Introduction to Statistics

• Statistics is a branch of mathematics for collecting, organizing, analyzing, interpreting, and presenting numerical data
• Statistics aid understanding patterns and trends, and provides tools for decision-making under uncertainty

Scope of Statistics

• Statistics applies to business and economics in market research, demand forecasting, and financial analysis
• Statistics applies to social sciences in the analysis of social issues, population studies, and public opinion surveys
• Statistics applies to natural and physical sciences for scientific experiments, quality control, and research
• Statistics applies to medicine and healthcare in clinical trials, epidemiology, and medical research
• Statistics applies to engineering and manufacturing for quality control and production planning
• Statistics applies to government and public policy for census, economic policies, and budgeting

Importance of Statistics

• Statistics aids decision-making based on data
• Statistics are used in forecasting and planning for economic planning, production scheduling, and business strategies
• Statistics provides methods to analyze experimental data for scientific research
• Statistics helps industries maintain product quality for quality control
• Statistics are used in the finance, insurance, and investment sectors to assess risks for risk management

Limitations of Statistics

• The accuracy of statistical conclusions depends on the data quality
• Correlation does not imply causation
• Data can be manipulated, which could mislead
• Statistical methods rely on assumptions that may not always hold
• It is difficult to apply to subjective concepts like emotions and opinions

Types of Data

• Primary Data is collected firsthand for a specific purpose
• Secondary Data is data collected by someone else

Primary Data Collection Methods

• Direct Personal Investigation involves a researcher collecting data personally
• Indirect Oral Investigation involves data collected through interviews and surveys
• Schedules and Questionnaires involve data collected through structured forms
• Observation Method involves data collected by observing subjects
• Experiments involves data collected through controlled experiments

Secondary Data Collection Methods

• Government Publications include census, economic surveys, etc
• Company Records include annual reports and sales records
• Research Papers and Journals include academic and industrial studies
• Websites and Databases are online data sources

Frequency Distribution

• Frequency distribution is a table that organizes data into categories or groups to show how often each category occurs

Steps to Prepare a Frequency Distribution

• Collect raw numerical data
• Determine the range by finding the difference between the highest and lowest values
• Decide the number of classes, typically between 5 and 20
• Find class width by dividing the range by the number of classes
• Create equal-sized group class intervals
• Tally data points in each class to count frequencies

Graphical Presentation of Frequency Distribution

• A histogram is a bar graph that represents the frequency distribution of continuous data (x-axis=class intervals, y-axis=frequency)
• A frequency polygon is a line graph that connects midpoints of histogram bars
• An ogive is used to present cumulative frequency data
• A bar diagram represents categorical data using bars
• A pie chart is a circular diagram where each sector represents a category

Types of Series

• Individual Series is data presented without grouping
• Discrete Series is data arranged in classes, with frequencies assigned
• Continuous Series is data grouped into class intervals

Measures of Central Tendency

• Measures of central tendency help determine the center or typical value in a dataset

Arithmetic Mean (AM)

• Arithmetic Mean is the sum of all values divided by the number of observations
• AM = (Sum X)/N
• The demerit of AM is that it is affected by extreme values

Geometric Mean (GM)

• Geometric Mean is the nth root of the product of n numbers
• GM = (Product X)^(1/N)
• The demerits of GM are that it is complex to compute, zero or negative values make it invalid

Harmonic Mean (HM)

• The Harmonic Mean is the reciprocal of the arithmetic mean of reciprocals
• HM = N / (Sum 1/X)
• The demerit of HM is that heavily affected by small values

Median

• Median is the middle value when data is arranged in order
• If N is odd: Median = Middle value
• If N is even: Median = Average of two middle values
• A merit of Median is that it is not affected by extreme values. A demerit is that it ignores data distribution

Mode

• Mode is the most frequently occurring value
• A merit of mode is that it is useful for categorical data. Demerits are that it is not unique and may not exist

Quartiles

• Quartiles divide data into four equal parts
• Q1 (First Quartile) = 25th percentile
• Q2 (Median) = 50th percentile
• Q3 (Third Quartile) = 75th percentile

Measures of Dispersion

• Dispersion measures the spread of data

Range

• Range is the difference between the highest and lowest values
• Range = Xmax - Xmin
• The demerit of range is that it ignores distribution

Quartile Deviation

• Quartile Deviation measures spread in the middle 50% of data
• QD = (Q3 - Q1) / 2

Mean Deviation

• Mean Deviation is the average absolute deviation from the mean or median
• MD = (Sum |X - M|) / N

Standard Deviation (SD)

• Standard Deviation is the square root of variance
• Sigma = Square root of (Sum (X - X bar)^2) / N
• The demerit of Standard Deviation is it is complex to compute

Absolute and Relative Measures

• Absolute Measures are expressed in original data units (Range, SD)
• Relative Measures are ratios or percentages (Coefficient of Variation)