BUSINESS STATISTICS IDescriptive 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 Nominal

Cross-sectional data Histograms Pie charts Percentage histograms Bar charts Ogives Stem and leaf plots Box plotsTime-series data Line charts Complex pie

Bar charts 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-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