Quantative methods for economics

population

the complete set of units (people, firms, etc.) we want

to study

sample

a subset of the population examined to learn about the

population.

representative sample

a sample that mirrors the population on

relevant characteristics

sampling bias 

systematic under- or over-representation of some

population members.

statistic

either a function applicable to data or the result of that

function, i.e. a number

parameter

a numerical characteristic of a population that a

statistic aims to estimate

Qualitive (categorical) data

the result of categorising or describing attributes of a population

Quantative (numerical) data

the result of counting or measuring attributes of a population

variable 

a characteristic of a unit being observed that may assume more than one set of values for each member of the population 

numerical variable

takes on values with equal units such as petals per flower

categorical variable

place a person or thing into a category such as colour of flower

data

the observed value of the variable(s)

quantative discrete variables

take on only certain numerical values, e.g. calls per week 

quantative continuous variables

take on all values in a defined range, e.g, length, weight, time

median

middle value seperating the greater and lesser halves of a data set

mode

most frequent value in a data set

function

a rule that assigns to each input exactly one output. it comes with a domain (allowed inputs) and a codomain (possible outputs). The set of outputs actually attained is the range. (image)

Domain and range

a function that maps every element in the domain to exactly 1 element in the range. Although each input can be sent to only one output, 2 different inputs can be sent to same output

statistical functions

when we aggregate data, we take a high dimensional domain and map it to a low dimensional range.

bar graph

the length of the bar for each category is proportional to the number or percent of individuals in each category. Bars may be vertical or horizontal. include the zero in the bar chart

simple random sampling

picking individuals out of proportion with equal chance

the problem, sampling bias

some members of population are not as likely to be chosen as others and we do not account for it

common type of sampling bias

self-selection, exclusion, survivorship

simple random sample

any group of individuals is equally likely to be chosen as any other groups of individuals

proportionate stratisified sample

divide the population into groups called strata and then take a proportionate number from each stratum. Advantage: sample is representative along the characteristic used for stratisfication

disproportionate stratisfied sample

over-sample (pick individuals with a higher chance from) groups with large variance, e.g, smaller groups. Leads to biased results if not adjusted

cluster sample

divide population into clusters (groups) and then randomly select some of the clusters. Include all the members from these clusters

convinience sample

use results that are readily available (already collected). cheaper but might be biased 

distribution

a description of how often each outcome occurs

empirical cumulative distribution function(ECDF)

A standard representation for an empirical (observed) distribution

Histograms

divides the span of our data into non-overlapping bins of the same size. Then, for each bin, we count the number of values that fall into that interval. The histogram plots these counts as bars with the base of the bar defined by the intervals, histograms are preffered over EDCFs as they are easier to interpret

Smooth density plots

basically smoothing out the edges of a histogram

Advantage of smooth density plot

prettier and easier to compate several distributions as less messy

Disadvantage of smooth density plot

Interpretation slightly more difficult, form is dependent on underlying smoothing, never a good idea to use methods we dont understand well 

what does it mean when data is not pretty?

it means asymetric for instance or carrying outlines

percentiles

the values for which p=0.01, 0.02,…0.99 of the data are less than or equal to that value respectively

median (percentile)

the most often used percentile is the 50% percentile, called the median

quaritiles

these are the percentiles at p= 0.025, 0.5, 0.75

range

the difference between the largest value and smallest value

box plot

provides a 5 number summary for data composed of the range along with quarities

stratification

often when we divide observations into groups based on the values of one or more variables associated with these observations

variance 

a measure of variation in the population. It is defined as the sum of squared deviations from the mean divided by the number of units 

standard deviation

the square root of the variance

what is the function of standard deviation?

it provides a numerical measure of the overall amount of variation in a data set, always positive or zero, it is small when the data are all concentrated close to the mean, exhibiting little variation or spread, can also be used to determine whether a particular data value is close to or far from the mean