Statistics and Probability - Introduction - PDF

What is Statistics? Statistic is an applied branch of mathematics that involves collection of data, analysis, interpretation and presentation of data to make decisions. It is a mathematical analysis...

What is Statistics? Statistic is an applied branch of mathematics that involves collection of data, analysis, interpretation and presentation of data to make decisions. It is a mathematical analysis representing quantitative models.  To summarize the What is quantitative meaning, model? It is an quantitative interpretation of models hold inherent data sets using flaws which may mathematical surpass or exceed formulas to identify into decision making, the trends. that could cause a potential mishap. Example of Statistics: The latest sales data have just came in, and your employer wants you to prepare a report on how the company could improve its What should you look business. for? We have to look for relevant information about how much was the previous or latest sales, what are the reasons why sales went down and why we have to do analysis to help sales to go up. What is Probability? Probability is primarily a theoretical branch of mathematics concerning description of how likely an event is to occur. It allows us to use information and data to make intelligent statements and forecast future events. It tells us how often some event will happen after repeated trials. Example: On rolling a dice, you get 6 possible outcomes. Each possibility only has one outcome, so each dice has probability of 1/6 For example, the probability of getting a number 2 on the dice is 1/6 Objectives Statistics: To define and quantify data need to be collected. To analyze the obtained data. To learn how to use statistical data on drawing conclusion. Probability: To define the principal concepts about probability. To express the concept of probability To help us understand which choices are safe and which  Although Statistics and Probability have fundamental differences the way we see and discussed, these two are related areas of So how we can define mathematics which concerns Statistics and Probability? with analyzing the relative frequency of events. And both subjects are important, relevant, and useful to us. What is Statistic and Statistics & Probability?Probability is the branch of mathematics that includes collection of analysis and interpretation of mathematical data use to draw conclusions and decisions by considering the laws governing random events. Importance of Statistics and Probability Importance of Statistics and Probability: 1. Weather Forecasting 2. Research 3. Insurance companies 4. Business and Traders 5. Medical Field 6. Production Companies 1. Weather forecasting On weather forecasting, everybody is concern to know what will be the weather for the next day, and Statistics and Probability helps Weather Forcasters to predict future weather. In safety, oftentimes, we rely on announcements from certain agencies or departments to send out reports regarding information on natural calamities like typhoons. Without records and formulas, forecasters will be unable to predict when the latest storm is coming. We won’t be able to make the necessary precautions when a thunderstorm or strong typhoon strikes. Statistics, in this sense, gives us a heads up so we can prepare better in case of emergencies. 2. Researchers They use their statistical skills to collect relevant data to complete their research works.  Reason 1: Statistics allows researchers to design studies such In the field of that the findings research, statistics from the studies is important for the can be following reasons: extrapolated (estimate or conclude and define) from a larger population. Reason 2: Statistics allows Reason 3: researchers to Statistics allows perform hypothesis researchers to tests to determine create confidence if some claim about intervals to capture a new drug, new uncertainty around procedure, new population manufacturing estimates. method is true. 3. Insurance Companies Insurance companies use statistical models to calculate risk of giving insurance on individual application.  Actuary is a Actuaries use person with mathematics expertise in and statistics to For example, if the fields of estimate Without an auto economics, financial impact Actuaries, insurance statistics and of uncertainty there will be policy holder mathematics, and help clients some risk that goes into debt, minimize risk. who helps in They asses and will cause he may be risk manage the insurance more likely to assessment risks of financial company to file a false and estimation investments, potential claim on his of premiums insurance buncruptcy. vehicle to for an policies and make money. insurance other potentially business. risky ventures. 4. Businessmen and Traders Statistics is the key on how traders and businessmen invest and make money in financial markets.  Statistical analysis is the process of collecting and analyzing data When managers to identify patterns analyze statistical and trends. research in Statistical research business, they in business enables determine how to managers to proceed in areas analyze past including auditing, performance, financial analysis predict future and marketing business research. environments and lead organizations effectively. 5. Medical Field On medical field, scientists use statistical valid rates of effectiveness before they prescribe drugs. Vaccines are tested for long time. Actually it has to take 10 to 15 years to make research and test. In epidemiology, proba bility theory is used to understand the relationship between exposures and the risk of health effects. The methods and tools of biostatistics are extensively used to understand disease development, uncover the etiology (or the causes of a disease), and evaluate the development of new strategies of prevention and control of the disease. 6. Production Companies Production companies makes quality testing using statistical samples to make sure that they sell best quality products. 7. Politics During the election period, they used to make prediction on who will win using statistical data from surveys, and collection of information of each candidates. Statisticians are getting information that fuels political theory, campaign strategy, and policy development. Random - is lack of definite intention without any planning. It can be attained by chance or accidental. Variable - it is an element or value that is not consistent and liable to change. Random Variable - is a variable whose value is dependent to the outcome of a well-defined random event or experiment. Two kinds of variable 1. Qualitative Variable is variable that pertains to the quality. 2. Quantitative Variable is variable that capable of being measured or counted. Two types of variable 1. Discrete Variable 2. Continues Variable  1. Discrete Variable is a quantitative variable whose value can be attained through counting. It can be finite in number of possible values. Example:  Number of chairs in a classroom  Students under STEM strand  Number of people recovered from COVID-19  Number of children in the  family Textbooks required in Statistics  Ballpen in a box  2. Continuous Variable is a variable that can assume an infinity many, unaccountable numbers or real number of values. Example:  Age of a child  Weight of a person  Height of an elf  Frequency of earthquake  Distance travelled by a vehicle  Amount of rain that falls in a storm.  The speed of cars  Time to wake up Activity 1 Identify whether each variable is discrete or continuous: 1. Time required for a vehicle to cover a mile. 2. Number of airplanes in an airport. 3. Population of ants inside a cave 4. Bees in a beehive 5. Volume of water in a pond 6. Exact age of a baby 7. Gray Hair strand of an old woman 8. Altitude of a mountain 9. Dosage of medicine 10. Body Temperature Activity 1: 1. Give the importance of Statistics and Probabilities in your own life and explain in brief. 2. Give 20 examples of Discrete and Continuous Variables. Possible values of Random Variable – are values that are obtained from functions that are assign a real number to each point of a sample space. Sample Space – is the set of all possible outcomes of the events. Probability Distribution Function – is a function P(X) that shows the relative probability that each outcome of an experiment will happen. Illustrative Example: If a basketball team will play for three consecutive games and if W stands for a win and L stands for a loss, the possible sample space of the results of 3 consecutive games is: [WWW, WWL, WLW, LWW, WLL, LWL, LLW, LLL] Note: We can assign numeric values of these outcomes as 1,2,3,4,5,6,7,8. Thus there are 8 possible outcomes, or 8 elements in the sample space as illustrated by a tabulation below:  Random Probability Sample Space Variable (x) Distribution Function [P(X)] WWW 3 1 8 WWL 2 WLW 2 3 8 LWW 2 WLL 1 LWL 1 3 8 LLW 1 LLL 0 1 8 Probability Mass Function – is a probability function of a discrete function of a discrete random variable. Discrete Probability Distribution – is a table of values that shows the probability of any of the outcomes of an experiment. Activity 1: Construct a sample space of tossing a three unbiased coins. Determine the discrete Probability Distribution.  Random Probability Sample Space Variable (x) Distribution Function [P(X)] TTT 0 1 8 TTH 1 THT 1 3 8 HTT 1 THH 2 3 HTH 2 8 HHT 2 HHH 3 1 8 Probability Histogram P(X) 3/8 1/8 0 0 1 1 1 2 2 2 3 (x) Exercise 1: Consider tossing a pair of unbiased coins. Construct its sample space and assign possible values of the sample points or Random Variables (x) and make a Probability Distribution Function P(X) for getting a Head. Sample Random Probability Space Variable (x) Distribution Function [P(X)] 1 4 TT 0 2 HT 1 4 TH 1 1 4 HH 2 Probability Histogram – shows the relative probabilities of the sample points in the form of a bar graph. P(X) 2/4 1/4 0 (x) 0 1 1 2  Mean Variance and Standard Deviation Mean and variance are fundamental concepts in statistics used to describe the characteristics of data sets. 1. Mean (Average) The mean is the average of a set of numbers. It is a measure of central tendency that represents the "typical" value in a data set. Formula: µ= where: n = total numbe of data points = sum of all the values Example: For the data set 1,2,3,4,5: Using the formula µ = µ= µ= µ=3 Mean, Variance and Standard Deviation 1. Mean (Average) µ The mean is the average of a set of numbers. It is a measure of central tendency that represents the "typical" value in a data set. Formula: Mean 2. Variance – is a measure of spread or dispersion. It measures the variation of the values of a random variable from the mean. 𝝈 𝟐 The symbol used for the variance in The variance measures how spread out the values in a data set are around the mean. A larger variance indicates that the data points are more dispersed, while a smaller variance suggests they are closer to the mean. Formula (Population Variance): = 3. Standard Deviation It is the square root of the variance. = Illustrative Example: The report of a weather bureau of the forecast of the number of typhoons entering yhe country’s area of responsibility is manifested in the following probability distribuiton: No. of typhoons per Probability (P(X = month x) 0 0.15 1 0.35 2 0.30 3 0.10 4 0.10 a.What is the average number of typhoon entering the country per month? b.What is the standard deviation? a. Mean ( No. of Probability Mean ( Typhoons 0 0.15 0 ( 0.15) 0.00 1 0.35 1 (0.35) 0.35 2 0.30 2 (0.30) 0.60 3 0.10 3 (0.10) 0.30 4 0.10 4 (0,10) 0.4) ∑ 𝝁=𝟏.𝟔𝟓 Variance) ( No. of Probability (P(X Variance Typhoons = x) 0 0.15 0.4084 1 0.35 0.1479 2 0.30 0.03675 3 0.10 0.18225 4 0.10 0.5452 ∑ 𝝈 =𝟏.𝟑𝟐 𝟐 Standard Deviation: = = 1.15