Statistical Concepts & Market Returns

ratio scales

Most refined level of measurement. Provides ranking and equal differences between scale values, and a true zero point as the origin. (ex. $) LOS7.a

geometric mean return formula

1 + RG = [(1+R1) + (1+R2) + ... + (1 + Rn)] ^ 1/n LOS7.e

parameter

A characteristic of a population. LOS7.b

histogram

A graphical presentation of the absolute frequency distribution. Frequency is on the vertical axis, interval is on the horizontal axis. Allows us to see where most of the observations are concentrated. LOS7.d

sample

A subset of the population of interest. LOS7.a

frequency distribution

A tabular presentation of statistic data that aids the analysis of large sets. LOS7.b

relative dispersion

Amount of variability in a distribution relative to a reference point or benchmark. Commonly measured with a coefficient of variation (CV). LOS7.i

mean absolute deviation (MAD) formula

Average of the absolute values of the deviations of individual observations of the arithmetic mean. MAD = (Σ |Xi - x̅|) / n LOS7.g

coefficient of variation (CV)

CV = standard deviation of x / average value of x measures amount of dispersion in a distribution relative to the distribution's mean. Enables us to make a direct comparison of dispersion across difference sets of data. In an investment setting, the CV is used to measure the risk (variability) per unit of expected return (mean). LOS7.i

relative frequency

Calculated by dividing the frequency of each return interval by the total number of observations. The percentage of total observations falling within each interval. LOS7.c

cumulative absolute frequency / cumulative relative frequency

Calculated by summing the absolute relative frequencies at the lowest interval to the highest. LOS7.c

uni-modal / bimodal / trimodal

Data sets that have one / two / three values that occur most frequently. LOS7.e

population variance formula

Defined as the average of the squared deviations from the mean. σ^2 = Σ (Xi - μ)^2 / N LOS7.g

platyjurtic

Describes a distribution that is less peaked than a normal distribution. LOS7.l

leptokurtic

Describes a distribution that is more peaked than a normal distribution. LOS7.l

range

Distance between the largest and smallest value in the data set. range = max value - min value LOS7.g

sample skewness

Equal to the sum of the cubed deviations from the mean divded by the cubed standard deviation and the number of observations. Sk = 1/n * ( Σ (Xi - x̅)^3 / s^3 ) LOS7.l

arithmetic means

Examples are population mean and sample mean. Most widely used measure of central tendency. The sum of the deviations of each observation in the data set from the mean is always zero. LOS7.e

Chebyshev's inequality

For any set of observations, whether sample or population data and regardless of the shape of hte distribution, the percentage of the observations that lie within k standard deviations of the mean is at least 1 - 1/k^2 for all k > 1. LOS7.h

Quantile

General term for the value at or below which a stated proportion of the data in a distribution lies. Ex. Quartiles (quarters), quintiles (fifths), deciles (tenths), percentile (hundredths). LOS7.f Ly = (n + 1) * y/100

Analyzing investment returns & Arithmetic / Geometric Means

Geometric means = for multiple years Arithmetic Mean = for the next year LOS7.g

mesokurtic

Has the same kurtosis as a normal distirbution. LOS7.l

ordinal scales

Higher level of measurement than nominal scale. Each observation is assigned to a category, and the categories are then ordered with respect to a characteristic. LOS7.a

measures of central tendency

Identify the center, or average (mean) of the data set. LOS7.e

nominal scales

Level of measurement that contains the least information (ex. numbered from 1-10 in no order). LOS7.a

frequency polygon

Like a histogram. The midpoint of each interval is plotted on the horizontal axis, the absolute frequency is plotted on the vertical axis. Each point is connected with a straight line. LOS7.d

sample variance

Measure of dispersion that applies when we are evaluating sample n observations from the population. s^2 = Σ (Xi - x̅)^2 / (n-1) LOS7.g

sample kurtosis

Measured using deviations raised to the fourth power. Sample kurtosis = 1/n * ( Σ (Xi - x̅)^4 / s^4 ) LOS7.l

inferential statistics

Pertains to the procedures used to make forecasts, estimates, or judgements about a large set of data on the basis of the statistical characteristics of a smaller set (a sample). LOS7.a

interval scales

Provide relative ranking, plus assurance that differences between scale values are equal. (ex. temperature) LOS7.a

measures of location

Quantiles and measures of central tendency are known collectively as Measures of Location. LOS7.f

weighted mean formula

Recognizes that different observations may have a disproportionate influence on the mean. x̅w = Σ wiXi - (w1X1 + w2X2 + ... + wnXn) LOS7.e

skewness

Refers to the extent to which a distribution is not symmetrical. Positively skewed (many outliers in the upper region, or right tail) or negatively skewed (many outliers in the lower left tail). Mean is affected more than median and mode and is pulled in direction of skew. LOS7.j

Sharpe measure (Sharpe ratio)

Sharpe ratio = (rp - rf) / σp rp = portfolio return rf = risk free return σp - standard deviation of portfolio returns Measures the excess return per unit of risk. LOS7.i

population standard deviation

Square root of the population variance. σ = [ Σ (Xi - μ)^2 / N ] ^1/2 LOS7.g

sample standard deviation

Square root of the sample variance. s = [ Σ (Xi - x̅)^2 / (n-1) ] ^1/2 LOS7.g

median

The midpoint of a data set when the data is arranged in ascending or descending order. Important because mean can be affected by outliers. LOS7.e

population

The set of all possible members of a stated group. LOS7.a

mode

The value that occurs most frequently in a data set. LOS7.e

harmonic mean

Used for certain computations, such as the average cost of shares purchased over time. N / ( Σ 1 / Xi) LOS7.e

sample statistic

Used to mature a characteristic of a sample. LOS7.b

descriptive statistics

Used to summarize the important characteristics of large data sets. LOS7.a

geometric mean (average) formula

Used when calculating investment returns over multiple periods or when measuring compound growth rates. G = (X1 * X2 * X3 * ... * Xn) ^ 1/n LOS7.e

dispersion

Variability around the central tendency. LOS7.g

symmetrical distribution

identical on both sides LOS7.j

kurtosis

measure of the degree to which a distribution is more or less peaked than a normal distribution. Kurtosis for all normal distributions is 3. LOS7.l

outliers

observations with extraordinarily large values, either positive or negative. LOS7.j

sample mean (average) formula

x̅ = Σ Xi / n LOS7.e

sum of mean deviations formula

Σ Xi - x̅ = 0 LOS7.e

population mean (average) formula

μ = Σ Xi / N LOS7.e