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BUSINESS STATISTICS I Descriptive Statistics & Data Collection Descriptive Statistics – Graphic Guidelines • Pie charts – nominal variables, eg. ‘religion’; cross-sectional • • • • data Bar charts – nominal or interval variables, eg. ‘religion’ or ‘margin debt’; time series or cross sectional data Line graphs – interval variables, eg. margin debt; time series data Histograms – interval variables, eg. golf scores; cross sectional data – depicts the SHAPE of a frequency distribution • Stem and Leaf Plot– quick and dirty histogram • Ogive – depicts a cumulative percentage frequency distribution Scatter diagram – two interval variables, eg. Margin vs, the market value Graphic Deception – some widely used methods • Graphs without a scale on one axis • Captions or titles intended to influence • Reporting only absolute changes in value and not percentage changes • Changing the scale of the vertical axis with breaks or truncations • Changing the scale of the horizontal axis • Changing the width as well as the height of bars or pictogram figures Summary of data types and available graphic techniques Interval Cross-sectional data Histograms Percentage histograms Ogives Stem and leaf plots Box plots Time-series data Line charts Bar charts Nominal Pie charts Bar charts Complex pie or bar charts Describing the frequency distribution for interval, cross sectional data • Shape • Center • Spread Describing distributions • SHAPE • Graphs • • • • • Histograms Percentage histograms Ogives Stem and leaf plots Box plots • Words • Symmetric, skewed, bell shaped, flat, peaked Descriptive statistics – • CENTER • Quantitative measures • Mean (arithmetic) • Median • Mode • Geometric mean • Mid-point of the range Descriptive statistics – • Numeric Measures – cont’d. • SPREAD (dispersion) • Range • Symmetric distributions • Standard deviation • Variance • Coefficient of variation • Skewed distributions • Quartiles • Min • Max • Interquartile range • Percentiles Z Scores and t-scores • Measures distance from the mean in standard deviations • Eg. T score for bone density – 1 to 2.5 standard deviations below the norm (mean) for a 23 year old indicates osteopenia; 2.5 or more indicates osteoporosis • (X-m)/s = z score • (X – Xbar)/s = t score Empirical Rule • For mound shaped distributions • About 68% of observations are within one standard deviation of the mean • About 95% of observations are within two standard deviations of the mean • Almost all (99.7%) observations are within three standard deviations of the mean Chebysheff’s Rule • For all distributions • Let k be greater than or equal to 1 • At least 1-(1/k2) of the observations are within k standard deviations of the mean • Examples • K=1 zero observations may be within one standard deviation of the mean • K=2 3/4th’s of observations must be within two standard deviations of the mean • K=3 8/9th’s of observations must be within three standard deviations of the mean Sampling • ‘Scientific sampling’ is random sampling • • • • Simple random samples Systematic random samples Stratified random samples Random cluster samples • What? • Why? • How? What is random sampling? • Simple random sample -Every sample with the same number of observations has the same probability of being chosen • Choose first sample member randomly • Stratified random sample – Choose simple random samples from the mutually exclusive strata of a population • Cluster sample – Choose a simple random sample of groups or clusters Why sample randomly? • To make valid statistical inferences to a population • Conclusions from a non-probability sample can be questioned • Conclusions from a self-selected sample are SLOP How can samples be randomly chosen? • Random number generators (software) • Ping pong balls in a hopper • Other mechanical devices • Random number tables • Slips of paper in a ‘hat’ With or without replacement