CFA Level 1 - Quantitative Methods Part 1
What is a histogram?
-Presentation of absolute frequency distribution -Bar chart of continuous data that has been classified into a frequency distribution
What is a frequency polygon?
-Presentation of absolute frequency distribution -Like a histogram, but only a line indicating interval changes
How is a conditional probability expressed?
-Probability of A given the occurrence of B -P(A|B)
How to interpret kurtosis?
-Relative to normal distribution, which is 3 -Positive excess kurtosis = leptokurtic -Negative excess kurtosis = playkurtic
What is a frequency distribution?
-Tabular presentation of statistical data that aids in analysis of large data sets -Summarizes data by assigning it to specified groups or intervals -Data can be measured using any type of scale
Why does the MAD computation use absolute values of each deviation?
The sum of actual deviations from the arithmetic mean is ZERO!!!
What is the mode?
Most frequently occurring in a data set
What are objective probabilities?
1. Empirical probability 2. Priori probability
What is absolute frequency?
Actual number of observations that fall within a given interval
What are exhaustive events?
All possible outcomes
What is geometric mean?
-Number of observations root of product of all observations -(Product of all observations)^1/n
What is the harmonic mean?
# of observations/(Sum of 1/each observation)
What is the formula for the position of an observation in a given percentile?
(Number of data points + 1)*given percentile/100
How to calculate median for even number sets?
(Sum of 2 middle observations)/2
How to computer sample skewness?
(Sum of all [Obs.i-Mean of Obs.])^3/(standard deviation of obs.)^3
How to construct a histogram?
-Absolute frequencies on vertical access -Intervals are scaled on horizontal access
When calculating geometric mean for a returns data set, what is required?
-Add 1 to each value under the radical sign -Subtract 1 from the result -[(1+obs1)*(1+ob2)(1+...)]^n/1
What are nominal scales?
-Least information -Obs. in no specific order, just assigned a number (no ranking, just given the sample)
How is the geometric mean relative to the arithmetic mean?
-Less than or equal to the arithmetic mean -Difference increases as dispersion of observations increase
What is the median?
-Midpoint of a data set when ARRANGED in ascending or descending order -Half the observations above, half below
How to construct a frequency polygon?
-Midpoint of each interval is plotted on horizontal axis -Absolute frequency on vertical axis -Connect each point w/ a straight line
What are ratio scales?
-Most refined level of measurement -Provide ranking, equal difference between scale values, and also have a true zero point as origin ($)
What are the steps to putting together a frequency distribution?
1. Define the intervals 2. Tally the observations 3. Count the observations
What are the 2 defining properties of probability?
1. Occurrence of any event is (O =< Event =< 1)
What are examples of quantiles?
1. Quartiles (4th) 2. Quintiles (5th) 3. Decile (10th) 4. Percentile (100th)
What is relative dispersion?
Amount of variability in a distribution relative to a reference or benchmark
What is empirical probability?
Analyzing past data
When is harmonic mean used?
Average cost of shares purchased over time
What is the mean absolute deviation?
Average of absolute values of deviations of individual observations from the arithmetic mean
What is the population variance?
Average of the squared deviations from the mean
What is the understatement of sample variance denominator referred to as?
Biased estimator of the population variance
What is it called when 2 (3) values appear more frequently?
Bimodal (trimodal, etc.)
What is the relation b/n quantiles and measures of central tendency?
Both are measures of location
What are mutually exclusive events?
Cannot happen at the same time
What is a parameter?
Characteristic of a population
What is a sample statistic?
Characteristic of a sample
What is another name for intervals?
Classes
How is relative dispersion frequently measured?
Coefficient of variation (in investing, think risk per unit of expected mean)
What is priori probability?
Determined using formal reasoning and inspection process
What is an issue with the population variance?
Difficult to interpret
What is negatively skewed distribution?
Disproportionately large amount of outliers in lower region or left tail
What is a relative frequency?
Dividend absolute frequency by total number of observations
What are ordinal scales?
Every obs. assigned one of several categories, then ranked based on assigned categoryq
What is skewness?
Extent to which distribution is not symmetrical
What is Chebyshev's inequality?
For any set of observations (sample or population and any shape), the percent of obs.. that lie within k standard deviations of the mean is at least (1-(1/k)^2) for all K > 1
What is the rank in value of means when the values of observations are not all equal?
Harmonic mean < Geometric mean < Arithmetic mean
What is the modal interval?
Interval with the greatest frequency
What does distributional symmetry imply?
Intervals of losses & gains will exhibit the same frequency
What is it called when a distributed is peaked?
Kurtosis
How to calculate excess Kurtosis?
Kurtosis minus 3 (computer kurtosis for all normal distributions is 3).
What is excess kurtosis?
More or less kurtosis than the normal distribution
What is leptokurtic?
More peaked than normal distribution
What is subjective probability?
Least formal method of developing probabilities and involves personal judgment
What is playkurtic?
Less peaked or flatter than a normal distribution
What is positive skewed distribution?
Many outliers in the upper region or right tail
How are assets weighted in calculating the return for a portfolio?
Market value of each asset relative to the entire portfolio
What is the range?
Maximum value - minimum value
What is kurtosis?
Measure of degree to which a distribution is more or less "peaked"
What are the types of scales in order of precision?
N - Nominal O - Ordinal I - Interval R - Ratio scale
Can a data set have numerous arithmetic means?
No
When is arithmetic and geometric mean the same?
No variability in the observations
Why do we use # of observations for sample variance -1 but not population variance?
Not using n-1 will underestimate the population parameter for small sample sizes
What is an outcome?
Observed value of random variable
What is conditional probability?
Occurrence of on event affects the probability of occurrence of another event
What is unique about arithmetic mean?
Only measure of central tendency for which sum of deviations from the mean is zero
What does the addition of 1 to each value for geometric mean do?
Prevents negative returns from being negative under the radical sign
What is unconditional probability?
Probability of an event regardless of past or future occurrence of other events
How to move from odds to probability?
Probability of occurrence/(1+probability of occurrence)
How to move from probability to odds?
Probability of occurrence/(1-probability of occurrence)
What are inferential statistics?
Procedures used to make forecasts, estimates, or judgments about a large set of data on basis of a smaller set (sample)
What is a limitation of geometric mean?
Product under the radical sign MUST be positive, otherwise doesn't work
What is an interval scale?
Provides relative ranking, like ordinal, but assures that the difference in scale value (intervals) is equal (i.e. temperature example)
What is required for position of an observation in a given percentile?
Rank all data points in ascending order
How to calculate sample kurtosis?
Same as sample skewness but now raise to the 4th power rather than 3rd
What is mesokurtic?
Same kurtosis as normal distribution
What is an event?
Single outcome or set of outcomes
What is it called when a distribution lacks symmetry?
Skewness
What is the population standard deviation?
Square root of population variance
What is the sample standard deviation?
Square root of the sample variance
How to calculate coefficient of variation?
Standard deviation of x/Mean of x
What is the arithmetic mean?
Sum of observations divided by number of observations
What is cumulative absolute/relative frequency? (Figure 7.2)
Summing absolute or relative frequencies starting at lowest interval and progressing through the highest
When are mean, median, and mode equal?
Symmetrical distribution
What is a random variable?
Uncertain quantity/number
What is it called when 1 value appears more frequently?
Unimodal
What are descriptive statistics?
Used to summarize import characteristics of large data sets
What are quantile?
Value at or below which a stated proportion of the data in a distribution lies
What is dispersion?
Variability around the central tendency
What is the return for a portfolio?
Weighted average of the returns of individual assets in the portfolio
When is kurtosis considered large?
When the absolute value of excess kurtosis is equal to or greater than 1
When is median a better measurement of central tendency than the mean?
When there are outliers
Can there be multiple modes or none at all?
Yes!
Can any quantile be expressed as a percentile?
Yes. For example, the 3rd quartile (3/4 of the data) represents 75% of the data.
What is weakness of an interval scale?
Zero does not necessarily indicate total absence of what is being measured, making ratios pointless
How to calculate mean absolute deviation?
[Sum of all ABS(Obs1-Mean)]/# of observations
How is a sample mean shown?
x̄
How to calculate population variance?
{[Sum of all (Obs1-Mean)]^2}/# of observations
How to calculate sample variance
{[Sum of all (Obs1-Mean)]^2}/(# of observations - 1)
How is a population mean shown?
μ
What is symbol for population standard deviation?
σ