Module 1. Intro to Stat Analysis PDF

An Introduction to Statistical Analysis Ariel F. Melad Faculty, Cagayan State University Revised: August 2024  Self – assessment Survey 1. What did you know about...

An Introduction to Statistical Analysis Ariel F. Melad Faculty, Cagayan State University Revised: August 2024  Self – assessment Survey 1. What did you know about STATISTICS? 2. What do you want to knowFeel in free to comment your STATISTICS? answer on the picture.  Objectives At the end of this module, students are expected to: define statistics and statistical analysis identify different types of variables employ and compare the different methods in collecting data collect statistical data both quantitative and qualitative classify data according to level of measurement develop tables and charts for categorical and numerical data identify appropriate sampling method to use in a given population apply statistics in business and economics apply the statistical analysis process  Introduction to Statistical Analysis TOPICS: 1.1 Definitions of Statistics and Statistical Analysis 1.2 Why study Statistics? 1.3 Major Areas of Statistics 1.4 Types of Data 1.5 Measurement Scales 1.6 Collection of Data 1.7 Organization and Presentation of Data 1.8 The Statistical Analysis Statistics???  Introduction to Statistical Analysis 1.1 Definitions of Statistics Statistics is a science that deals with the collection, organization, presentation, analysis and interpretation of data to make decisions. Plural form: set of numerical data  Introduction to Statistical Analysis 1.2 Definition of Statistical Analysis Statistical Analysis is a science concerned with the organization and interpretation of data according to well- defined, systematic, and mathematical procedures and rules.  Introduction to Statistical Analysis Major characteristics of statistics: Statistics are the aggregates of facts. It means a single figure is not statistics. For example, national income of a country for a single year is not statistics but the same for two or more years is statistics. Statistics are affected by a number of factors. For example, sale of a product depends on a number of factors such as its price, quality, competition, the income of the consumers, and so on.  Introduction to Statistical Analysis Statistics must be reasonably accurate. Wrong figures, if analyzed, will lead to erroneous conclusions. Hence, it is necessary that conclusions must be based on accurate figures. Statistics must be collected systematically. If data are collected haphazardly, they will not be reliable and will lead to misleading conclusions. Lastly, statistics should be placed concerning each other. If one collects data unrelated to each other, then such data will be confusing and will not lead to any logical conclusions. Data should be comparable over time and over space.  1.2 Why Study Statistics? General Uses of Statistics aids in decision making ✓provides comparison ✓explains action that have taken place ✓justifies a claim or assertion ✓predicts future outcome ✓estimates unknown quantities summarizes data for public use  1.2 Why Study Statistics? Data give us information that we need. They serve as our basis for planning and decision making. In business, decision makers use statistics to: Present and describe business data and information properly. Draw conclusions about large groups of individuals or items, using information collected from subsets of the individuals or items. Make reliable forecasts about a business activity. Improve business processes.  Introduction to Statistical Analysis Why do we collect data? A marketing research analyst needs to assess the effectiveness of a new television advertisement. A pharmaceutical manufacturer needs to determine whether a new drug is more effective than those currently in use. An operations manager wants to monitor a manufacturing process to find out whether the quality of the product being manufactured is conforming to company standards. An auditor wants to review the financial transactions of a company in order to determine whether the company is in compliance with generally accepted accounting principles.  Two Major Areas of Statistics Descriptive Statistics comprises on methods concerned with collecting, summarizing, and describing a set of data as to yield meaningful information. Inferential Statistics comprises on methods concerned with analysis of subset of data leading to predictions or inferences or broad generalization about the entire set of data.  Introduction to Statistical Analysis Examples: Descriptive Statistics Inferential Statistics A housewife wants to determine A housewife would like to predict, the average monthly amount she based on here last year’s grocery spent on groceries in the past 6 bill, the average weekly amount months. she will spent on groceries for this year. A politician wants to know the A politician would like to estimate, exact percentages of votes cast based on an opinion poll, his for him in the last election. chance for reelection in the upcoming election. A bowler wants to find his A bowler wants to estimate his bowling average for the past 6 chance of winning a game based games. on his performance and his opponents.  Introduction to Statistical Analysis Two players in our team have been AFK during our last 2 games. Two players in our team have been practicing their mage heroes for the upcoming season. Maya Loan releases hit 47 billion pesos to a million borrowers. [Philippine star,2024] Hence, most Filipinos do not have money.  Introduction to Statistical Analysis Population vs. Sample Population a collection of all the elements under consideration in a statistical inquiry Sample a part (or subset) of the population from which data are collected Example: A manufacturer of kerosene heaters wants to determine if customers are satisfied with the performance of their heaters. Toward this goal, 5,000 of his 200,000 customers are contacted and each is asked “Are you satisfied with the performance of the kerosene heater you purchased?” Identify the population and sample.  Introduction to Statistical Analysis Example 1. A candidate for political office hires a polling firm to assess his chances in the upcoming election. The population consists of all voters in the candidate’s district. The sample consists of all voters in the candidate’s district interviewed by the polling firm. Example 2. The researcher would like to determine the number of Small and Medium Enterprises (SMEs) in Region 2. The population consists of all SMEs in the region while sample consist of all SMEs particularly in Cagayan only.  Introduction to Statistical Analysis Slovin’s Formula: Where, n = sample size N= population size e = desired margin of error  Introduction to Statistical Analysis Example: An entrepreneur will conduct a survey to find out the acceptability of the new product in the market. If there are 30,000 residents in the district and they want 5% margin of error, how many people living there should be respondents in the survey? Solution:  Definition of Terms Parameter vs. Statistic Parameter a numerical characteristic of the population Statistic a numerical characteristic of the sample Example: In order to estimate the true proportion of employees at a certain office who smoke cigarettes, the personnel department polled a sample of 200 employees and determined that the proportion of employees from the sample who smoke cigarettes is 0.12. What are the parameter and statistic in this situation?  Definition of terms Variable or data refers to the characteristics or property whereby members of the group or set vary from another. A characteristic or information of interest that is observable or measurable from every individual or object under consideration. Example: Gender, Age, Occupation, Profit of the organization, expenditure, number of customers, employee’s ID number, etc.  1.4 Types of Data Data Categorical Scale Nominal Ordinal Discrete Continuous  Types of Data 1. Qualitative or Categorical (non-numeric) measures a quality or characteristics. Examples: Employment status, Marital Status, Political Party, Eye Color, etc. 2. Quantitative or Scale (numeric) measures a numerical quantity or amount. It answer the question “how much” or “how many”. Examples: family income, expenditures, profit, number of customers, height, etc. Types of Quantitative Data Discrete data assumes only a finite or countable number of values. Example: number of customer in the selling area, Class size, number of household, number of sibling, etc. Continuous data assumes infinitely many values corresponding to the points on a line interval or whose set of values is uncountable. Examples: profit of the company, family income, height, weight etc.  1.5 Measurement Scales Measurement is the process of determining the value or label of the variable of interest based on what have been observed. 1. Nominal Scale used with variables that are qualitative in nature. The data collected are simply labels, categories or nameless without any implicit or explicit ordering of the categories or explicit ordering of the labels. It is the lowest level of measurement.  1.5 Measurement Scales Examples: Sex /Gender The complexion of students Hair Color of students Marital status  Measurement Scales 2. Ordinal scale has a relative low level of property of magnitude, but it does not have the property of equal intervals between the adjacent units. This is concerned with the ranking or order of the objects measured. The level of measurement is higher than nominal. Examples: Winners of a contest Faculty Rank Military Rank Student class designation Student grades Product satisfaction  Measurement Scales 3. Interval scale has its property of magnitude and equal interval between two adjacent units, but it does not have an absolute zero point, that is the number zero is arbitrarily assigned and does not mean the absence of the characteristic under consideration. The level of measurement is higher than the ordinal. Examples: Temperature in Celsius scale IQ of zero does not mean total absence of knowledge. Military time 4. Ratio scale is the highest level of measurement scale. It has all the properties of an interval scale, that is, it has magnitude and equal intervals plus the absolute zero point. Furthermore, the number zero indicates the absence of the characteristic under consideration. Examples: The reaction time to a particular drug The number of visits to a Doctor, The weight loss of on diet individual, The average score of CAT score of CAS students  Definitions Dependent vs. Independent variable Dependent variable is sometimes called criterion variables while the Independent variable is sometimes called predictor variables or a variable that can be controlled or manipulated. independent variable - presume cause dependent variable - presume effect.  Independent vs. Dependent Example 1. Suppose the investigator is interested in the relationships between two variables: the effect of information about the gender of a job applicant on hiring decisions made by personnel managers. The hiring decision is the dependent variable because it is thought to depend on the information about the gender of the applicant, while the gender of the applicant is independent because it is assumed to influence the dependent variable and does not “depend” on the other variable.  Independent vs. Dependent Example 2. “Study on the effect of Psychological stress on blood pressure” the Independent variable is the amount Psychological stress an individual feeling and the dependent variable is the individual’s blood pressure.  Independent vs. Dependent The effect of online advertising on sales Lung cancer due to smoking cigarettes The size of the store and amount of expenditure 1.6 Collection of Data 2 types of data Primary data refers to the information gathered directly from an original source or based on the direct or firsthand experience secondary data refers to the information which is previously gathered by previous individual.  1.6 Collection of Data Methods of Collecting Data 1. Interview Method is a person-to-person exchange between the interviewee and the interviewer. It provides consistent and more precise information. Methods of Collecting Data 2 types: Direct Method. The researcher personally interviews the respondent. Basically the researcher visits the interviewee for a specific time available or scheduled time by the interviewee. A focus group discussion (FGD) can also be used to interview a group of respondents. The method is appropriate if the information is only minimal and less number of samples.  Methods of Collecting Data Indirect Method. The researcher uses telephone or any device to interview the respondents. This method is quite expensive especially if there so many respondents. This method is considered biased because people with no telephone cannot have the chance to be included as sample in the study. Disadvantages: Time consuming, Expensive and has limited coverage.  Methods of Collecting Data 2. Questionnaire Method is a written response given to prepared questions. Questionnaire is a list of well-planned questions to illicit answers to the problems of the study.  Questionnaire Method Multiple choice. Questions that contain several alternatives. Dichotomous. Questions having two afterlives available. Multiple choice and dichotomous are close-ended questions. Open-ended. Questions where respondent are free to formulate his own answer and expand on the subject of the question.  Questionnaire Method Pitfalls encountered in constructing survey questionnaire: 1.incorrect ordering of questions 2.double-barreled questions 3.sensitive or threatening questions 4.unrealistic questions 5.incomplete or non-exhaustive listing 6.biased wording  Examples of Pitfalls in Question Construction Incorrect ordering of questions Q7. How would you rate the scent of lotion X? Q8. What do you think of lotion X? Improvement: ________________________________________ ________________________________________ Double-barreled questions “Does your department have a special recruitment policy for ethnic minorities and women?” Improvement: ____________________________________ _____________________________________  Sensitive or Threatening Questions “Do you steal office supplies? Yes _____ No _____ Improvement: ____________________________________________ ____________________________________________ Unrealistic questions “What brand of perfume do you think you will be using three years from now?” Improvement: ____________________________________________ ____________________________________________ Incomplete/non-exhaustive listing “Did you learn about the brand from TV, radio, newspaper, or friends?” Improvement: ____________________________________________ ____________________________________________  Biased Wording “Do you think that decent, low-cost funerals are sensible? Improvement: ____________________________________________ ____________________________________________ Leading questions “The majority of physicians in the Philippines feel that smoking is harmful; do you agree?” Improvement: ________________________________________________ ______________________________________________________  Methods of Collecting Data 3. Registration Method is a method of gathering information or data is enforced by the law. The examples of data gathered using this method are data obtained from National Statistics Office, DepEd, CHED, SEC, Supreme Court and many government agencies.  Methods of Collecting Data 4. Observation Method where the investigator observes the behavior of the subject and their outcomes. The subject usually cannot talk or write and it requires a proper recording of the behavior at the appropriate time and situation. It is commonly used in psychological and anthropological studies.  Methods of Collecting Data 5. Experiment method is a method of gathering data when the objective is to determine the cause and effect relationship of a phenomenon under controlled conditions. Key variables: (1) independent variable - treatment (2) dependent variable - measurement  Other variable: Extraneous variables are not part of the experiment but can influence the result.  Sampling Techniques 1. Random sampling techniques is a process of selecting samples in such a way that all individuals in the defined population have an equal chance of being selected as sample, the process being called is randomization 2. Non Random Sampling is a process of selecting samples wherein not all members are given equal chances to be chosen as a sample  Random Sampling a. Simple Random Sampling (SRS) Lottery Sampling. It is the most common and the easiest method of random sampling where a group of sample is selected from a population. The names of respondents are written in a pieces of papers then rolled and place in bowl. The respondents who are included in the study are those names written on the piece of paper picked at random from the bowl. Use of table of random numbers. The samples can be selected using table of random numbers or using random numbers from a scientific calculator.  Random Sampling b. Stratified Sampling is a process of selecting sample in such a way that identified sub-groups or strata in the population are presented in the sample in the same proportion that they exist in the population. Population must be first divided into homogeneity to avoid the possibility of drawing samples whose member come only from one stratum. It is often called stratified proportional sampling.  Random Sampling Steps: Identify and define the population. Determine the sample size using Slovin’s Formula. Identify the sub-groups (or strata) Classify all members of the population as members of the identified strata. Randomly select samples.  Random Sampling Illustration: Consider the data on Population of Barangay Malaya according to socio-economic status. Sample Strata Population Percentage Computation size, n High income 500 0.05 385*.05 19 Middle Income 2,500 0.25 385*.25 96 Low Income 7,000 0.7 385*.7 270 TOTAL 10,000 385  Random Sampling c. Cluster Sampling sometimes referred to as an area sampling because it is frequently applied on geographical basis. On this basis, districts or blocks of a municipality or city are selected. The sample clusters may be chosen by SRS or by systematic sampling. These districts or blocks constitute the cluster. The number of clusters C in the population is called the size of the population of clusters. Clusters may be of equal or unequal sizes.  Random Sampling Example 1. A community has a lower, middle and upper income residents living side by side, we may use this community as a source of a sample to study the different socio-economic status. By concentrating on this particular area, we can save time, effort and money.  Random Sampling Example 2. A forester wishes to estimate the average height of coconut trees on a plantation. The plantation is divided into 386 plots. An SRS of 20 is selected and all trees on the sampled plots are measured.  Random Sampling d. Systematic Sampling with a random start is a method of selecting a sample by taking every k th unit from an ordered population, the first unit being selected at random. Here k is called the sampling interval and 1/k the sampling fraction.  Random Sampling Sample Selection Procedure (for n that is a divisor of N) 1.Number the units of the population consecutively from 1 to N. 2. Determine k by the formula. k = N / n = population size/sample size 3. Select the random start r (from 1 to k). The unit corresponding to r is the first unit of the sample. 4. The other units of the sample correspond to the labels, r + k, r + 2k,..., r+(n-1)k  Random Sampling Example 1. A medical investigator is interested in obtaining information about the average number of times N = 15,000 specialists prescribed a certain drug in the previous year. To obtain a sample of n = 500 specialists, we could select one specialist at random from the first k = 30 names appearing on the list and then select every 30th name thereafter until a sample of size 500 is selected.  Random Sampling e. Multi-Stage sampling is a technique uses several stages or phases in getting the sample. The population is first divided into a number of first-stage or primary units, from which a sample is drawn. Within the sampled first-stage units, a sample of second-stage or secondary units is drawn.  Random Sampling Example. The following is one possible set up for school surveys. Sample Stage Primary Department Secondary Program Third stage Major Fourth state Year level  Non-Random Sampling a. Purposive Sampling is based on the criteria or a certain characteristics lay down by the researcher. People who satisfy the criteria are interviewed.  Non-Random Sampling Example 1. A researcher may want to find out the central bank circular, instead of interviewing the executives of all banks, he purposely can choose to interview the key executives of only five biggest banks in the country.  Non-Random Sampling Example 2. A researcher wants to catch out the profitability of SMEs in Tuguegrao, hence, the researcher may randomly select SME owners from the chosen area.  Non-Random Sampling b. Quota Sampling is relatively quick and inexpensive method to operate. Each interviewer is given definite instruction about the section of the public he is to question, but the final choice of the actual person is left to his own convenience or preference. Example 1. A researcher want to determine the favorite teams of the PBA. His quota is say 100 basketball fans. The researcher may reach his quota by going to the ARANETA coliseum to enjoy watching the game at the same time interview thrilled viewers during the game. c. Convenience Sampling is done where in a researcher may get his sample at his own convenience.  Non-Random Sampling Example 1. A researcher wants to find out the most cost- effective fertilizer that could give a better yield among farmers. Hence, the researcher may gather data from his barangay because this is the most accessible for him to gather the data.  Non-Random Sampling d. Accidental or incidental sampling is a process of getting a subject of study that is only available during the period.  Non-Random Sampling Example 1. If the researcher wants to determine the top seller brand of toothpaste in the region, then the researcher would identify the pick shopping hours in a certain mall and stand by at the exit gate and interview.  Non-Random Sampling e. Snowball sampling can happen in a number of ways, but generally it is when a group of people recommends potential participants for a study, or directly recruits them for the study. Those participants then recommend additional participants, and so on, thus building up like a snowball rolling down a hill.  Non-Random Sampling If you are trying to recruit people who are difficult to identify or have to meet certain criteria to participate, then snowball sampling can be used to ease data collection  Non-Random Sampling Cases Wherein Nonprobability Sampling is Useful Researchers do not have enough resources to implement probability sampling. There are very few respondents who are willing to be interviewed. It is extremely difficult to locate or identify subjects. Organization and Presentation of Data Methods of Presenting Data 1.Textual method 2.Tabular method 3.Graphical method  Organization and Presentation of Data 1. Textual Method. Presents data in narrative form to describe the data collected. Example 1: The top three prospective investing countries for the third quarter of 2015 include the Netherlands, Japan and South Korea. The Netherlands pledged PhP 27.7 billion or 56.9 percent share during the quart er while Japan and South Korea committed PhP 4.1 billion and PhP 3.6 billion, or 8.4 percent and 7.5 percent of the total approved FI, respectively.[source: psa.gov.ph, 2015]  Organization and Presentation of Data Example 2. Most Filipino couples prefer to tie the knot in May. Around 12% of all weddings in 2015 or 50,469 ceremonies were in held in May, recent data from the Philippine Statistics Authority(PSA) showed.  Organization and Presentation of Data 2. Tabular Method. Presents data in condensed form by arranging them systematically in rows and columns.A statistical table that can be constructed to present a data collected is the Frequency Distribution Table (FDT).  Organization and Presentation of Data Some Guidelines in Presenting Tables A title should be provided which includes the type of information given and any relevant dates. Row and column labels should be precise. Categories should not overlap. The units of measure must be clearly stated. Show any relevant total, subtotals, percentages, etc. Indicate if the data were taken from another publication by including a source note. Formal tables should be self-explanatory, although they may be accompanied by a paragraph of interpretation or a paragraph directing attention to important figures. Organization and Presentation of Data Table 3. Average monthly wages of 143 employees of ABC Company Monthly Wages (Rs) No. of Workers 800-1,000 18 1,000-1,200 25 1,200-1,400 30 1,400-1,600 34 1,600-1,800 26 1,800-2,000 10 Total 143 Source: ABC Company, Western INDIA Organization and Presentation of Data Table 1. Blood Types of 25 Patients Afflicted with certain Disease Blood Type Number of Patients Percentage O 9 36% A 5 20% B 6 24% AB 5 20% Total 25 100% Source: CARAGA General Hospital, Philippines  Frequency distribution table Frequency distribution table (FDT) for Quantitative type of data If the classes are numerical and discrete, we construct our FDT as follows: Frequency distribution table  Frequency distribution table Step 3. Compute the class size (i) or class width by rounding top the nearest value whose precisions is the same as those of the raw data i = R/K Step 4. Determine the classes by starting with the lowest class with the lowest value in our data or the highest class with the highest score.  Frequency distribution table Step 5. Sort or tally the data into these classes. Step 6. Count the number of tallies in each class and write them under frequency column. We usually omit the tally column in presenting the FDT.  Frequency distribution table Step 7. Build additional columns to obtain other information about the distributional characteristics of the data. These are Class Boundaries (CB) Class Mark or Midpoint (X) Relative Frequency (RF) Cumulative frequencies CF Example: ABC Company, Inc. Suppose we are given the number of goods delivered for a given 40 outlets. The numbers below are called raw data 120 133 180 138 140 150 170 153 161 149 124 168 148 139 161 142 130 143 137 147 165 138 147 167 156 148 128 118 146 150 149 129 142 158 152 150 175 151 142 157  Example: ABC Company, Inc. Solution: Step 1. R = 180-118 = 62 Step 2. K = 40 = 6.32 = 6 (round off) Step 3. C = 62/6 = 10.33 = 11(round up) Step 4, 5 and 6  Example: ABC Company, Inc. Solution: Step 1. R = 100.8-59.2 = 41.6 Step 2. K = 25 = 5 (round off) Step 3. C = 41.6/5 = 8.32 = 9(round up) Step 4, 5 and 6 Example: ABC Company, Inc. Classes CB f X RF% CF 118 - 128 117.5 – 128.5 4 123 10.0% 4 40 129 - 139 128.5 – 139.5 7 134 17.5% 11 36 140 - 150 139.5 – 150.5 15 145 37.5% 26 29 151 - 161 150.5 – 161.5 8 156 20.0% 34 14 162 - 172 161.5 – 172.5 4 167 10.0% 38 6 173 - 183 172.5 – 183.5 2 178 5% 40 2  Example: ABC Company, Inc. Classes CB f X RF% CF 59.2 - 68.1 68.2 - 77.1 77.2 - 86.1 86.2 - 95.1 95.2 - 104.1  Stem-and-leaf plot 11 8 12 0489 13 039878 14 086293879227 15 600812307 16 15187 17 50 18 0 19  Stem-and-leaf plot 2 3 4 5 9.2, 6 6.6,6.7,6.8,5.3, 7 0.2,1,5.4,8.5,8.5,2,9.9,6.2, 8 6.9,2.5,0.8,7.8 9 2.8,2.9,7.3,2.8,4.1 10 0.8,  Interpretation ABC Company, Inc. has recorded the highest number of goods delivered in 40 outlets is 140 – 150 which is estimated almost 38% of its outlets. There were 11 or 27.5% outlets which is below the average number of goods delivered while 14 or 35% are above the average number of deliveries. The highest number of goods delivered is 180 while the lowest number is 118. [source: ABC Company, Inc.]  Graphical Presentation of Data Qualities of a Good Graph Accuracy. A graph should not be deceptive, distorted, misleading, or in any way susceptible to wrong interpretations as a result of inaccurate or careless construction. Simplicity. The basic design of a graph should be simple, straightforward, not loaded with irrelevant or trivial symbols and ornamentation.  Graphical Presentation of Data Clarity. The graph should be easily read and understood; there should be a forceful and unmistakable focus on the message that the graph is trying to communicate. Appearance. A good graph is one that one that is designed and constructed to attract and hold attention by holding a neat, dignified, and professional appearance.  Common Types of Graphs 1.Line graph 2.Bar chart/Column chart 3.Pie chart 4.Pictograph 5.Scatter graph 1. Line Graph is a graphical presentation of data especially useful for showing trends over a period of time. Example of LINE graph Example of Multiple line graph Example of Multiple line graph  BAR GRAPH 2. Bar graph consist of series of rectangular bars where the length of the bar represents the magnitude to be demonstrated.  BAR GRAPH Use horizontal arrangement of the individual bars when comparison of categories is being made. Use vertical arrangement of the individual bars when chronological comparisons are being made.  Bar Graph a. Column charts can be used for vertical arrangement of the individual bars when chronological comparisons are being made. The emphasis is on the magnitude of data set. BAR GRAPH Source: Philippine Statistics Authority, National Accounts BAR GRAPH b. Horizontal Bar Charts can be used for qualitative types of data in a given specific time. It is use to compare the magnitudes of the different categories of a qualitative variable. Bar Graph  Pie Graph c. Pie graph provides easy measurement and fast presentation of nominal data divided into a few categories. It is a circular graph that is useful in showing how quantity is distributed among a group of categories. Pie Graph  Pie Graph P E R C E N T AG E D IS T R IB U T IO N O F T O T AL F AM IL Y E X P E N D IT U R E S , P H IL IP P IN E S : 1 9 9 4 H ouse Furnishings & Equipm ents O therExpenditures 3% T axes Paid 2% M edical C are 1% 2% R ecreation 0% Education 4% Food 50% C lothing, Footw ear, etc. 4% Personal C are & Effects 3% H ousehold O perations 3% T ransportation & C om m unication 5% Fuel,Light & W ater 6% H ousing Alcoholic B everages 15% T obacco 1% 1%  Pictograph d. Pictogram used pictures or symbols to represent certain quantity or volume. Similar to bar charts except that the bars are replaced by pictures with each object or character represents a certain amount quantity.  Pictograph Steps in constructing a pictograph Step 1: Encode the data in the Microsoft excel as shown below then highlight the entire data including the label. Step 2. Click insert ribbon. Step 3. From the next menu, click bar section followed by clicking the second choice (stack bar) from the 2-D Bar section.  Pictograph Step 4. Right- click the bars on the graph then select format data series. Step 5. Click Fill section. Under fill section, select picture or texture fill. Step 6. Insert a picture from the file, clipboard or clipart then click stack section. Step 7. Insert legend, title, etc. as shown below to finish the graph.  Histogram e. Histogram is a special type of bar graph of a frequency distribution table. Class boundaries are represented by the width of the bars and the frequencies that fall within the classes are represented by the height of the bars. Histogram  Scatter plot f. Scatterplot. A graph used to examine whether a relationship exists between quantitative variables. For instance, whether sale is related to advertising, whether starting salary is related is related to undergraduate grade point average, or whether the price of a stock is related to the company’s profit per share. Scatter plot Scatter plot  The Statistical Analysis The Research Study Process  The Statistical Analysis The study analysis process  The Statistical Analysis How statistical analysis can help you?  References Altares, et.al. Elementary Statics with Computer Applications. Rex Book Store. 2012 Amid, Diego M. Fundamentals of Statistics. Lorimar Publishing Co.,Inc. 2005 Cabrero, Salamat and Sta. Maria. Business Statistics. Anvil Publishing Company.2013. Coronel, et.al. Statistics for University Students, A comprehensive Approach. National Book Store. 2004. Mansfield, E. Statistics for Busienss and Economics Methods and Applications , 3rd Editon. W.W. Norton and Company , INC. 1987 Nalangan, L.C. and Casinillo M.C. Laboratory Manual in Statistics 1 (Elementary Statistics). Rex Book Store. 2009. Parreno E.B. and Jimenez, R.O. Basic Statistics, 2nd edition. C&E Publishing, Inc. 2014.  References Russell, J. and Boggis, E.The Statistics Tutor’s Quick Guide to Commonly Used Statistical Tests. Ellen Marshall, University of Sheffield. www.statstutor.ac.uk. Samuels, P. (n.d). Statistical Methods, An introduction. Birmingham City University. Retrived from http://www.statstutor.ac.uk/resources/uploaded/1introduction3.pdf Sison and Ereno. Computation Mathematics for Teachers (MATH F). University of the Philippines Open University. 1998 Surinder Kundu. Business Statistics, 2014 Weires, R.M. Introduction to Business Statistics, 7th edition. Cengage Learning Asia Pte Ltd. 2015. Zamora-Reyes, C.O and Ladao-Saren, L.B. Elementary Statistics Txt/Workbook. National Book Store. 2003. A module on Basic Statistics.University of the Philippines Statistical Center Research Foundation (UPSCRF) Internet references: http://www.riosalado.edu/web/oer/WRKDEV10020011_INTER_0000_v1/lessons/Mod05_M eanMedianMode.shtml http://highered.mheducation.com/sites/dl/free/0070164959/791053/Chapter_12.ppt