supposed to do, on the job. His tasks and responsibilities are not clear. The employee is not sure what he is expected to do. It creates confusion in the minds of the worker and results in stress.

3. Rotating Work Shifts

Stress may occur in those individuals who work on different work shifts. Employees may be expected to work on day shift for some days and then on the night shift. That may create problems in adjusting to the shift timings, and it can affect not only personal life but also family life of the employee.

4. Role Conflict

It takes place when people have different expectations from the person performing a particular role. It can also occur if the job is not as per expectation, or when a job demands a certain type of behavior that is against the person’s moral values.

5. Occupational Demands

Some jobs are more demanding than others. Jobs that involve risk, and danger are more stressful. Research findings indicate, job that cause stress needs constant monitoring of equipments and devices, unpleasant physical conditions, making decisions, etc.

6. Lack of Participation in Decision-making

Many experienced employees feel that management should consult them on matters affecting their jobs. In reality, the superiors hardly ask the concerned employees before taking a decision. That develops a feeling of being neglected, which may lead to stress.

7. Work Overload

Excessive workload leads to stress as it puts a person under tremendous pressure. Work overload may take two different forms:

• Qualitative work overload implies performing a job that is complicated or beyond the employee’s capacity.
• Quantitative work overload is a result of many activities performed in a prescribed time.

8. Work Underload

In this, case, too little work or very easy work is expected on the part of the employee. Doing less work or jobs of routine and simple nature would lead to monotony and boredom, which can lead to stress.

9. Poor Working Conditions

Employees may be subject to poor working conditions. It would include bad lighting and ventilation, unhygienic sanitation facilities, excessive noise, and dust, presence of toxic gasses and fumes, inadequate safety measures, etc. All these unpleasant conditions create physiological and psychological imbalance in humans thereby causing stress.

10. Lack of Group Cohesiveness

Every group is characterized by its cohesiveness, although they differ widely in its degree. Individuals experience stress when there is no unity among work group members. There are mistrust, jealousy, frequent quarrels, etc., in groups and this lead to stress to employees.

11. Interpersonal and Intergroup Conflict

These conflicts take place due to differences in perceptions, attitudes, values and beliefs between two or more individuals and groups. Such conflicts can be a source of stress for group members.

12. Organizational Changes

When changes occur, people have to adapt to those changes, and this may cause stress. Stress is higher when changes are significant or unusual like transfer or adoption of new technology.

13. Lack of Social Support

When individuals believe that they have the friendship and support of others at work, their ability to cope with the effects of stress increases. If this kind of social support is not available, then an employee experiences more stress.

Job Stress also called occupational stress, work-related stress is a negative response (stress) that occurs in workplaces due to various demands or situations people find themselves in and not having enough resources to deal with it. Perceptions of loss and harm result in an individual’s stress response being triggered. Having inadequate coping resources is typically at the root of this reaction. The greater the emphasis on the consequences of failing, the greater will be the stress response.

Causes of Stress

Common causes of work-related stress include:

• Sudden deadlines or demanding job
• Harassment or bullying
• Stereotypes
• Gender discrimination
• Social isolation
• Work-home conflict
• Violence
• High risk jobs
• Unpleasant relationship with boss and/or co-workers
• Constant work contact

Factors such as the following impact one’s response to stress:

• Personality factors
• Level of job expertise
• Social support (family, co-workers & friends)
• Health status and disability
• Gender, ethnicity, and age
• Financial demands
• Life conditions outside the workplace

Signs and Symptoms of Stress

It is vital that you pay attention to signs of stress and take measures to deal with it effectively. Here are some indications of the presence of stress. Keep in mind that some of your symptoms may also be due to certain illnesses:

Problems with emotional health

• Mood changes
• Increased anxiety
• Depressed or pessimistic thought and feeling
• Low self-esteem
• Increased sense of irritation, sensitive or easily hurt
• Loss of motivation

Problems with mental health

• Confusion
• Concentration problems
• Poor memory

Problems with physical health

• High blood pressure
• Sweating
• Stress related rashes
• High cholesterol
• Muscle tension
• Sleep problems

Other behavioral changes

• More than normal absence from work
• Arriving late to work more than usual
• Changes in eating habits
• Increased use of drugs, alcohol, or nicotine
• Behavior effected by mood changes
• Changes in sleep patterns

Prolonged stress can lead to physical and mental illness. Therefore, it is important that you do not ignore stress symptoms for too long. Be sure to see your general physician and talk to your human resources manager or boss.

Consequences of Stress (Frone, Kelloway, & Barling, 2005)

Stress itself is not necessarily harmful, but persistent and prolonged stress can be harmful. As seen in the below graphic, not all stress is bad. However, prolonged stress without adequate resources to help one deal with adversities can lead to harmful outcomes.

Consequences occur at both the individual and the organizational level:

Individual Level Consequences

• Poor psychological and mental health outcomes (e.g., anger, depression, anxiety, posttraumatic stress syndrome, burnout, etc.)
• Impaired physiological processes (e.g., cardiovascular reactivity, elevated levels of various hormones, impaired immune function)
• Physical disease outcomes (e.g., hypertension, stroke, cancer, ulcers and gastrointestinal disorders, musculoskeletal disorders, migraine headaches)
• Detrimental behavioral outcomes (sleep disturbance; alcohol, tobacco, and illicit drug use; poor eating habits; intimate partner violence)

Organization Level Consequences

• Poor psychological and emotional outcomes (e.g., job dissatisfaction, low organizational commitment)
• Indicators of poor physical health (absence due to illness, workers’ compensation claims)
• Work-related behavioral impairment (injuries, poor job performance, on-the-job substance use)

Everyone who has ever held a job has, at some point, felt the pressure of work-related stress. Any job can have stressful elements, even if you love what you do. In the short-term, you may experience pressure to meet a deadline or to fulfill a challenging obligation. But when work stress becomes chronic, it can be overwhelming — and harmful to both physical and emotional health.

Unfortunately, such long-term stress is all too common. In fact, APA’s annual Stress in America survey has consistently found that work is cited as a significant source of stress by a majority of Americans. You can’t always avoid the tensions that occur on the job. Yet you can take steps to manage work-related stress.

Common Sources of Work Stress

Certain factors tend to go hand-in-hand with work-related stress. Some common workplace stressors are:

• Low salaries
• Excessive workloads
• Few opportunities for growth or advancement
• Work that isn’t engaging or challenging
• Lack of social support
• Not having enough control over job-related decisions
• Conflicting demands or unclear performance expectations.

Effects of Uncontrolled Stress

Work-related stress doesn’t just disappear when you head home for the day. When stress persists, it can take a toll on your health and well-being.

A stressful work environment can contribute to problems such as headache, stomachache, sleep disturbances, short temper and difficulty concentrating. Chronic stress can result in anxiety, insomnia, high blood pressure and a weakened immune system. It can also contribute to health conditions such as depression, obesity and heart disease. Compounding the problem, people who experience excessive stress often deal with it in unhealthy ways such as overeating, eating unhealthy foods, smoking cigarettes or abusing drugs and alcohol.

Taking Steps to Manage Stress

1. Track your stressors

Keep a journal for a week or two to identify which situations create the most stress and how you respond to them. Record your thoughts, feelings and information about the environment, including the people and circumstances involved, the physical setting and how you reacted. Did you raise your voice? Get a snack from the vending machine? Go for a walk? Taking notes can help you find patterns among your stressors and your reactions to them.

2. Develop healthy responses

Instead of attempting to fight stress with fast food or alcohol, do your best to make healthy choices when you feel the tension rise. Exercise is a great stress-buster. Yoga can be an excellent choice, but any form of physical activity is beneficial. Also make time for hobbies and favorite activities. Whether it’s reading a novel, going to concerts or playing games with your family, make sure to set aside time for the things that bring you pleasure. Getting enough good-quality sleep is also important for effective stress management. Build healthy sleep habits by limiting your caffeine intake late in the day and minimizing stimulating activities, such as computer and television use, at night.

3. Establish boundaries

In today’s digital world, it’s easy to feel pressure to be available 24 hours a day. Establish some work-life boundaries for yourself. That might mean making a rule not to check email from home in the evening, or not answering the phone during dinner. Although people have different preferences when it comes to how much they blend their work and home life, creating some clear boundaries between these realms can reduce the potential for work-life conflict and the stress that goes with it.

4. Take time to recharge

To avoid the negative effects of chronic stress and burnout, we need time to replenish and return to our pre-stress level of functioning. This recovery process requires “switching off” from work by having periods of time when you are neither engaging in work-related activities, nor thinking about work. That’s why it’s critical that you disconnect from time to time, in a way that fits your needs and preferences. Don’t let your vacation days go to waste. When possible, take time off to relax and unwind, so you come back to work feeling reinvigorated and ready to perform at your best. When you’re not able to take time off, get a quick boost by turning off your smartphone and focusing your attention on non-work activities for a while.

5. Learn how to relax

Techniques such as meditation, deep breathing exercises and mindfulness (a state in which you actively observe present experiences and thoughts without judging them) can help melt away stress. Start by taking a few minutes each day to focus on a simple activity like breathing, walking or enjoying a meal. The skill of being able to focus purposefully on a single activity without distraction will get stronger with practice and you’ll find that you can apply it to many different aspects of your life.

6. Talk to your supervisor

Employee health has been linked to productivity at work, so your boss has an incentive to create a work environment that promotes employee well-being. Start by having an open conversation with your supervisor. The purpose of this isn’t to lay out a list of complaints, but rather to come up with an effective plan for managing the stressors you’ve identified, so you can perform at your best on the job. While some parts of the plan may be designed to help you improve your skills in areas such as time management, other elements might include identifying employer-sponsored wellness resources you can tap into, clarifying what’s expected of you, getting necessary resources or support from colleagues, enriching your job to include more challenging or meaningful tasks, or making changes to your physical workspace to make it more comfortable and reduce strain.

7. Get some support

Accepting help from trusted friends and family members can improve your ability to manage stress. Your employer may also have stress management resources available through an employee assistance program (EAP), including online information, available counseling and referral to mental health professionals, if needed. If you continue to feel overwhelmed by work stress, you may want to talk to a psychologist, who can help you better manage stress and change unhealthy behavior.

In probability theory, the sample space (also called sample description space or possibility space) of an experiment or random trial is the set of all possible outcomes or results of that experiment. A sample space is usually denoted using set notation, and the possible ordered outcomes are listed as elements in the set. It is common to refer to a sample space by the labels S, Ω, or U (for “universal set”). The elements of a sample space may be numbers, words, letters, or symbols. They can also be finite, countably infinite, or uncountably infinite.

For example, if the experiment is tossing a coin, the sample space is typically the set {head, tail}, commonly written {H, T}. For tossing two coins, the corresponding sample space would be {(head,head), (head,tail), (tail,head), (tail,tail)}, commonly written {HH, HT, TH, TT}. If the sample space is unordered, it becomes {{head,head}, {head,tail}, {tail,tail}}.

For tossing a single six-sided die, the typical sample space is {1, 2, 3, 4, 5, 6} (in which the result of interest is the number of pips facing up).

A subset of the sample space is an event, denoted by E. Referring to the experiment of tossing the coin, the possible events include E={H} and E={T}.

A well-defined sample space is one of three basic elements in a probabilistic model (a probability space); the other two are a well-defined set of possible events (a sigma-algebra) and a probability assigned to each event (a probability measure function).

Another way to look as a sample space is visually. The sample space is typically represented by a rectangle, and the outcomes of the sample space denoted by points within the rectangle. The events are represented by ovals, and the points enclosed within the oval make up the event.

Equally likely outcomes

Some treatments of probability assume that the various outcomes of an experiment are always defined so as to be equally likely. For any sample space with N equally likely outcomes, each outcome is assigned the probability 1/N. However, there are experiments that are not easily described by a sample space of equally likely outcomes for example, if one were to toss a thumb tack many times and observe whether it landed with its point upward or downward, there is no symmetry to suggest that the two outcomes should be equally likely.

Though most random phenomena do not have equally likely outcomes, it can be helpful to define a sample space in such a way that outcomes are at least approximately equally likely, since this condition significantly simplifies the computation of probabilities for events within the sample space. If each individual outcome occurs with the same probability, then the probability of any event becomes simply:

P ( e v e n t ) = number of outcomes in event/ number of outcomes in sample space

For example, if two dice are thrown to generate two uniformly distributed integers, D1 and D2, each in the range [1…6], the 36 ordered pairs (D1, D2) constitute a sample space of equally likely events. In this case, the above formula applies, such that the probability of a certain sum, say D1 + D2 = 5 is easily shown to be 4/36, since 4 of the 36 outcomes produce 5 as a sum. On the other hand, the sample space of the 11 possible sums, {2, …,12} are not equally likely outcomes, so the formula would give an incorrect result (1/11).

Another example is having four pens in a bag. One pen is red, one is green, one is blue, and one is purple. Each pen has the same chance of being taken out of the bag.

The sample space S= {red, green, blue, purple}, consists of equally likely events. Here, P(red)=P(blue)=P(green)=P(purple)=1/4.

Sample event

A sample event refers to subset pertaining to sample space that comprises of sample space, empty set (A event that is impossible and possesses zero probability) and singleton set (it is also known as an elementary event). Other event stands to be proper subset pertaining to the sample space which generally comprises of different elements.
Event under the probability theory refers to set of outcome ascertained out of the experiment to which the assigning of probability is done. One outcome might be considered as element pertaining to multiple events that are different from one another. A complimentary event is defined by the event which comprises of event that is not happening and together it defines Bernoulli trial.

Sample event basically refers to something that happens or occurs. For instance flipping of count stands to be an event. Passing by the bench in the park while walking is also counted as an event. There are 1 or more outcomes associated with every event. For example, coin flipping stands to be an event, however, attaining a tail stands to be the outcome of such event. Walking in park stands to be an event, however identifying the friend in park stands to be the outcome associated with the event.

Thus it can be said event stands to be subset pertaining to sample space that comprises of sample space, none or all of the outcome. The event will be considered as a simple event in case it has only 1 sample point. If there are 2 or more sample points underlying an event the same will be considered as compound events. The event will be known as null space in case there is no sample point underlying an event.

In probability theory, an event is a set of outcomes of an experiment (a subset of the sample space) to which a probability is assigned. A single outcome may be an element of many different events, and different events in an experiment are usually not equally likely, since they may include very different groups of outcomes. An event defines a complementary event, namely the complementary set (the event not occurring), and together these define a Bernoulli trial: did the event occur or not?

Typically, when the sample space is finite, any subset of the sample space is an event (i.e. all elements of the power set of the sample space are defined as events). However, this approach does not work well in cases where the sample space is uncountably infinite. So, when defining a probability space it is possible, and often necessary, to exclude certain subsets of the sample space from being event.

A simple example

If we assemble a deck of 52 playing cards with no jokers, and draw a single card from the deck, then the sample space is a 52-element set, as each card is a possible outcome. An event, however, is any subset of the sample space, including any singleton set (an elementary event), the empty set (an impossible event, with probability zero) and the sample space itself (a certain event, with probability one). Other events are proper subsets of the sample space that contain multiple elements. So, for example, potential events include:

• “Red and black at the same time without being a joker” (0 elements),
• “The 5 of Hearts” (1 element),
• “A King” (4 elements),
• “A Face card” (12 elements),
• “A Spade” (13 elements),
• “A Face card or a red suit” (32 elements),
• “A card” (52 elements).

Since all events are sets, they are usually written as sets (e.g. {1, 2, 3}), and represented graphically using Venn diagrams. In the situation where each outcome in the sample space Ω is equally likely, the probability P {\displaystyle P} of an event A is the following formula:

The expected value (or mean) of X, where X is a discrete random variable, is a weighted average of the possible values that X can take, each value being weighted according to the probability of that event occurring. The expected value of X is usually written as E(X) or m.

• E(X) = S x P(X = x)

So the expected value is the sum of: [(each of the possible outcomes) × (the probability of the outcome occurring)].

In more concrete terms, the expectation is what you would expect the outcome of an experiment to be on average.

Example

What is the expected value when we roll a fair die?

There are six possible outcomes: 1, 2, 3, 4, 5, 6. Each of these has a probability of 1/6 of occurring. Let X represent the outcome of the experiment.

Therefore P(X = 1) = 1/6 (this means that the probability that the outcome of the experiment is 1 is 1/6)
P(X = 2) = 1/6 (the probability that you throw a 2 is 1/6)
P(X = 3) = 1/6 (the probability that you throw a 3 is 1/6)
P(X = 4) = 1/6 (the probability that you throw a 4 is 1/6)
P(X = 5) = 1/6 (the probability that you throw a 5 is 1/6)
P(X = 6) = 1/6 (the probability that you throw a 6 is 1/6)

E(X) = 1×P(X = 1) + 2×P(X = 2) + 3×P(X = 3) + 4×P(X=4) + 5×P(X=5) + 6×P(X=6)

Therefore E(X) = 1/6 + 2/6 + 3/6 + 4/6 + 5/6 + 6/6 = 7/2

So the expectation is 3.5 . If you think about it, 3.5 is halfway between the possible values the die can take and so this is what you should have expected.

Expected Value of a Function of X

To find E[ f(X) ], where f(X) is a function of X, use the following formula:

• E[ f(X) ] = S f(x)P(X = x)

Example

For the above experiment (with the die), calculate E(X²)

Using our notation above, f(x) = x²

f(1) = 1, f(2) = 4, f(3) = 9, f(4) = 16, f(5) = 25, f(6) = 36
P(X = 1) = 1/6, P(X = 2) = 1/6, etc

So E(X²) = 1/6 + 4/6 + 9/6 + 16/6 + 25/6 + 36/6 = 91/6 = 15.167

The expected value of a constant is just the constant, so for example E(1) = 1. Multiplying a random variable by a constant multiplies the expected value by that constant, so E[2X] = 2E[X].

A useful formula, where a and b are constants, is:

• E[aX + b] = aE[X] + b

[This says that expectation is a linear operator].

Variance

The variance of a random variable tells us something about the spread of the possible values of the variable. For a discrete random variable X, the variance of X is written as Var(X).

• Var(X) = E[ (X – m)² ]            where m is the expected value E(X)

This can also be written as:

• Var(X) = E(X²) – m²

The standard deviation of X is the square root of Var(X). 

Note that the variance does not behave in the same way as expectation when we multiply and add constants to random variables. In fact:

• Var[aX + b] = a²Var(X)

You is because: Var[aX + b] = E[ (aX + b)² ] – (E [aX + b])² .

= E[ a²X² + 2abX + b²] – (aE(X) + b)²
= a²E(X²) + 2abE(X) + b² – a²E²(X) – 2abE(X) – b²
= a²E(X²) – a²E²(X) = a²Var(X)

Decision theory (or the theory of choice not to be confused with choice theory) is the study of an agent’s choices. Decision theory can be broken into two branches: normative decision theory, which analyzes the outcomes of decisions or determines the optimal decisions given constraints and assumptions, and descriptive decision theory, which analyzes how agents actually make the decisions they do.

Decision theory is closely related to the field of game theory and is an interdisciplinary topic, studied by economists, statisticians, psychologists, biologists, political and other social scientists, philosophers, and computer scientists.

Empirical applications of this rich theory are usually done with the help of statistical and econometric methods.

Normative and descriptive

Normative decision theory is concerned with identification of optimal decisions where optimality is often determined by considering an ideal decision maker who is able to calculate with perfect accuracy and is in some sense fully rational. The practical application of this prescriptive approach (how people ought to make decisions) is called decision analysis and is aimed at finding tools, methodologies, and software (decision support systems) to help people make better decisions.

In contrast, positive or descriptive decision theory is concerned with describing observed behaviors often under the assumption that the decision-making agents are behaving under some consistent rules. These rules may, for instance, have a procedural framework (e.g. Amos Tversky’s elimination by aspects model) or an axiomatic framework (e.g. stochastic transitivity axioms), reconciling the Von Neumann-Morgenstern axioms with behavioral violations of the expected utility hypothesis, or they may explicitly give a functional form for time-inconsistent utility functions (e.g. Laibson’s quasi-hyperbolic discounting).

The prescriptions or predictions about behavior that positive decision theory produces allow for further tests of the kind of decision-making that occurs in practice. In recent decades, there has also been increasing interest in what is sometimes called “behavioral decision theory” and contributing to a re-evaluation of what useful decision-making requires.

The Law of Total Probability states that the payoff for a strategy is the sum of the payoffs for each outcome multiplied by the probability of each outcome.

A simple example illustrates this law. Suppose there is an interaction in which you could either win or lose. There are two outcomes (win and loss), each with its own probability. According to the Law of Total Probability, the payoff is:

(probability of winning) × (payoff if you win) + (probability of losing) × (payoff if you lose)

To make this easier to write, we’ll represent the probability of an event as P_event, so now we have:

P_win × (payoff for win) + P_lose × (payoff for loss)

How do we know the probability of each outcome? Since we want to find the average payoff for all players of the strategy, we imagine the probability for an average member of the population, that is, one who is of average size, fighting ability, and so on. In this simple example, that means that the probabilities of winning and losing are equal, at ½. (You could also reason that each interaction has a winner and a loser, so there are equal numbers of winners and losers in the population, making the probability of each outcome the same.)

When there are more than two possible outcomes, there are more terms in the sum:

P_{Outcome 1} × (payoff for Outcome 1) + P_{Outcome 2} × (payoff for Outcome 2) + … + P_{Outcome N} × (payoff for Outcome N)

For example, in a betting game that depends on the suit of a card that is drawn from a full deck, your payoff would be ¼(payoff for club) + ¼(payoff for spade) +¼(payoff for diamond) + ¼(payoff for heart).
Or, for a more complex example, consider a game in which you roll dice and you get one payoff if the number is 1-3, another if it is 4-5, and another if it is 6. Your payoff would be (1/2)×(payoff for 1-3) + (1/3)×(payoff for 4-5) + (1/6)×(payoff for 6).

You may have noticed that the probabilities add to 1 in all of these examples. This is no accident, and when calculating average payoffs, the probabilities must always add to 1.

Let’s take as an example animals fighting over a resource. For simplicity, we’ll say that the resource value is v and that the cost of losing a fight is c. Whenever two animals fight, there is a winner, who gets the resource, and a loser, who gets nothing and incurs a cost. The probability of winning and the probability of losing are equal, at ½. Thus half the population gets v and half gets −c. The payoff is:

P_win × (payoff for win) + P_lose × (payoff for loss), which is ½×v + ½×−c, or v/2−c/2.

How does it work in less extreme cases? After all, the winner may not get everything and the loser nothing. A winner might get most of the resource and the loser the rest, with the costs being similarly divided. Even when we consider these cases, however, the average outcome is still v/2−c/2. For every winner who gets ¾, a loser gets ¼, which averages to ½. If a winner gets 2/3, the loser gets 1/3, again averaging to ½, and so on for any other division of the resource and cost between winner and loser.

This illustrates that, when thinking about payoffs, we can usually simplify our reasoning and still get the right answer. In this case, we simplified things by making the outcome all-or-none, v or −c.

When filling out a payoff matrix, you need to do this calculation for each pair of strategies. Of course, the probabilities may differ depending on the strategies. The above example was for two animals using the same simple strategy, fighting. With other strategies, calculating the probabilities may be trickier.

Conditional Strategies

Some strategies are conditional, in that the user of the strategy acts differently depending on circumstances. For example, “fight if I’m larger than my opponent, but back off if I’m smaller” or “fight to keep ownership of a resource, but don’t fight if someone else already owns it” are both conditional strategies. The action depends on a condition such as size or ownership.

The total payoff depends on the probability of each condition being met and on the outcome of each action. So if Condition 1 leads to Action 1, Condition 2 leads to Action 2, and so on, the total payoff is:

P_{Condition 1} × (payoff for Action 1) + P_{Condition 2} × (payoff for Action 2) + … + P_{Condition N} × (payoff for Action N)

Of course, the payoff for each action may also involve probabilities. This sounds complicated, but it’s not difficult if you break it down.

1. Figure out the payoff for each action, using probability if necessary, like we did above for the simple fighting strategy.
2. Determine the probability of the condition that causes each action.
3. Multiply the probability of each condition (from step 2) by the payoff of the action that it causes (from step 1).
4. Add the probability × payoff pairs.
5. To fill the payoff matrix, repeat this for each pair of strategies.

Let’s do this for the “fight to keep ownership of a resource, but don’t fight if someone else already owns it” strategy when paired against a simple “always fight” strategy.

Our conditional strategy has two possible actions, fight and not-fight. What is the payoff of each against a fighting strategy? We already solved the fight vs. fight payoff above, which is v/2−c/2. What about not-fight vs. fight? If we don’t fight, we simply get nothing and incur no cost of losing a fight, so that payoff is 0.

According to the fixed base methods, the base remains the same and unchangeable throughout the series. But as the time passes some items may be added in the series while some may be deleted. It, therefore, becomes tough to compare the result of the current conditions with that of the past period. Thus, in such a situation changing the base period is more appropriate. Chain Index Numbers method is one such method.

Under this method, firstly we express the figures for each year as a percentage of the preceding year. These are known as Link Relatives. We then need to chain them together by successive multiplication to form a chain index.

Thus, unlike fixed base methods, in this method, the base year changes every year. Hence, for the year 2001, it will be 2000, for 2002 it will be 2001, and so on. Let us now study this method step by step.

Steps in the construction of Chain Index Numbers

1. Calculate the link relatives by expressing the figures as the percentage of the preceding year. Thus,

Link Relatives of current year = (price of current year/price of previous year) × 100

2. Calculate the chain index by applying the following formula:

Chain Index = (Current year relative × Previous year link relative) / 100

Advantages of Chain Index Numbers Method

1. This method allows the addition or introduction of the new items in the series and also the deletion of obsolete items.
2. In an organization, management usually compares the current period with the period immediately preceding it rather than any other period in the past. In this method, the base year changes every year and thus it becomes more useful to the management.

Disadvantages of Chain Index Numbers Method

1. Under this method, if the data for any one of the year is not available then we cannot compute the chain index number for the subsequent period. This is so because we need to calculate the link relatives, which are not possible to be calculated in this case.
2. In case an error occurs in the calculation of any of the link relatives, then that error gets compounded and all the subsequent link relatives will also become incorrect. Thus, the entire series will give a misrepresented picture.

Time series datasets can contain a seasonal component.

This is a cycle that repeats over time, such as monthly or yearly. This repeating cycle may obscure the signal that we wish to model when forecasting, and in turn may provide a strong signal to our predictive models.

• The definition of seasonality in time series and the opportunity it provides for forecasting with machine learning methods.
• How to use the difference method to create a seasonally adjusted time series of daily temperature data.
• How to model the seasonal component directly and explicitly subtract it from observations.

Seasonality in Time Series

Time series data may contain seasonal variation.

Seasonal variation, or seasonality, are cycles that repeat regularly over time.

A repeating pattern within each year is known as seasonal variation, although the term is applied more generally to repeating patterns within any fixed period.

Introductory Time Series with R

A cycle structure in a time series may or may not be seasonal. If it consistently repeats at the same frequency, it is seasonal, otherwise it is not seasonal and is called a cycle.

Benefits to Machine Learning

Understanding the seasonal component in time series can improve the performance of modeling with machine learning.

This can happen in two main ways:

• Clearer Signal: Identifying and removing the seasonal component from the time series can result in a clearer relationship between input and output variables.
• More Information: Additional information about the seasonal component of the time series can provide new information to improve model performance.

Both approaches may be useful on a project. Modeling seasonality and removing it from the time series may occur during data cleaning and preparation.

Extracting seasonal information and providing it as input features, either directly or in summary form, may occur during feature extraction and feature engineering activities.

Types of Seasonality

There are many types of seasonality; for example:

• Time of Day.
• Daily.
• Weekly.
• Monthly.
• Yearly.

As such, identifying whether there is a seasonality component in your time series problem is subjective.

The simplest approach to determining if there is an aspect of seasonality is to plot and review your data, perhaps at different scales and with the addition of trend lines.

Removing Seasonality

Once seasonality is identified, it can be modeled.

The model of seasonality can be removed from the time series. This process is called Seasonal Adjustment, or Deseasonalizing.

A time series where the seasonal component has been removed is called seasonal stationary. A time series with a clear seasonal component is referred to as non-stationary.

There are sophisticated methods to study and extract seasonality from time series in the field of Time Series Analysis. As we are primarily interested in predictive modeling and time series forecasting, we are limited to methods that can be developed on historical data and available when making predictions on new data.

In this tutorial, we will look at two methods for making seasonal adjustments on a classical meteorological-type problem of daily temperatures with a strong additive seasonal component. Next, let’s take a look at the dataset we will use in this tutorial.