2

Logistics

• Bring name placards to class
• Pre-class assignment 1 was due 15 minutes before the class
• Students are allowed up to 2 skips without penalty
• Assignments are graded pass/fail based on effort
• There is an in-class worksheet to complete during the class
• The worksheet must be picked up from the front of the class
• Students can take a photo or scan of the completed worksheet
• The worksheet must be submitted on MyCourses before the next class
• The worksheet is graded pass/fail based on accuracy

Today's Learning Objectives

• Descriptive Statistics
  • Including mean, variance, and standard deviation
  • Including median and quantiles
  • Including density functions
  • Including joint distributions
• Sample and Population
  • Including representativeness
  • Including sampling error

Descriptive Statistics

• Summarisation of information to make data understandable
• The objective is to reduce the amount of numbers while losing as little information as possible
• Stata's summarize command summarises and details data, and allows shortened formats (eg. sum earn, d)
• Descriptive statistics provides key data:
  • Sample mean which equals x = n 1/n ΣX(i=1)
  • Single number measures of variation
  • Selected quantiles

Measures of Variation

• Variance is: Var(x) = 1/n Σ(x-x̄)²
• Standard deviation is: SD(x) = √Var(x)
• Coefficient of variation involves normalizing the standard deviation with the mean to compare across variables:
  • CV(x) = SD(x) / x̄

Quantiles

• Quantile Q(p) represents the value below which a fraction p of observations fall
• Some quantiles have specific names, like the median
• Q(.5) is the median, where 50% of observations lie below this value
• Quartiles include Q(.25), Q(.5), and Q(.75)
• Deciles include Q(.1), Q(.2), ..., Q(.9)
• Percentiles include Q(.01), Q(.02), ..., Q(.99)
• The width of a distribution is characterized using percentile ratios
  • 90/10 ratio: Q(.9)/Q(.1) = 15
  • 90/50 ratio: Q(.9)/Q(.5) = 2.1
  • 50/10 ratio: Q(.5)/Q(.1) = 7

Density Functions (Discrete Variables)

• If the distribution of random variable X is discrete, its density function = fX(x) = P(X=x)
• The functions measure the probability that the random variable X takes a specific value x
• Conditions:
  • fX(x) ≥ 0
  • ∑ fX(x) = 1
• The probability that X takes a value within set A is P(X∈A) = ∑ fX(x) X∈A
• A histogram represents the empirical counterpart of a density function for a discrete variable

Histograms

• The height of each bar in a histogram represents the fraction of observations with a particular value
• The observations are divided into bins to draw a histogram of them
  • Each observation is allocated to a single bin, ensuring all observations are assigned
  • The width of the bin represents the range of values that observations within that bin can take
  • Changing the number of bins allows different perspectives on same data

Density Functions (Continuous Variables)

• The probability that X takes a value within a set A, if X is continuous, is P(X∈A) = ∫ fX(x)dx
• Continuous variables can take infinite values, therefore, X taking a specific value approaches zero P(X=x) = ∫fX(x)dx = 0
• For a probabilistic variable with small variation (h>0):
  • fX(x) ≈ P(X=x±h/2)
  • (X=x±h/2) means (x - h/2 ≤ X ≤ x+h/2)
  • This is the basis for the defintion of a kernel

Kernel Density Estimation

• A kernel density estimator represents a local average for a value - fₙ(x) = 1/n Σ K(x-xᵢ)
• bandwidth (h): Indicates how much data around x is used
• Kernel function (Kₙ): weights observations within the bandwidth
• Stata chooses an “optimal” bandwidth by default
  • Larger bandwidth disregards more data

Cumulative Density Function (CDF)

• The CDF for a continuous variable provides the fraction of observations with values of x below t with: Fx(t) = ∫ fX(s)ds
• For instance, in a standardized normal distribution:
  • Fx(-1) = 0.159
  • Fx(0) = 0.5
• Plotting all of these points will show the entire function for various values of x

Population and Sample

• Population: the entire group of N units about which conclusions are drawn
• Sample: a specific group of n units selected from the population to collect data
  • Aim is inference of the population
  • This refers to deducing or concluding information from evidence/reasoning instead of explicit statements
• sampling bias: sample must represent population
• sampling error: exceptional observations must be sampled by chance

Sampling Bias: 1936 US Presidential Election Polls

• Literary Digest sent 10 million "straw” ballots, asking who people intended to vote for in the upcoming election
  • 2.4 million ballots were returned
  • 57% indicated they would vote for Landon
  • 43% indicated they would vote for Roosevelt
• Fact: Roosevelt won the election with 62% of the vote
• George Gallup also conducted a poll
  • Sample size was just 50,000
  • Prediction: 56% for Roosevelt

Sampling Error

• Random sampling removes bias
  • each object in population has same probability of being selected into the sample
  • difference between a sample statistic and population parameter arising by chance

Sampling Error Example

• The population mean income among 15-64 year olds in Finland in 2010 (N ≈ 3.5M) was €26,144 µx = 1/n Σxᵢ
• Sample Average: x̄ = 1/n Σxᵢ
• The larger the smaple size, the closer the sample average tends to be to the population mean
• Sample averages are distributed relatively symmetrically around population mean
• These Properties are also known as:
  • The law of Large Numbers
  • Central Limit Theorem

Cross Tabulation (Joint Distribution)

• Table displays small data sets of two variables
• Each observation has:
  • Rows = values of Y
  • Columns = values of X
  • Cells = observations of value (y, x)
• Cross tabulation cells report the share of observations with value (y, x)
• fXY(x, y) = P(X = x, Y = y) shows that the empirical counterpart of the joint density function
• Probability: the random variable X takes the value x and that random variable Y takes the value y

Summary Of Key Concepts

• Concepts
  • density function, including CDF
  • joint distributions
• Things to worry about when using samples
  • representativeness
  • sampling error

The next lecture features:

• Conditional descriptive statistics
  • and applications to make sense of recent research on inequality

Upcoming Assignments

• In-class worksheet 1 due on MyCourses before next lecture
  • You may upload a photo/scan (preferred) OR
  • can turn in paper copy in-person beginning of next lecture
• Pre-class assignment 2 = due 15 minutes before next lecture
• Homework 1 = due Jan 15
  • attend exercise session 1 for practical tools
  • you have the conceptual tools to get started

Explore the basics of histograms, including the meaning of bar heights and how they are constructed. Understand the effects of changing the number of bins and the importance of proper data allocation. Learn about representing income distribution and probability in continuous random variables.