Measures of Dispersion Jacek Wallusch _________________________________ Statistics for International Business Lecture 2: Measures of Dispersion

Getting Started ____________________________________________________________________________________________ risk, uncertainty Steven Obanno: Do you believe in God Mister Le Chiffre? Le Chiffre: No, I believe in a reasonable rate of return. Steven Obanno: I want no risk in my portfolio. Le Chiffre: Agreed. Probability distribution and uncertainty and risk – this topic will be reconsidered soon Statistics: 2 Steven Obanno and Le Chiffre, Cassino Royale

frequency of a class devided by the sample size. Getting Started ____________________________________________________________________________________________ frequency distribution Frequency Distribution: tabular summary of a set of data showing the frequency (or number) of items in each of several non-overlapping classes. Relative Frequency Distribution: frequency of a class devided by the sample size. Statistics: 2 A histogram displays the frequencies or relative frequencies (vertical axis) and the variable of interest (horizontal axis). File: redskins.xls

Cumulative frequency: Getting Started ____________________________________________________________________________________________ frequency distribution Items: Frequency: Relative frequency: Cumulative frequency: Cumulative relative frequency: A a B b a + b Sum: a + b = T Statistics: 2 T – sample size

Getting Started ____________________________________________________________________________________________ dispersion Ogive Statistics: 2 Ogive and gothic architecture

Example: Washington Redskins and yearly salaries Getting Started ____________________________________________________________________________________________ dispersion Ogive cumulative relative frequency Example: Washington Redskins and yearly salaries salary clusters Statistics: 2 Ogive depicts the cumulative distribution or cumulative relative frequency distribution

Getting Started ____________________________________________________________________________________________ dispersion Ogive and mean value Statistics: 2 Dotted, dashed line – cumulative relative frequency, solid line – mean value (arithmetic)

to measure disparities Measures of Dispersion ____________________________________________________________________________________________ usage Application: to measure volatility to measure disparities histogram (spread around the mean) ogive (cumulative distribution) variance and/or standard deviation Statistics: 2

Measures of Dispersion ____________________________________________________________________________________________ variance population variance estimate the mean value; estimate the deviations from the mean value; estimate the sum of squared deviations sample variance Statistics: 2 T – number of observations, m – population mean

estimate the square root of variance Measures of Dispersion ____________________________________________________________________________________________ standard deviation stadard deviation of a population estimate the square root of variance standard deviation of a sample population vs. sample: estimate the population st.dev. for Redskins’ defence Statistics: 2 standard deviation is measured in the same units as the original data

Getting Started ____________________________________________________________________________________________ dispersion Regional disparities Statistics: 2 Interpret the histograms first. Then, estimate the sample standard deviation.

estimate the population standard deviation for the Redskins’ defence Measures of Dispersion ____________________________________________________________________________________________ standard deviation Population vs. Sample: estimate the population standard deviation for the Redskins’ defence Statistics: 2 standard deviation is measured in the same units as the original data

a ratio of the interquartile range to the median Volatility ____________________________________________________________________________________________ measuring variation Coefficient of variation: normalised measure of dispersion, a ratio of the standard deviation to the average Quartile coefficient of variation: a ratio of the interquartile range to the median Statistics: 2 REMARKS [1] normalised measure: suitable for variables expressed in different units; [2] suitable for variables expressed on ratio scale;

Dispersion ____________________________________________________________________________________________ detecting outliers z-score: estimate the mean value; estimate the standard deviation; divide the deviation from the mean by the standard deviation; outliers: suggested z-score for no outliers Statistics: 2 Outlier: an observation numerically distant from the other observations in a sample

Dispersion ____________________________________________________________________________________________ z-Score z-score for the defensive players Statistics: 2 database: Redskins.xls