Fundaments of Statistics

Fundaments of Statistics

1. Inference: The objective is to infer the values of θ based on the information of the sample

2. Estimators: Evaluate data from the sample to obtain estimations for the population's parameters.

3. DATA

   • Cross-Sectional Data: a group of observations, taken at a given point of time.
     • Random Sampling
   • Time Series Data: Observations on a variable over time. Ex: Stock prices, money supply, consumer price index.
   • Pooled Cross Sections: Both cross-sectional and time series features. Ex: In 1985 a random sample of households survey for income, savings. In 1990, a new random sample of households with the same questions.
   • Panel or Longitudinal Data: Time series for each cross-sectional member in the data set. Example educational history for a set of individuals followed over a ten-year period.

4. Normal Distribution:μ = mean, σ = standard deviation, σ 2 = variance.

   • Central Limit Theorem: under mild conditions, the mean of a large number of random variables independently drawn from the same distribution is distributed approximately normally, irrespective of the form of the original distribution

5. Estimation Methods

   • Method of Moments: is a method of estimation of population parameters such as mean, variance, median, etc. (which need not be moments), by equating sample moments with unobservable population moments and then solving those equations for the quantities to be estimated.
   • Maximum likelihood estimation: is a method of estimating the parameters of a statistical model. When applied to a data set and given a statistical model, maximum-likelihood estimation provides estimates for the model's parameters.
   • Least Squares Method: is a standard approach to the approximate solution of overdetermined systems, i.e., sets of equations in which there are more equations than unknowns. "Least squares" means that the overall solution minimizes the sum of the squares of the errors made in the results of every single equation.

6. Statistic Inference

   • Confidence Interval: It is an observed interval (i.e. it is calculated from the observations), in principle different from sample to sample, that frequently includes the parameter of interest if the experiment is repeated.
     • Confidence Level: How frequently the observed interval contains the parameter is determined by the confidence level. the proportion of such intervals that contain the true value of the parameter will match the confidence level The most common: 90, 95, 99%
   • Statistical Hypothesis test for one mean: We look for a qualitative affirmation to see if the evidence is consistent or not with the hypothesis.
     • Two types of errors. a type I error (or error of the first kind) is the incorrect rejection of a true null hypothesis. A type II error (or error of the second kind) is the failure to reject a false null hypothesis
   • Hypothesis testing for equal means: We want to know if the evidence is consistent with two populations having the same mean.
   • ANOVA: Hypothesis testing for two or more equal means. used to analyze the differences between group means and their associated procedures (such as "variation" among and between groups

7. Descriptive Statistics

   • Describes the data in a sample, data already happened and are particular for the sample, not necessary generalizable to all the population
   • Histogram: Graph in which we measure the frequencies of a variable for certain discrete ranges
   • Measures of Central Tendency
     • Mean: the sum of all measurements divided by the number of observations in the data set
     • Medium: the middle value that separates the higher half from the lower half of the data set.
     • Mode: the most frequent value in the data set.
   • Variability Measures
     • Range: The range is simply the highest score minus the lowest score.
     • Variance: the variance is defined as the average squared difference of the scores from the mean.
     • Standard Deviation: is simply the square root of the variance
   • Other measures
     • Skewness: The skewness for a normal distribution is 0, and any symmetric data should have a skewness near zero. Negative values for the skewness indicate data that are skewed left and positive values for the skewness indicate data that are skewed right.
     • Kurtosis: The kurtosis for a standard normal distribution is 3.