VIEWS: 42 PAGES: 45 POSTED ON: 4/16/2011
STATISTICS-INTRODUCTION • Role of Statistics in Managerial Decisions • Nature of Data, Population data ,Sample data. • Frequency Distribution 0 You use statistics daily without even realizing it!!! You use statistics very often without even realising it !!!!! Examples ?????? 1 Statistics is used to help determine Which product I should sale (Demand stats) How much you pay for insurance (Mortality Stat) Whether drugs are approved for use (Drug trials) Which cars you buy (Reliability ratings, crash tests) Which products are on you grocery shelf (focus groups), and where they are located (Big Bazzar & Snacks Shop are right next to each other…what a concept!!!) What politicians claim as their “firm beliefs” (opinion polls). Favorites to win in sports. Whether it will rain And , on and on. 2 Statistics…..Defn Many people think of statistics as large amounts of numerical data, e.g. share prices, GDP statistics, runs scored by Sachin etc etc Definition : Statistics refers to the range of techniques and procedure for collecting data, summarizing data, classifying data, analyzing data, interpreting data, displaying data and making decisions based on data. Definition: By Statistics, we mean aggregate of facts, affected to a marked extent by multiplicity of causes, numerically expresses, enumerated or estimated accordingly to a reasonable standards of accuracy, collected in a systematic manner for a predetermined purpose and placed in relation to each other 3 Characteristics of Statistics Statistics are the aggregate of facts Statistics are affected to a marked extent by multiplicity of causes Statistics are numerically expressed Statistics are expressed according to reasonable standards of accuracy Statistics should be collected with reasonable standards of accuracy Statistics should be placed in relation to each other 4 Why Study Statistics It presents the facts in a definite & clear terms. It gives the concise shape to the mass of figures and develops meaning from the data It helps to compare between two sets of figures It helps in formulating & testing hypothesis It helps in understanding & predicting the future events, from the past & current data It helps in formulation of suitable policies It helps in understanding the complex happenings Statistics are widely used in business. Usage continues to increase as the business world becomes larger, more complex, and more quantitative. 5 Limitations of Statistics Statistics does not study individual observations. It is only concerned with groups of observations Statistics deals with quantitative characteristics. It does not deal with qualitative characteristics such as beauty, honesty, sharpness, brightness, poverty, intelligence etc Statistical laws are true only on averages Statistics does not reveal the entire story Statistics is only one of the methods of studying the problem Statistics can be misused Statistical data should be uniform & homogeneous. 6 Decision Making - Businesses Accounting Public accounting firms use statistical sampling procedures when conducting audits for their clients. Economics Economists use statistical information in making forecasts about the future of the economy or some aspect of it. 7 Decision Making - Businesses Marketing Electronic point-of-sale scanners at retail checkout counters are used to collect data for a variety of marketing research applications. Production A variety of statistical quality control charts are used to monitor the output of a production process. 8 Decision Making - Businesses Finance Financial advisors use price-earnings ratios and dividend yields to guide their investment recommendations. 9 Uses & Abuses of Statistics Most of the time, samples are used to infer something (draw conclusions) about the population. However, occasionally the conclusions are inaccurate or inaccurately portrayed for the following reasons: Sample is too small. Even a large sample may not represent the population. Unauthorized personnel are giving wrong information that the public will take as truth. A possibility is a company sponsoring a statistics research to prove that their company is better. Visual aids may be correct, but emphasize different aspects. Specific examples include graphs which don't start at zero thus exaggerating small differences and charts which misuse area to represent proportions. Precise statistics or parameters may incorrectly convey a sense of high accuracy. Misleading or unclear or incomplete information may be shared. 10 Misleading Statistical Presentation These two graphs represent sales…who has seen faster sales growth? 16000 14000 14000 13500 12000 13000 10000 12500 8000 12000 6000 11500 4000 11000 2000 10500 0 10000 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 These are actually the same numbers with different scales along the side. 11 Branches of Statistics The academic discipline of statistics can be divided into two major branches: – Descriptive statistics – Inferential statistics. 12 Descriptive Statistics Deals with summarizing and presenting data in a readable, easily understood form. It is tabular, graphical, and numerical methods used to summarize data Techniques: • Visualizing and Summarizing Data: Raw Data, Data Array, Distribution • Characterizing Distributions with Numerical and Graphical Tools: Histogram, Ogive, Measures of Central Tendency: mean, median, mode; Measures of Dispersion: Range, standard deviation, variance, etc. • Exploring the Relationship between Two variables: Scatter Diagrams, Correlation Coefficients, Frequency Tables 13 Inferential Statistics Drawing conclusions about a population based on information from a sample. Statistical Inference is the process of using information obtained from analyzing a sample to make estimates about characteristics of the entire population. It is a discipline that allows us to estimate unknown quantities by making some elementary measurements. Using these estimates we can then make Predictions and Forecast the Future Statistical Inference with Hypothesis Testing: null and alternative hypotheses, one-tailed vs. two-tailed tests, test statistics, p-value, statistical significance, decision rules • The Concept of Risk and Power: risks involved, type I and II errors, confidence level and power of test • Statistical Inference with Confidence Intervals: how it works, when to use it • Equivalence of the Hypothesis Testing and the Confidence Interval Approaches • Statistical Inference for a Single Sample or Group: Hypothesis Testing vs. Confidence Interval Approach 14 Population & Sample Population Sample 15 Population & Sample Population: The complete set of data elements is termed the population. It is a set of all items in a particular study Sample: A sample is a portion of a population selected for further analysis. It is the subset of population Parameter: A parameter is a characteristic of the whole population Statistic: A statistics is a characteristic of the sample, presumably a measurable Remember: Parameter is to Population as Statistic is to Sample 16 Why Sample Why Sample? Less time consuming than a census Less costly to administer than a census More practical to administer than a census of the targeted population Case of Sampling Survey Opinion Polls 17 Data – Data are the facts and figures that are collected, summarized, analyzed, and interpreted. A collection of data is called „data set‟ and a single observation is called a „data element‟ – Data can be further classified as being qualitative (Attribute) or quantitative (Variable). – Variables: Weight, height etc……Two types….Continuous & Discrete Continuous Variable is the variable, which can take any value within the given interval . E.g. Weight….50.0, 50.2, 50.5, 51.0 etc Discrete variable is the variable which can take isolated values e.g. No of patients visiting a doctor e.g. 50, 51 etc – Attribute: Honesty, Integrity etc 18 Data Types Data Numerical Categorical (Quantitative) (Qualitative) Discrete Continuous 19 Primary Data Data can be classified as Primary Data or Secondary Data Primary data are those which are collected for a specific purpose directly from the field and hence are original in nature. This is collected by or on behalf of the person or persons who are going to make the use of the data. Once the data have been collected, processed & published, it becomes the secondary data for the subsequent usage by different people for other application in different connection Methods for Primary Data Collection • Direct Personal Interview • Observations • Indirect Oral Interviews • Information from agents/correspondents • Mailed Questionnaire Method 20 Secondary Data Secondary data are such numerical information, which have been already collected by some agency for specific purpose and are subsequently compiled from that source for the application in different connections. There are many advantages of using secondary data • It is inexpensive • Large quantity of data available from wide range of sources • The data may be available for many number of years, and hence we can understand trend and may forecast the futuristic information 21 Data Sources Primary Secondary Data Collection Data Compilation Print or Electronic Observation Survey Experimentation 22 Descriptive Statistics 23 Data Processing Techniques •Raw Data •Data Array •Discrete Frequency Distribution •Continuous Frequency Distribution 24 Raw Data & Data Array Raw Data: •Information before it is arranged & analysed is raw data. It is called raw, as it is unprocessed by any statistical methods •Example Data Array: •It involves arranging the values in either ascending or descending order •Example 25 Numerical 1 – Data Array Raw Data 14 26 2 34 8 13 27 37 9 12 39 42 45 30 32 24 24 30 20 23 14 18 30 33 24 34 30 10 22 14 Prepare data array. 26 Numerical 1 – Solution Data Array. 2 8 9 10 12 13 14 14 14 18 20 22 23 24 24 24 26 27 30 30 30 30 32 33 34 34 37 39 42 45 27 Discrete Distribution •In the discrete frequency distribution, after arranging the values in ascending order, we count the frequency i.e. number of times each value has appeared in the data set by using tally marks •Discrete distribution is also known as ungrouped FD. •Numerical 28 Numerical 2 - Discrete FD Marks Tally Frequency Marks Tally Fequency Marks Marks 2 1 24 3 8 1 26 1 9 1 27 1 10 1 30 4 12 1 32 1 13 1 33 1 14 3 34 2 18 1 37 1 20 1 39 1 22 1 42 1 23 1 45 1 29 Continuous Frequency Distribution •Continuous Frequency Distribution •In this, all the values are classified in groups or classes, hence this type of distribution is known as grouped or continuous frequency distribution •Class Limits •Class Interval •Class Frequency •Class Mid Point or Class Mark 30 Class Limits Class Limits The two boundaries of the class are known as Class Limits. The Class Limits are the lowest and the highest value that can be included in the class. e.g. 10-20…In this class, 10 is the lower limit and 20 is the upper limit The lower limit of the class is that value below which no observation can be included in the class. The upper limit of the class is that value above which no observation can be included in the class. 31 Class Interval Class Interval The difference between the upper limit and lower limkt of the class is known as class interval or class width of that class. e.g. Class 10-20 has the CI of 10. In case, for the classification, the number of classes are not given, then the number of classes can be determined by (A) using the Sturge‟s formula No of Classes (K) = 1 + 3.322 log N (B) K shall be the smallest exponent of number 2 i.e. „2 power K should be greater than or equal to N. where N is the total no of observations Note: Normally, classes should be between 5 & 15. 32 Class Interval Formula for the Class Interval: Class Interval (i) = (Next unit value after the largest value in the data – Smallest value in the data)/No of Classes e.g. If the marks of 30 students range between 10 & 40 and if we want to divide in 3 classes, then Class Interval (i) = (41-10)/3 = 10.33 i.e. 11 The classes become 10-21, 21-32, 32-43. 33 Exclusive / Inclusive Method There are 2 methods of classifying the data according to class intervals. Exclusive Method: In this, the class intervals are so fixed that the upper limit of the class is the lower limit of the next class. In other words, in exclusive method, upper limits are excluded from that class. E.g. 10-20, 20-30, 30-40 etc. This is more suitable for continuous variable. Inclusive Method: In this type, the upper limits are included in the class. E.g. 10-19, 20-29, 30-39 etc. This is more suitable for discrete variable. Correction Factor = (Lower Limit of 2nd Class – Upper Limit of 1st Class)/2 34 Correction Factor In case of inclusive type, for getting the correct CI, we need to add the correction factor to upper limit of the classes and subtract the same from the lower limit of the classes. Correction Factor = (Lower Limit of 2nd Class – Upper Limit of 1st Class)/2 e.g. 10-19, 20-29 etc etc Class Correction factor = (20-19)/2 = 0.5 and hence the class becomes 9.5-19.5 and hence the CI becomes 10 35 Inclusive to exclusive Convert the following inclusive classes into exclusive classes Inclusive Type Exclusive type 10-14 9.5-14.5 15-19 14.5-19.5 20-24 19.5-24.5 25-29 24.5-29.5 36 Constructing FD Step 1: Decide on the type (Inclusive / Exclusive) and number of classes for dividing the data by using Sturge‟s formula. (If given in the numerical, then go to step 2 directly. Step 2: Sort the data into different classes and count the frequency Step 3: Illustrate the data in the chart 37 Numerical 3 – Continuous FD Step 1: Calculate the No of Classes (Sturge‟s formula) No of Classes (K) = 1 + 3.322 log N = 1 + 3.322 log 30 = 1 + 3.322 (1.477) = 5.9 = 6 Also 2 power 5 = 32, hence by using the other formula, the number of classes shall be 5. Let‟s use 6 classes and proceed. i.e. K = 6. Step 2: Sort the data points into classes and count the no of points in each class. Now Class Interval width = (Next unit value after Largest value –Smallest value)/K = (46-2)/6 = 44/6 = 7.33 i.e. approx 8. Hence the classes shall be 2-9, 10-17, 18-25, 26-33, 34-41, 42-49. 38 Numerical 3 – Continuous FD Class Tally Marks Frequency 2–9 3 10 – 17 6 18 – 25 7 26 – 33 8 34 – 41 4 42 – 49 2 39 Numerical 4 The following set of the data represents the Km per litre of 40 similar motor cycles. 40.5, 39.7, 40.6, 39.9, 40.9, 38.9, 41.4, 40.5, 41.0, 38.8, 39.6, 40.4, 39.9, 40.2, 40.8, 40.7, 40.6, 41.7, 40.8, 39.1, 40.1, 40.7, 40.1, 40.7, 40.7, 39.8, 39.3, 39.6, 40.5, 41.3, 41.0, 39.9, 40.4, 40.9, 40.1, 41.2, 40.2, 40.0, 39.4, 40.6. Construct the frequency distribution to this data taking classes as 38.5-39.0, 39.0-39.5 etc 40 Numerical 4 Classes Tally Marks Frequency 38.5-39.0 2 39.0-39.5 3 39.5-40.0 7 40.0-40.5 8 40.5-41.0 14 41.0-41.5 5 41.5-42.0 1 41 Numerical 5 The credit office of a departmental store gave the following statements for the payment due to 40 customers. Construct the frequency table of the balances due, taking the class interval as Rs 50 and under Rs 200, Rs 200 and under Rs 350 etc. Also find the relative frequencies & percentage frequencies. 337, 570, 99, 759, 487, 352, 115, 60, 521, 95, 563, 399, 625, 215, 360, 178, 827, 301, 501, 199, 110, 501, 201, 99, 637, 328, 539, 150, 417, 250, 451, 595, 422, 344, 186, 681, 397, 790, 272, 514. 42 Numerical 5 Classes Tally Frequency Relative Percentage Marks Frequency Frequency 50-200 10 10/40 = 0.25 0.25*100=25 200-350 8 0.2 20 350-500 8 0.2 20 500-650 10 0.25 25 650-800 3 0.075 7.5 800-950 1 0.025 2.5 40 1.00 100 43 Numerical 6 The following data shows the time spent by the passenger at the airport before he can enter into the check in lounge. Construct the frequency distribution using the suitable classes. Also find the relative and percentage frequency 34, 40, 23, 28, 31, 40, 25, 33, 47, 32, 44, 34, 38, 31, 33, 42, 26, 35, 27, 31, 29, 40, 31, 30, 34, 31, 38, 35, 37, 33, 24, 44, 37, 39, 32, 36, 34, 36, 41, 39, 29, 22, 28, 44, 51, 31, 44, 28, 47, 31. 44