Statistics Fundamentals: Measures of Central Tendency, Dispersion, Shape, and Probability
Measures of Position
Central Position Measures: The Arithmetic Mean
Central position measures: they describe the global behavior of the data distribution and locate the center of frequencies, that is, the place where the parts of the population are located.
Arithmetic mean: it is the sum of the values of the data multiplied by their relative frequencies. It is sensitive to extreme values.
Properties:
- The sum of the deviations from the mean is zero.
- Grouping.
- Translation and change of scale.
Geometric Mean and Harmonic Mean
Geometric mean: it is the product of all the values raised to the power of the total number of values (the nth root). The values must be positive. It is used for percentages, rates, and index numbers.
Harmonic mean: it is used for speeds, prices, yields, currency exchanges, etc.
Mode, Median, and Percentiles
Mode: it is the value with the highest frequency. It is the most common value. A distribution may have several modes.
Median: it is the value that divides an ordered set of observations into two equal parts. It minimizes the sum of absolute deviations.
Quantiles: they are the values that divide an ordered set of observations into a number of equal intervals.
Measures of Dispersion
Range
Dispersion measures: they summarize in a single number the variability of the data.
Range: it is the difference between the extreme values of the variable. In grouped data, the first and last intervals are used.
Mean absolute deviation: it can be taken with respect to the arithmetic mean, the median, etc. It measures the average representativeness of the mean. It is minimal when taken with respect to the median.
Variance and Standard Deviation
Variance: it is the arithmetic mean of the squared deviations from the arithmetic mean.
Properties:
- It cannot be negative.
- It is invariant under translation of the origin.
- It changes with a change of scale, but it is not invariant under a change of scale.
Standard deviation: it is the positive square root of the variance. It is expressed in the same units as the data.
Relative Measures of Dispersion
Coefficient of variation (aperture coefficient): it indicates the extreme values of a variable. It applies to wages, etc.
Relative range: it is the ratio between the range and the arithmetic mean.
Coefficient of variation: it is the ratio between the standard deviation and the arithmetic mean. It is used to compare the dispersion of different variables. The smaller it is, the more homogeneous the data.
Moments About the Mean and Non-Central Moments
Moments: they summarize information about the frequency distribution in relation to some aspect or property. The arithmetic mean and the variance are particular cases.
Measures of Shape
Skewness Coefficient (Fisher)
Shape measures: values that describe how the distribution is represented in a frequency graph.
Skewness coefficient (Fisher): it is the most important one used. It is dimensionless and independent of the origin and scale changes.
Kurtosis Coefficient (Fisher)
It is applied to almost unimodal, symmetric distributions. It refers to the bell shape of the Gaussian curve. It is dimensionless and independent of the origin and scale changes.
Measures of Concentration
Lorenz Curve
Concentration measures: they are values that are lower than or equal to a certain degree of the total distribution of the variable. These variables are positive.
Lorenz curve: it is the graphical representation of the cumulative relative frequencies against the first polygon.
Gini Index
Gini index: it is a value between 0 and 1 that indicates the degree of concentration.
Median and Mediala
Mediala: it is the value of the variable that divides the sum of the parts into two equal halves when the observations are ordered.
Relationship Between Concentration and Dispersion
Concentration: it is the inequality in the distribution of the total of the variable. It is greater or lesser depending on the degree to which the total is unequally distributed.
Dispersion: it is the variability of the data. It is greater or lesser depending on the representativeness of the average.
Linear Correlation
Linear Correlation Coefficient
Covariance: it is the relationship between two variables, X and Y. If it is positive, the variables vary in the same direction. If it is negative, they vary in opposite directions. If there is statistical independence, sxy = 0.
Linear correlation coefficient: it is an objective measure of the joint variability of two variables. It is bounded between -1 and 1.
Index Numbers
Simple and Synthetic Indices
Index numbers: they describe and compare economic magnitudes.
Simple index: it is the quotient I = gt / g0.
Properties:
- Circular: it is compared with other values, which are based on a period when we refer to it.
- Inversion: it is important to refer to 1 as a distinguishing criterion.
- Chaining over time: it is obtained by multiplying the index for the period t with respect to period 0, and the intermediate indices with respect to the previous period.
- Multiplication: the simple index of 1 for the other is the product of the simple indices.
- Division: the simple index of 1 over the other is the quotient of the simple indices.
Synthetic index: it is the index that summarizes in a single number the complex magnitude of simple indices.
Sauerbeck index: it is the arithmetic mean of the simple indices.
Bradstreet and Dudot index: it is the sum of the components of the current period divided by the sum of the components of the base period.
Laspeyres index: it is the arithmetic mean of the simple indices weighted with base period factors.
Paasche index: it is the harmonic mean of the simple indices weighted with current period factors.
Fisher index: it is the geometric mean of the Laspeyres and Paasche indices.
Price and Quantity Indices
Laspeyres price index: its calculation consists of the budget of the base-year prices for the base year and the current year, and comparing them with each other.
Paasche price index: the period considered for the current budget determines its value, and the current and base years are compared with each other.
Laspeyres quantity index: its value is computed from the budget of the base and current years, and compared with each other.
Paasche quantity index: the current and base period budgets are considered, and its value is determined from the base-year prices compared with each other.
Probability
Probability Definitions
Laplace probability definition: it applies when all the elementary events have the same probability of occurring.
Frequency definition of probability: probability is approximated by the relative frequency of an event over a large number of repetitions.
Concept of Probability
Sample space: it is the set of all the possible results of a random experiment.
Event space: it is the set of all the subsets of the sample space.
Definition of probability: given a finite sample space and an event space, a probability is a function that assigns a number to each event.
Computing probability: it is the probability of the intersection of various events.