Tag: Time Series Analysis

Arithmetic Mean

The arithmetic mean,’ mean or average is calculated by summ­ing all the individual observations or items of a sample and divid­ing this sum by the number of items in the sample. For example, as the result of a gas analysis in a respirometer an investigator obtains the following four readings of oxygen percentages:

He calculates the mean oxygen percentage as the sum of the four items divided by the number of items here, by four. Thus, the average oxygen percentage is

Mean = 61.3 / 4 =15.325%

Calculating a mean presents us with the opportunity for learning statistical symbolism. An individual observation is symbo­lized by Y_i, which stands for the ith observation in the sample. Four observations could be written symbolically as Yi, Y₂, Y₃, Y₄.

We shall define n, the sample size, as the number of items in a sample. In this particular instance, the sample size n is 4. Thus, in a large sample, we can symbolize the array from the first to the nth item as follows: Y₁, Y₂…, Y_n. When we wish to sum items, we use the following notation:

The capital Greek sigma, Ʃ, simply means the sum of items indica­ted. The i = 1 means that the items should be summed, starting with the first one, and ending with the nth one as indicated by the i = n above the Ʃ. The subscript and superscript are necessary to indicate how many items should be summed. Below are seen increasing simplifications of the complete notation shown at the extreme left:

Properties of Arithmetic Mean:

1. The sum of deviations of the items from the arithmetic mean is always zero i.e.

∑(X–X) =0.

2. The Sum of the squared deviations of the items from A.M. is minimum, which is less than the sum of the squared deviations of the items from any other values.
3. If each item in the series is replaced by the mean, then the sum of these substitutions will be equal to the sum of the individual items.                       

Merits of A.M:

1. It is simple to understand and easy to calculate.
2. It is affected by the value of every item in the series.
3. It is rigidly defined.
4. It is capable of further algebraic treatment.
5. It is calculated value and not based on the position in the series.

Demerits of A.M:

1. It is affected by extreme items i.e., very small and very large items.
2. It can hardly be located by inspection.
3. In some cases A.M. does not represent the actual item. For example, average patients admitted in a hospital is 10.7 per day.
4. M. is not suitable in extremely asymmetrical distributions.

Weighted Mean

In some cases, you might want a number to have more weight. In that case, you’ll want to find the weighted mean. To find the weighted mean:

1. Multiply the numbers in your data set by the weights.
2. Add the results up.

For that set of number above with equal weights (1/5 for each number), the math to find the weighted mean would be:
1(*1/5) + 3(*1/5) + 5(*1/5) + 7(*1/5) + 10(*1/5) = 5.2.

Sample problem: You take three 100-point exams in your statistics class and score 80, 80 and 95. The last exam is much easier than the first two, so your professor has given it less weight. The weights for the three exams are:

• Exam 1: 40 % of your grade. (Note: 40% as a decimal is .4.)
• Exam 2: 40 % of your grade.
• Exam 3: 20 % of your grade.

What is your final weighted average for the class?

1. Multiply the numbers in your data set by the weights:

   .4(80) = 32

   .4(80) = 32

   .2(95) = 19

2. Add the numbers up. 32 + 32 + 19 = 83.

The percent weight given to each exam is called a weighting factor.

Weighted Mean Formula

The weighted mean is relatively easy to find. But in some cases the weights might not add up to 1. In those cases, you’ll need to use the weighted mean formula. The only difference between the formula and the steps above is that you divide by the sum of all the weights.

The image above is the technical formula for the weighted mean. In simple terms, the formula can be written as:

Weighted mean = Σwx / Σw

Σ = the sum of (in other words…add them up!).
w = the weights.
x = the value.

To use the formula:

1. Multiply the numbers in your data set by the weights.
2. Add the numbers in Step 1 up. Set this number aside for a moment.
3. Add up all of the weights.
4. Divide the numbers you found in Step 2 by the number you found in Step 3.

In the sample grades problem above, all of the weights add up to 1 (.4 + .4 + .2) so you would divide your answer (83) by 1:
83 / 1 = 83.

However, let’s say your weighted means added up to 1.2 instead of 1. You’d divide 83 by 1.2 to get:
83 / 1.2 = 69.17.

Combined Mean

A combined mean is a mean of two or more separate groups, and is found by:

1. Calculating the mean of each group,
2. Combining the results.

Combined Mean Formula

More formally, a combined mean for two sets can be calculated by the formula :

Where:

• x_a = the mean of the first set,
• m = the number of items in the first set,
• x_b = the mean of the second set,
• n = the number of items in the second set,
• x_c the combined mean.

A combined mean is simply a weighted mean, where the weights are the size of each group.

Bayes’ Theorem is a way to figure out conditional probability. Conditional probability is the probability of an event happening, given that it has some relationship to one or more other events. For example, your probability of getting a parking space is connected to the time of day you park, where you park, and what conventions are going on at any time. Bayes’ theorem is slightly more nuanced. In a nutshell, it gives you the actual probability of an event given information about tests.

“Events” Are different from “tests.” For example, there is a test for liver disease, but that’s separate from the event of actually having liver disease.

Tests are flawed:

Just because you have a positive test does not mean you actually have the disease. Many tests have a high false positive rate. Rare events tend to have higher false positive rates than more common events. We’re not just talking about medical tests here. For example, spam filtering can have high false positive rates. Bayes’ theorem takes the test results and calculates your real probability that the test has identified the event.

Bayes’ Theorem (also known as Bayes’ rule) is a deceptively simple formula used to calculate conditional probability. The Theorem was named after English mathematician Thomas Bayes (1701-1761). The formal definition for the rule is:

In most cases, you can’t just plug numbers into an equation; You have to figure out what your “tests” and “events” are first. For two events, A and B, Bayes’ theorem allows you to figure out p(A|B) (the probability that event A happened, given that test B was positive) from p(B|A) (the probability that test B happened, given that event A happened). It can be a little tricky to wrap your head around as technically you’re working backwards; you may have to switch your tests and events around, which can get confusing. An example should clarify what I mean by “switch the tests and events around.”

Bayes’ Theorem Example

You might be interested in finding out a patient’s probability of having liver disease if they are an alcoholic. “Being an alcoholic” is the test (kind of like a litmus test) for liver disease.

A could mean the event “Patient has liver disease.” Past data tells you that 10% of patients entering your clinic have liver disease. P(A) = 0.10.

B could mean the litmus test that “Patient is an alcoholic.” Five percent of the clinic’s patients are alcoholics. P(B) = 0.05.

You might also know that among those patients diagnosed with liver disease, 7% are alcoholics. This is your B|A: the probability that a patient is alcoholic, given that they have liver disease, is 7%.

Bayes’ theorem tells you:

P(A|B) = (0.07 * 0.1)/0.05 = 0.14

In other words, if the patient is an alcoholic, their chances of having liver disease is 0.14 (14%). This is a large increase from the 10% suggested by past data. But it’s still unlikely that any particular patient has liver disease.

Conditional Probability refers to the probability of an event occurring given that another event has already occurred. It measures how the occurrence of one event affects the likelihood of another event. In many real-life situations, events are not independent, and the probability of one event depends on the outcome of another. Conditional probability helps analyze such relationships and provides a more accurate understanding of uncertain situations.

This concept is widely used in business, economics, finance, insurance, medicine, and statistics. It helps organizations make informed decisions by considering available information and understanding how different events are connected.

Definition

Conditional Probability is the probability of an event occurring under the condition that another related event has already taken place.

The probability of the occurrence of an event A given that an event B has already occurred is called the conditional probability of A given B:

The same is explained in Figure 2.15 using the sample spaces related to the events A and B, assuming that there are few sample points common to these two events. Part 1 of the figure shows the total sample space related to the experiment as in the form of rectangle and the sample space related to the event A as a circle. Similarly part 2 of the figure shows the total sample space and the sample space related to event B. As explained earlier in conditional probability the total sample space is restrained to the sample space that is related to event B (which has already occurred). The same is shown in part 3 of Figure 2.15. Now the sample space for event A (B is the total sample space available) is nothing but the sample points related to event A and falling in the sample space. This is nothing but the intersection of the events A and B and is shown in part 3 of the figure as the hatched area.  

Figure 2.15: Representation of conditional probability using the Venn diagrams

For example, there are 100 trips per day between two places X and Y. Out of these 100 trips 50 are made by car, 25 are made by bus and the other 25 are by local train. Probabilities associated to these modes are 0.5, 0.25, and 0.25, respectively. In transportation engineering both the bus and the local train are considered as public transport so the event space associated to this is the summation of the event spaces associated to bus and local train. Probability of choosing public transportation is 0.5. Now if one is interested in finding the probability of choosing bus given public transportation is chosen the conditional probability is useful in finding that.

Characteristics of Conditional Probability

• Depends on the Occurrence of Another Event

A key characteristic of conditional probability is that it depends on the occurrence of another event. Unlike simple probability, which measures the likelihood of an event independently, conditional probability considers additional information. The probability of an event changes when another related event has already occurred. For example, the probability of a customer purchasing a printer may increase if the customer has already purchased a laptop. This dependency makes conditional probability highly useful in analyzing real-world situations where events are interconnected and influence one another.

• Measures Relationships Between Events

Conditional probability helps measure and understand the relationship between two or more events. It shows how the occurrence of one event affects the likelihood of another event occurring. By analyzing these relationships, businesses and researchers can identify patterns and dependencies within data. For example, a retailer may study whether customers who buy one product are more likely to buy another. This characteristic makes conditional probability valuable in market research, risk assessment, and forecasting. It provides insights into event interactions that simple probability cannot capture effectively.

• Based on Joint Probability

Another important characteristic is that conditional probability relies on joint probability. To calculate conditional probability, the probability of both events occurring together must be known. Joint probability provides the foundation for determining how likely one event is when another has already occurred. This relationship ensures that conditional probability is mathematically consistent and accurate. By using joint probability, analysts can examine event dependencies in a systematic manner. This characteristic highlights the close connection between different probability concepts and their role in statistical analysis.

• Applicable to Dependent Events

Conditional probability is particularly useful when dealing with dependent events. Dependent events are events where the occurrence of one influences the probability of another. In many business and real-world situations, events are not independent. For example, customer purchasing decisions may depend on previous purchases or promotional offers. Conditional probability helps quantify these dependencies and provides more realistic probability estimates. This characteristic makes it an essential tool for understanding situations where outcomes are interconnected and cannot be analyzed accurately using independent probabilities alone.

• Provides Updated Probability Estimates

Conditional probability allows probabilities to be updated when new information becomes available. Instead of relying solely on initial estimates, it incorporates additional data to produce revised probability values. This characteristic is especially important in dynamic environments where circumstances change over time. For example, a bank may reassess the probability of loan repayment after receiving updated information about a customer’s financial status. By adjusting probabilities based on current information, conditional probability improves the accuracy and relevance of decision-making and forecasting processes.

• Supports Better Decision-Making

A significant characteristic of conditional probability is its ability to support informed decision-making. By considering specific conditions and relevant information, it provides more accurate estimates of future outcomes. Managers, investors, and policymakers use conditional probability to evaluate alternatives and assess risks. For example, a business may determine the likelihood of achieving sales targets under certain market conditions. This information enables decision-makers to choose strategies that maximize opportunities and minimize risks. Consequently, conditional probability plays an important role in effective planning and management.

• Forms the Foundation of Advanced Statistical Methods

Conditional probability serves as the basis for many advanced statistical and analytical techniques. Concepts such as Bayes’ Theorem, predictive modeling, machine learning, and statistical inference all rely on conditional probability principles. By understanding how probabilities change under specific conditions, analysts can develop sophisticated models for forecasting and decision support. This characteristic demonstrates the importance of conditional probability in both theoretical and applied statistics. Its role as a foundational concept makes it essential for advanced research and data analysis across numerous disciplines.

• Widely Applicable in Real-Life Situations

Conditional probability has broad applicability in business, finance, insurance, healthcare, engineering, and many other fields. Real-world events are often dependent on specific conditions, making conditional probability highly relevant. Businesses use it to analyze customer behavior, assess risks, and forecast demand. Insurance companies use it to estimate claim probabilities based on customer profiles. Financial institutions apply it in credit risk analysis and investment decisions. This widespread applicability demonstrates its practical value and importance. As a result, conditional probability is one of the most widely used concepts in probability and statistics.

Applications of Conditional Probability in Business

Regression Line is the line that best fits the data, such that the overall distance from the line to the points (variable values) plotted on a graph is the smallest. In other words, a line used to minimize the squared deviations of predictions is called as the regression line.

There are as many numbers of regression lines as variables. Suppose we take two variables, say X and Y, then there will be two regression lines:

• Regression line of Y on X: This gives the most probable values of Y from the given values of X.
• Regression line of X on Y: This gives the most probable values of X from the given values of Y.

The algebraic expression of these regression lines is called as Regression Equations. There will be two regression equations for the two regression lines.

The correlation between the variables depend on the distance between these two regression lines, such as the nearer the regression lines to each other the higher is the degree of correlation, and the farther the regression lines to each other the lesser is the degree of correlation.

The correlation is said to be either perfect positive or perfect negative when the two regression lines coincide, i.e. only one line exists. In case, the variables are independent; then the correlation will be zero, and the lines of regression will be at right angles, i.e. parallel to the X axis and Y axis.

The regression lines cut each other at the point of average of X and Y. This means, from the point where the lines intersect each other the perpendicular is drawn on the X axis we will get the mean value of X. Similarly, if the horizontal line is drawn on the Y axis we will get the mean value of Y.

Co-efficient of Regression

The Regression Coefficient is the constant ‘b’ in the regression equation that tells about the change in the value of dependent variable corresponding to the unit change in the independent variable.

If there are two regression equations, then there will be two regression coefficients:

• Regression Coefficient of X on Y:

The regression coefficient of X on Y is represented by the symbol b_xy that measures the change in X for the unit change in Y. Symbolically, it can be represented as:

The b_xy can be obtained by using the following formula when the deviations are taken from the actual means of X and Y:When the deviations are obtained from the assumed mean, the following formula is used:

• Regression Coefficient of Y on X:

The symbol b_yx is used that measures the change in Y corresponding to the unit change in X. Symbolically, it can be represented as:

In case, the deviations are taken from the actual means; the following formula is used:
The _byx can be  calculated by using the following formula when the deviations are taken from the assumed means:

The Regression Coefficient is also called as a slope coefficient because it determines the slope of the line i.e. the change in the independent variable for the unit change in the independent variable

Scatter Diagram Method is the simplest method to study the correlation between two variables wherein the values for each pair of a variable is plotted on a graph in the form of dots thereby obtaining as many points as the number of observations. Then by looking at the scatter of several points, the degree of correlation is ascertained.

The degree to which the variables are related to each other depends on the manner in which the points are scattered over the chart. The more the points plotted are scattered over the chart, the lesser is the degree of correlation between the variables. The more the points plotted are closer to the line, the higher is the degree of correlation. The degree of correlation is denoted by “r”.

The following types of scatter diagrams tell about the degree of correlation between variable X and variable Y.

1. Perfect Positive Correlation (r = +1):

The correlation is said to be perfectly positive when all the points lie on the straight line rising from the lower left-hand corner to the upper right-hand corner.

2. Perfect Negative Correlation (r = -1):

When all the points lie on a straight line falling from the upper left-hand corner to the lower right-hand corner, the variables are said to be negatively correlated.

3. High Degree of +Ve Correlation (r = + High):

The degree of correlation is high when the points plotted fall under the narrow band and is said to be positive when these show the rising tendency from the lower left-hand corner to the upper right-hand corner.

4. High Degree of –Ve Correlation (r = – High):

The degree of negative correlation is high when the point plotted fall in the narrow band and show the declining tendency from the upper left-hand corner to the lower right-hand corner.

5. Low degree of +Ve Correlation (r = + Low):

The correlation between the variables is said to be low but positive when the points are highly scattered over the graph and show a rising tendency from the lower left-hand corner to the upper right-hand corner.

6. Low Degree of –Ve Correlation (r = + Low):

The degree of correlation is low and negative when the points are scattered over the graph and the show the falling tendency from the upper left-hand corner to the lower right-hand corner.

7. No Correlation (r = 0):

The variable is said to be unrelated when the points are haphazardly scattered over the graph and do not show any specific pattern. Here the correlation is absent and hence r = 0.

Thus, the scatter diagram method is the simplest device to study the degree of relationship between the variables by plotting the dots for each pair of variable values given. The chart on which the dots are plotted is also called as a Dotogram.