Creating Frequency Distribution Table (2014)

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Using Descriptive statistics we can organize the data to get the general pattern of the data and check where data values tend to concentrate and try to expose extreme or unusual data values. Let us start learning about the Frequency Distribution Table and its construction.

A frequency distribution is a compact form of data in a table that displays the categories of observations according to their magnitudes and frequencies such that similar or identical numerical values are grouped. The categories are also known as groups, class intervals, or simply classes. The classes must be mutually exclusive classes showing the number of observations in each class. The number of values falling in a particular category is called the frequency of that category denoted by $f$.

A Frequency Distribution Table shows us a summarized grouping of data divided into mutually exclusive classes and the number of occurrences in a class. Frequency distribution is a way of showing raw (ungrouped or unorganized) data into grouped or organized data to show results of sales, production, income, loan, death rates, height, weight, temperature, etc.

The relative frequency of a category is the proportion of observed frequency to the total frequency obtained by dividing observed frequency by the total frequency and denoted by $r.f.$.  The sum of r.f. column should be one except for rounding errors. Multiplying each relative frequency of class by 100 we can get the percentage occurrence of a class. A relative frequency captures the relationship between a class total and the total number of observations.

The Frequency Distribution Table may be made for continuous data, discrete data, and categorical data (for both qualitative and quantitative data). It can also be used to draw some graphs such as histograms, line charts, bar charts, pie charts, frequency polygons, Pareto Charts, Scatter diagrams, stem and leaf displays, etc.

Steps of Creating Frequency Distribution Table

  1. Decide about the number of classes. The number of classes is usually between 5 and 20. Too many classes or too few classes might not reveal the basic shape of the data set, also it will be difficult to interpret such frequency distribution. The maximum number of classes may be determined by the formula:
    \[\text{Number of Classes} = C = 1 + 3.3 log (n)\]
    \[\text{or} \quad C = \sqrt{n} \quad {approximately}\]where $n$ is the total number of observations in the data.
  2. Calculate the range of the data ($Range = Max – Min$) by finding minimum and maximum data values. The range will be used to determine the class interval or class width.
  3. Decide about the width of the class denoted by h and obtained by
    \[h = \frac{\text{Range}}{\text{Number of Classes}}= \frac{R}{C} \]
    Generally, the class interval or class width is the same for all classes. The classes all taken together must cover at least the distance from the lowest value (minimum) in the data set up to the highest (maximum) value. Also note that equal class intervals are preferred in frequency distribution, while unequal class intervals may be necessary in certain situations to avoid a large number of empty, or almost empty classes.
  4. Decide the individual class limits and select a suitable starting point for the first class which is arbitrary, it may be less than or equal to the minimum value. Usually, it is started before the minimum value in such a way that the midpoint (the average of lower and upper-class limits of the first class) is properly placed.
  5. Take an observation and mark a vertical bar (|) for a class it belongs. A running tally is kept till the last observation. The tally counts indicate five.
  6. Find the frequencies, relative frequency,  cumulative frequency, etc. as required.
Frequency Distribution Table
Frequency Distribution Table

A frequency distribution is said to be skewed when its mean and median are different. The kurtosis of a frequency distribution is the concentration of scores at the mean, or how peaked the distribution appears if depicted graphically, for example, in a histogram. If the distribution is more peaked than the normal distribution it is said to be leptokurtic; if less peaked it is said to be platykurtic.

Continuous Frequency Distribution Table

Further Reading: Frequency Distribution Table

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Objectives of Time Series Analysis (2014)

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There are many objectives of time series analysis. The one of major Objectives of Time Series is to identify the underlying structure of the Time Series represented by a sequence of observations by breaking it down into its components (Secular Trend, Seasonal Variation, Cyclical Trend, Irregular Variation).

Objectives of Time Series Analysis

The objectives of Time Series Analysis are classified as follows:

  1. Description
  2. Explanation
  3. Prediction
  4. Control

The description of the objectives of time series analysis is as follows:

Description of Time Series Analysis

The first step in the analysis is to plot the data and obtain simple descriptive measures (such as plotting data, looking for trends,  seasonal fluctuations, and so on) of the main properties of the series. In the above figure, there is a regular seasonal pattern of price change although this price pattern is not consistent. The Graph enables us to look for “wild” observations or outliers (not appear to be consistent with the rest of the data). Graphing the time series makes possible the presence of turning points where the upward trend suddenly changed to a downward trend. If there is a turning point, different models may have to be fitted to the two parts of the series.

Explanation

Observations were taken on two or more variables, making it possible to use the variation in a one-time series to explain the variation in another series. This may lead to a deeper understanding. A multiple regression model may be helpful in this case.

Prediction

Given an observed time series, one may want to predict the future values of the series. It is an important task in sales forecasting and is the analysis of economic and industrial time series. Prediction and forecasting are used interchangeably.

Control

When time series is generated to measure the quality of a manufacturing process (the aim may be) to control the process. Control procedures are of several different kinds. In quality control, the observations are plotted on a control chart and the controller takes action as a result of studying the charts. A stochastic model is fitted to the series. Future values of the series are predicted and then the input process variables are adjusted to keep the process on target.

Objectives of Time Series Analysis seasonal-effects
Image taken from: http://archive.stats.govt.nz

The figure shows that there is a regular seasonal pattern of price change although this price pattern is not consistent.

In quality control, the observations are plotted on the control chart and the controller takes action as a result of studying the charts.

A stochastic model is fitted to the series. Future values of the series are predicted and then the input process variables are adjusted to keep the process on target.

Learn more about Time Series on Wikipedia

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Primary and Secondary Data (2014)

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Data

Before learning about primary and Secondary Data, let us first understand the term Data in Statistics.

The facts and figures which can be numerically measured are studied in statistics. Numerical measures of the same characteristics are known as observation and collection of observations is termed as data. Data are collected by individual research workers or by organizations through sample surveys or experiments, keeping in view the objectives of the study. The data collected may be (i) Primary Data and (ii) Secondary Data.

Primary and Secondary Data in Statistics

The difference between primary and secondary data in Statistics is that Primary data is collected firsthand by a researcher (organization, person, authority, agency or party, etc.) through experiments, surveys, questionnaires, focus groups, conducting interviews, and taking (required) measurements, while the secondary data is readily available (collected by someone else) and is available to the public through publications, journals, and newspapers.

Primary and Secondary Data

Primary Data

Primary data means the raw data (data without fabrication or not tailored data) that has just been collected from the source and has not gone through any kind of statistical treatment like sorting and tabulation. The term primary data may sometimes be used to refer to first-hand information.

Sources of Primary Data

The sources of primary data are primary units such as basic experimental units, individuals, and households. The following methods are used to collect data from primary units usually and these methods depend on the nature of the primary unit. Published data and the data collected in the past are called secondary data.

  • Personal Investigation
    The researcher experiments or surveys himself/herself and collects data from it. The collected data is generally accurate and reliable. This method of collecting primary data is feasible only in the case of small-scale laboratories, field experiments, or pilot surveys and is not practicable for large-scale experiments and surveys because it takes too much time.
  • Through Investigators
    The trained (experienced) investigators are employed to collect the required data. In the case of surveys, they contact the individuals and fill in the questionnaires after asking for the required information, whereas a questionnaire is an inquiry form having many questions designed to obtain information from the respondents. This method of collecting data is usually employed by most organizations and it gives reasonably accurate information but it is very costly and may be time-consuming too.
  • Through Questionnaire
    The required information (data) is obtained by sending a questionnaire (printed or soft form) to the selected individuals (respondents) (by mail) who fill in the questionnaire and return it to the investigator. This method is relatively cheap as compared to the “through investigator” method but the non-response rate is very high as most of the respondents don’t bother to fill in the questionnaire and send it back to the investigator.
  • Through Local Sources
    The local representatives or agents are asked to send requisite information and provide the information based on their own experience. This method is quick but it gives rough estimates only.
  • Through Telephone
    The information may be obtained by contacting the individuals by telephone. It is Quick and provides the accurate required information.
  • Through Internet
    With the introduction of information technology, people may be contacted through the Internet and individuals may be asked to provide pertinent information. Google Survey is widely used as an online method for data collection nowadays. There are many paid online survey services too.

It is important to go through the primary data and locate any inconsistent observations before it is given a statistical treatment.

Secondary Data

Data that has already been collected by someone, may be sorted, tabulated, and has undergone a statistical treatment. It is fabricated or tailored data.

Sources of Secondary Data

The secondary data may be available from the following sources:

  • Government Organizations
    Federal and Provincial Bureau of Statistics, Crop Reporting Service-Agriculture Department, Census and Registration Organization etc.
  • Semi-Government Organization
    Municipal committees, District Councils, Commercial and Financial Institutions like banks etc
  • Teaching and Research Organizations
  • Research Journals and Newspapers
  • Internet

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Markov Chain an Introduction (2014)

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A Markov chain, named after Andrey Markov is a mathematical system that experiences transitions from one state to another, between a finite or countable number of possible states. Markov chain is a random process usually characterized as memoryless: the next state depends only on the current state and not on the sequence of events that preceded it. This specific kind of memorylessness is called the Markov property. Markov chains have many applications as statistical models of real-world processes.

If the random variables $X_{n-1}$ and $X_n$ take the values $X_{n-1}=i$ and $X_n=j$, then the system has made a transition $S_i \rightarrow S_j$, that is, a transition from state $S_i$ to state $S_j$ at the $n$th trial. Note that $i$ can equal $j$, so transitions within the same state may be possible. We need to assign probabilities to the transitions $S_i \rightarrow S_j$. Generally in the chain, the probability that $X_n=j$ will depend on the whole sequence of random variables starting with the initial value $X_0$.

The Markov chain has the characteristic property that the probability that $X_n=j$ depends only on the immediately previous state of the system. This means that we need no further information at each step other than for each $i$ and $j$,  \[P\{X_n=j|X_{n-1}=i\}\]
which means the probability that $X_n=j$ given that $X_{n-1}=i$: this probability is independent of the values of $X_{n-2},X_{n-3},\cdots, X_0$.

Let us have a set of states $S=\{s_1,s_2,\cdots,s_n\}$. The process starts in one of these states and moves successively from one state to another. Each move is called a step. If the chain is currently in state $s_i$ then it moves to state $s_j$ at the next step with a probability denoted by $p_{ij}$ (transition probability) and this probability does not depend upon which states the chain was in before the current state. The probabilities $p_{ij}$ are called transition probabilities ($s_i  \xrightarrow[]{p_{ij}} s_j$ ). The process can remain in its state, and this occurs in probability $p_{ii}$.

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An initial probability distribution, defined on $S$ specifies the starting state. Usually, this is done by specifying a particular state as the starting state.

A Markov chain is a sequence of random variables $X_1, X_2,\cdots,$ with the Markov property that, given the present state, the future and past state are independent. Thus
\[P(X_n=x|X_1=x_1,X_2=x_2\cdots X_{n-1}=x_{n-1})\]
\[\quad=P(X_n=x|X_{n-1}=x_{n-1})\]
Or
\[P(X_n=j|X_{n-1}=i)\]

Example: Markov Chain

A Markov chain $X$ on $S=\{0,1\}$ is determined by the initial distribution given by $p_0=P(X_0=0), \; p_1=P(X_0=1)$ and the one-step transition probability given by $p_{00}=P(x_{n+1}=0|X_n=0)$, $p_{10}=P(x_{n+1}=0|X_n=1)$, $p_{01}=1-p_{00}$ and $p_{11}=1-p_{10}$, so one-step transition probability in matrix form is $P=\begin{pmatrix}p_{00}&p_{10}\\p_{01}&p_{11}\end{pmatrix}$

Markov Chain

Markov chains are a powerful tool for modeling various random processes. However, it’s important to remember that they assume the Markov property, which may not always hold true in real-world scenarios.

Applications of Markov Chains

  • Information Theory: Used in data compression algorithms like Huffman coding.
  • Search Algorithms: Applied in recommender systems and website navigation analysis.
  • Queueing Theory: Helps model customer arrivals and service times in queues.
  • Financial Modeling: Financial analysts can use Markov chains to model stock prices or economic trends.
  • Game Design: Markov chains can be used to create video games with more realistic and interesting behavior for non-player characters.
  • Predictive Text: Smartphones that suggest the next word you are typing use a kind of Markov chain, where the probability of the next word depends on the current word.
  • Modeling weather: Markov chains can be used to represent the probabilities of transitioning between different weather states.
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References

  • https://en.wikipedia.org/wiki/Markov_chain
  • http://people.virginia.edu/~rlc9s/sys6005/SYS_6005_Intro_to_MC.pdf
  • http://www.dartmouth.edu/~chance/teaching_aids/books_articles/probability_book/Chapter11.pdf

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