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2.1: Data 101 and Descriptive Statistics

ECON 480 · Econometrics · Fall 2019

Ryan Safner
Assistant Professor of Economics
safner@hood.edu
ryansafner/metricsf19
metricsF19.classes.ryansafner.com

Review From 1.2: Two Big Problems with Data

  • We want to use econometrics to identify causal relationships and make inferences about them

  1. Problem for identification: endogeneity

  2. Problem for inference: randomness

Review from 1.2: Identification Problem: Endogeneity

  • An independent variable (X) is exogenous if its variation is unrelated to other factors that affect the dependent variable (Y)

  • An independent variable (X) is endogenous if its variation is related to other factors that affect the dependent variable (Y)

Review from 1.2: Inference Problem: Randomness

  • Data is random due to natural sampling variation

    • Taking one sample of a population will yield slightly different information than another sample of the same population

  • Common in statistics, easy to fix

  • Inferential Statistics: making claims about a wider population using sample data

    • We use common tools and techniques to deal with randomness

The Two Problems: Where We're Heading...Ultimately

Sample→⏟statistical inferencePopulation→⏟causal indentificationUnobserved Parameters

  • We want to identify causal relationships between population variables

    • Logically first thing to consider
    • Endogeneity problem

  • We'll use sample statistics to infer something about population parameters

    • In practice, we'll only ever have a finite sample distribution of data
    • We don't know the population distribution of data
    • Randomness problem

Data 101

Data 101 I

  • Data are information with context

  • Individuals are the entities described by a set of data

    • e.g. persons, households, firms, countries

Data 101 I

  • Variables are particular characteristics about an individual

    • e.g. age, income, profits, population, GDP, marital status, type of legal institutions

  • Observations or cases are the separate individuals described by a collection of variables

    • e.g. for one individual, we have their age, sex, income, education, etc.

  • individuals and observations are not necessarily the same:

    • e.g. we can have separate observations on the same individual over time

Categorical Data

  • Categorical data place an individual into one of several possible categories

    • e.g. sex, season, political party
    • may be responses to survey questions
    • can be quantitative (e.g. age, zip code)

  • R calls these factors (we'll deal with them much later in the course)

Categorical Data: Visualizing I

Summary of diamonds by cut
cut n frequency
Fair 1610 0.0298480
Good 4906 0.0909529
Very Good 12082 0.2239896
Premium 13791 0.2556730
Ideal 21551 0.3995365

  • Good way to represent categorical data is with a frequency table

  • Count (n): total number of individuals in a category

  • Frequency: proportion of a category relative to all data

Categorical Data: Visualizing II

  • Charts and graphs are always better ways to visualize data

  • A bar graph represents categories as bars, with lengths proportional to the count or relative frequency fo each category

ggplot(diamonds, aes(x=cut,
                     fill=cut))+
  geom_bar()+
  guides(fill=F)+
  theme_bw(base_family = "Fira Sans Condensed",
           base_size=20)

Categorical Data: Visualizing III

  • Avoid pie charts!

  • People are not good at judging 2-d differences (angles, area)

  • People are good at judging 1-d differences (length)

Categorical Data: Visualizing III

  • Avoid pie charts!

  • People are not good at judging 2-d differences (angles, area)

  • People are good at judging 1-d differences (length)

Categorical Data: Visualizing IV

  • Maybe a stacked bar chart
diamonds %>%
  count(cut) %>%
ggplot(., aes(x="", y=n, fill=cut))+
  geom_col()+
  geom_label(aes(x="", y=n, label=cut), position = position_stack(), color="white")+
  guides(fill=F)+
  theme_void()

Categorical Data: Visualizing IV

  • Maybe lollipop chart
diamonds %>%
  count(cut) %>%
  mutate(cut_name = as.factor(cut)) %>%
ggplot(., aes(x = cut_name, y = n, color = cut))+
 geom_point(stat="identity",
            fill="black",
            size=12)  +
  geom_segment(aes(x = cut_name, y = 0,
                   xend = cut_name,
                   yend = n), size = 2)+
  geom_text(aes(label = n),color="white", size=3) +
  coord_flip()+
  labs(x = "Cut")+
  theme_classic(base_family = "Fira Sans Condensed",
                base_size=20)+
  guides(color = F)

Categorical Data: Visualizing IV

  • Maybe a treemap
library(treemapify)
diamonds %>%
  count(cut) %>%
ggplot(., aes(area = n, fill = cut)) +
  geom_treemap() +
  guides(fill = FALSE) +
  geom_treemap_text(aes(label = cut), 
                    colour = "white",
                    place = "center",
                    grow = TRUE)

Quantitative Data I

  • Quantitative variables take on numerical values of equal units that describe an individual

    • Units: points, dollars, inches
    • Context: GPA, prices, height

  • We can mathematically manipulate only quantitative data

    • e.g. sum, average, standard deviation

Context is Key!

  • How variables are classified depends on the purpose of collecting and using the data

Context is Key!

  • How variables are classified depends on the purpose of collecting and using the data

Quick Check: What kind of data (categorical or quantitative) does each variable describe?

Context is Key!

  • How variables are classified depends on the purpose of collecting and using the data

Quick Check: What kind of data (categorical or quantitative) does each variable describe?

  1. Age measured in years

Context is Key!

  • How variables are classified depends on the purpose of collecting and using the data

Quick Check: What kind of data (categorical or quantitative) does each variable describe?

  1. Age measured in years

  2. Age measured in ranges (0-9 years, 10-19 years, 20-29 years, etc)

Context is Key!

  • How variables are classified depends on the purpose of collecting and using the data

Quick Check: What kind of data (categorical or quantitative) does each variable describe?

  1. Age measured in years

  2. Age measured in ranges (0-9 years, 10-19 years, 20-29 years, etc)

  3. The date a purchase was made

Context is Key!

  • How variables are classified depends on the purpose of collecting and using the data

Quick Check: What kind of data (categorical or quantitative) does each variable describe?

  1. Age measured in years

  2. Age measured in ranges (0-9 years, 10-19 years, 20-29 years, etc)

  3. The date a purchase was made

  4. Transaction ID

Context is Key!

  • How variables are classified depends on the purpose of collecting and using the data

Quick Check: What kind of data (categorical or quantitative) does each variable describe?

  1. Age measured in years

  2. Age measured in ranges (0-9 years, 10-19 years, 20-29 years, etc)

  3. The date a purchase was made

  4. Transaction ID

  5. The amount of money spent on a Super Bowl ad

Context is Key!

  • How variables are classified depends on the purpose of collecting and using the data

Quick Check: What kind of data (categorical or quantitative) does each variable describe?

  1. Age measured in years

  2. Age measured in ranges (0-9 years, 10-19 years, 20-29 years, etc)

  3. The date a purchase was made

  4. Transaction ID

  5. The amount of money spent on a Super Bowl ad

  6. Customer ratings

Context is Key!

  • How variables are classified depends on the purpose of collecting and using the data

Quick Check: What kind of data (categorical or quantitative) does each variable describe?

  1. Age measured in years

  2. Age measured in ranges (0-9 years, 10-19 years, 20-29 years, etc)

  3. The date a purchase was made

  4. Transaction ID

  5. The amount of money spent on a Super Bowl ad

  6. Customer ratings

  7. The number of correct answers on an exam

Discrete Data

  • Discrete data are finite, with a countable number of alternatives

  • Categorical: e.g. letter grades A, B, C, D, F

  • Quantitative: integers, e.g. SAT Score, number of children

Continuous Data

  • Continuous data are infinitely divisible, with an uncountable number of alternatives

    • e.g. weights, temperature, GPA

  • Many discrete variables may be treated as if they are continuous

    • e.g. SAT scores, wages

Discrete or Continuous?

Quick Check: What kind of data (discrete or continuous) does each variable describe?

Discrete or Continuous?

Quick Check: What kind of data (discrete or continuous) does each variable describe?

  1. Weight in pounds

Discrete or Continuous?

Quick Check: What kind of data (discrete or continuous) does each variable describe?

  1. Weight in pounds

  2. Price in dollars

Discrete or Continuous?

Quick Check: What kind of data (discrete or continuous) does each variable describe?

  1. Weight in pounds

  2. Price in dollars

  3. Grade (Letter)

Discrete or Continuous?

Quick Check: What kind of data (discrete or continuous) does each variable describe?

  1. Weight in pounds

  2. Price in dollars

  3. Grade (Letter)

  4. Grade (Percentage)

Discrete or Continuous?

Quick Check: What kind of data (discrete or continuous) does each variable describe?

  1. Weight in pounds

  2. Price in dollars

  3. Grade (Letter)

  4. Grade (Percentage)

  5. Temperature

Discrete or Continuous?

Quick Check: What kind of data (discrete or continuous) does each variable describe?

  1. Weight in pounds

  2. Price in dollars

  3. Grade (Letter)

  4. Grade (Percentage)

  5. Temperature

  6. Amazon Star Rating

Discrete or Continuous?

Quick Check: What kind of data (discrete or continuous) does each variable describe?

  1. Weight in pounds

  2. Price in dollars

  3. Grade (Letter)

  4. Grade (Percentage)

  5. Temperature

  6. Amazon Star Rating

  7. Number of customers

Spreadsheets

ID Name Age Sex Income
1 John 23 Male 41000
2 Emile 18 Male 52600
3 Natalya 28 Female 48000
4 Lakisha 31 Female 60200
5 Cheng 36 Male 81900

  • The most common data structure we use is a spreadsheet

    • Note, in R: a data.frame or tibble

  • A row contains data about all variables for a single individual

  • A column contains data about a single variable across all individuals

Spreadsheets II

  • It is common to use some notation like the following:

  • Let {x1,x2,⋯,xn} be a simple data series on variable X

    • n individual observations
    • xi is the value of the ith observation for i=1,2,⋯,n

Spreadsheets II

  • It is common to use some notation like the following:

  • Let {x1,x2,⋯,xn} be a simple data series on variable X

    • n individual observations
    • xi is the value of the ith observation for i=1,2,⋯,n

Quick Check: Let x represent the score on a homework assignment: 75,100,92,87,79,0,95

  1. What is n?
  2. What is x1?
  3. What is x6?

Datasets: Cross-Sectional

ID Name Age Sex Income
1 John 23 Male 41000
2 Emile 18 Male 52600
3 Natalya 28 Female 48000
4 Lakisha 31 Female 60200
5 Cheng 36 Male 81900

  • Cross-sectional data: observations of individuals at a given point in time

  • Each observation is a unique individual

  • Simplest and most common data

  • A "snapshot" to compare differences across individuals

Datasets: Time-Series

Year GDP Unemployment CPI
1950 8.2 0.06 100
1960 9.9 0.04 118
1970 10.2 0.08 130
1980 12.4 0.08 190
1985 13.6 0.06 196

  • Time-series data: observations of the same individuals over time

  • Each observation is an individual-year

  • Often used for macroeconomics, finance, and forecasting

  • Unique challenges for time series

  • A "moving picture" to see how individuals change over time

Datasets: Panel

City Year Murders Population Unemployment
Philadelphia 1986 5 3.700 8.7
Philadelphia 1990 8 4.200 7.2
D.C. 1986 2 0.250 5.4
D.C. 1990 10 0.275 5.5
New York 1986 3 6.400 9.6

  • Panel, or longitudinal dataset: a time-series for each cross-sectional entity

  • Must be the same cross-sectional entities over time

  • More common today for serious researchers

  • Unique challenges for panel data

  • A combination of "snapshot" comparisons and differences over time

Descriptive Statistics

Variables and Distributions

  • Variables take on different values, we can describe a variable's distribution (of these values)

  • We want to visualize and analyze distributions to search for meaningful patterns using statistics

Two Branches of Statistics

  • Two main branches of statistics:

  1. Descriptive Statistics: describes or summarizes the properties of a sample

  2. Inferential Statistics: infers properties about a larger population from the properties of a sample1

1 We'll encounter inferential statistics mainly in the context of regression later.

Histograms

  • A common way to present a quantitative variable's distribution is a histogram

    • The quantitative analog to the bar graph for a categorical variable

  • Divide up values into bins of a certain size, and count the number of values falling within each bin, representing them visually as bars

Histogram: Example

Example: a class of 13 students takes a quiz (out of 100 points) with the following results:

{0,62,66,71,71,74,76,79,83,86,88,93,95}

Histogram: Example

Example: a class of 13 students takes a quiz (out of 100 points) with the following results:

{0,62,66,71,71,74,76,79,83,86,88,93,95}

quizzes<-tibble(scores = c(0,62,66,71,71,74,76,79,83,86,88,93,95))

Histogram: Example

Example: a class of 13 students takes a quiz (out of 100 points) with the following results:

{0,62,66,71,71,74,76,79,83,86,88,93,95}

h<-ggplot(quizzes,aes(x=scores))+
  geom_histogram(breaks = seq(0,100,10),
                 color = "black",
                 fill = "#56B4E9")+
  scale_x_continuous(breaks = seq(0,100,10))+
  labs(x = "Scores",
       y = "Number of Students")+
  theme_bw(base_family = "Fira Sans Condensed",
           base_size=20)
h

Descriptive Statistics

  • We are often interested in the shape or pattern of a distribution, particularly:
    • Measures of center
    • Measures of dispersion
    • Shape of distribution

Mode

  • The mode of a variable is simply its most frequent value

  • A variable can have multiple modes

Mode

  • The mode of a variable is simply its most frequent value

  • A variable can have multiple modes

Example: a class of 13 students takes a quiz (out of 100 points) with the following results:

{0,62,66,71,71,74,76,79,83,86,88,93,95}

Mode

  • There is no dedicated mode() function in R, surprisingly

  • A workaround in dplyr: 

quizzes %>%
  count(scores) %>%
  arrange(desc(n))
## # A tibble: 12 x 2
##    scores     n
##     <dbl> <int>
##  1     71     2
##  2      0     1
##  3     62     1
##  4     66     1
##  5     74     1
##  6     76     1
##  7     79     1
##  8     83     1
##  9     86     1
## 10     88     1
## 11     93     1
## 12     95     1

Multi-Modal Distributions

  • Looking at a histogram, the modes are the "peaks" of the distribution

    • Note: depends on how wide you make the bins!

  • May be unimodal, bimodal, trimodal, etc

tibble(scores=c(0,33,33,33,33,35,62,66,71,71,74,76,79,83,86,88,93,95)) %>%
  count(scores) %>%
  arrange(desc(n))
## # A tibble: 14 x 2
##    scores     n
##     <dbl> <int>
##  1     33     4
##  2     71     2
##  3      0     1
##  4     35     1
##  5     62     1
##  6     66     1
##  7     74     1
##  8     76     1
##  9     79     1
## 10     83     1
## 11     86     1
## 12     88     1
## 13     93     1
## 14     95     1

Symmetry and Skew I

  • A distribution is symmetric if it looks roughly the same on either side of the "center"

  • The thinner ends (far left and far right) are called the tails of a distribution

Symmetry and Skew I

  • If one tail stretches farther than the other, distribution is skewed in the direction of the longer tail

Outliers

  • Outlier: extreme value that does not appear part of the general pattern of a distribution

  • Can strongly affect descriptive statistics

  • Might be the most informative part of the data

  • Could be the result of errors

  • Should always be explored and discussed!

Measures of Center

Arithmetic Mean (Population)

  • The natural measure of the center of a population's distribution is its "average" or arithmetic mean (μ)

μ=x1+x2+...+xNN=1NN∑i=1xi

  • For N values of variable x, "mu" is the sum of all individual x values (xi) from 1 to N, divided by the N number of values1

  • See today's class notes for more about the summation operator, Σ, it'll come up again!

1 Note the mean need not be an actual value of the data!

Arithmetic Mean (Sample)

  • When we have a sample, we compute the sample mean (ˉx)

ˉx=x1+x2+...+xnn=1nn∑i=1xi

  • For n values of variable x, "x-bar" is the sum of all individual x values (xi) divided by the n number of values

Arithmetic Mean (Sample)

  • When we have a sample, we compute the sample mean (ˉx)

ˉx=x1+x2+...+xnn=1nn∑i=1xi

  • For n values of variable x, "x-bar" is the sum of all individual x values (xi) divided by the n number of values

Example:

{0,62,66,71,71,74,76,79,83,86,88,93,95}

ˉx=113(0+62+66+71+71+74+76+79+83+86+88+93+95)ˉx=94413ˉx=72.62

Arithmetic Mean (Sample)

  • When we have a sample, we compute the sample mean (ˉx)

ˉx=x1+x2+...+xnn=1nn∑i=1xi

  • For n values of variable x, "x-bar" is the sum of all individual x values (xi) divided by the n number of values

Example:

{0,62,66,71,71,74,76,79,83,86,88,93,95}

ˉx=113(0+62+66+71+71+74+76+79+83+86+88+93+95)ˉx=94413ˉx=72.62

quizzes %>%
  summarize(mean=mean(scores))
## # A tibble: 1 x 1
##    mean
##   <dbl>
## 1  72.6

Arithmetic Mean: Affected by Outliers

  • If we drop the outlier (0)

Arithmetic Mean: Affected by Outliers

  • If we drop the outlier (0)

Example:

{62,66,71,71,74,76,79,83,86,88,93,95}

ˉx=112(62+66+71+71+74+76+79+83+86+88+93+95)=94412=78.67

Arithmetic Mean: Affected by Outliers

  • If we drop the outlier (0)

Example:

{62,66,71,71,74,76,79,83,86,88,93,95}

ˉx=112(62+66+71+71+74+76+79+83+86+88+93+95)=94412=78.67

quizzes %>%
  filter(scores>0) %>%
  summarize(mean=mean(scores))
## # A tibble: 1 x 1
##    mean
##   <dbl>
## 1  78.7

Median

{0,62,66,71,71,74,76,79,83,86,88,93,95}

  • The median is the midpoint of the distribution

    • 50% to the left of the median, 50% to the right of the median

  • Arrange values in numerical order

    • For odd n: median is middle observation
    • For even n: median is average of two middle observations

Mean, Median, and Outliers

Mean, Median, Symmetry, Skew I

  • A symmetric distribution has mean ≈ median
symmetric %>%
  summarize(mean = mean(x),
            median = median(x))
## # A tibble: 1 x 2
##    mean median
##   <dbl>  <dbl>
## 1     4      4

Mean, Median, Symmetry, Skew II

  • A left-skewed distribution has mean < median
leftskew %>%
  summarize(mean = mean(x),
            median = median(x))
##       mean median
## 1 4.615385      5

Mean, Median, Symmetry, Skew III

  • A right-skewed distribution has mean > median
rightskew %>%
  summarize(mean = mean(x),
            median = median(x))
## # A tibble: 1 x 2
##    mean median
##   <dbl>  <dbl>
## 1  3.38      3

Measures of Spread

Measures of Spread: Range

  • The more variation in the data, the less helpful a measure of central tendency will tell us
  • Beyond just the center, we also want to measure the spread
  • Simplest metric is range =max−min

Measures of Spread: 5 Number Summary I

A common set of summary statistics about a distribution is known as the "five number summary":

  1. Minimum value
  2. 25th percentile (Q1, the median of the first 50% of data)
  3. 50th percentile (median, Q2)
  4. 25th percentile (Q3, the median of the last 50% of data)
  5. Maximum value

Measures of Spread: 5 Number Summary I

A common set of summary statistics about a distribution is known as the "five number summary":

  1. Minimum value
  2. 25th percentile (Q1, the median of the first 50% of data)
  3. 50th percentile (median, Q2)
  4. 25th percentile (Q3, the median of the last 50% of data)
  5. Maximum value
# Base R summary command (includes Mean)
summary(quizzes$scores)
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##    0.00   71.00   76.00   72.62   86.00   95.00
quizzes %>% # dplyr
  summarize(Min = min(scores),
            Q1 = quantile(scores, 0.25),
            Median = median(scores),
            Q3 = quantile(scores, 0.75),
            Max = max(scores))
## # A tibble: 1 x 5
##     Min    Q1 Median    Q3   Max
##   <dbl> <dbl>  <dbl> <dbl> <dbl>
## 1     0    71     76    86    95

Measures of Spread: 5 Number Summary II

  • The nth percentile of a distribution is the value that places n percent of values beneath it
quizzes %>%
  summarize("37th percentile" = quantile(scores,0.37))
## # A tibble: 1 x 1
##   `37th percentile`
##               <dbl>
## 1              72.3

Boxplots I

  • Boxplots are a great way to visualize the 5 number summary

  • Height of box: Q1 to Q3 (known as interquartile range (IQR), middle 50% of data)

  • Line inside box: median (50th percentile)

  • "Whiskers" identify data within 1.5×IQR

  • Points beyond whiskers are outliers

    • common definition: Outlier>1.5×IQR

Comparisons I

  • Boxplots (and five number summaries) are great for comparing two distributions

Example:

Quiz 1:{0,62,66,71,71,74,76,79,83,86,88,93,95}Quiz 2:{50,62,72,73,79,81,82,82,86,90,94,98,99}

Comparisons II

quizzes_new %>% summary()
##     student       quiz_1          quiz_2     
##  Min.   : 1   Min.   : 0.00   Min.   :50.00  
##  1st Qu.: 4   1st Qu.:71.00   1st Qu.:73.00  
##  Median : 7   Median :76.00   Median :82.00  
##  Mean   : 7   Mean   :72.62   Mean   :80.62  
##  3rd Qu.:10   3rd Qu.:86.00   3rd Qu.:90.00  
##  Max.   :13   Max.   :95.00   Max.   :99.00

Aside: Making Nice Summary Tables I

  • I don't like the options available for printing out summary statistics

  • So I wrote my own R function that makes nice summary tables that uses dplyr and tidyr

  • One day I will release as a package; until then the .R file is saved here

# loads .R files to full functions from; make sure it's in YOUR working directory/Project!
source("../files/summaries.R") # MY path (this website's R Project)
# let's summarize variables "hwy" and "cty" from *mpg* dataset
mpg %>%
  summary_table(hwy, cty)
## # A tibble: 2 x 9
##   Variable   Obs   Min    Q1 Median    Q3   Max  Mean `Std. Dev.`
##   <chr>    <dbl> <dbl> <dbl>  <dbl> <dbl> <dbl> <dbl>       <dbl>
## 1 cty        234     9    14     17    19    35  16.9        4.26
## 2 hwy        234    12    18     24    27    44  23.4        5.95

Aside: Making Nice Summary Tables II

  • And when knitted in R markdown:
mpg %>%
  summary_table(hwy, cty) %>%
  knitr::kable(., format="html")
Variable Obs Min Q1 Median Q3 Max Mean Std. Dev.
cty 234 9 14 17 19 35 16.86 4.26
hwy 234 12 18 24 27 44 23.44 5.95
  • We'll talk more about using markdown and making final products nicer when we discuss your paper project (have you forgotten?)

Deviations

  • Every observation i deviates from the mean of the data: deviationi=xi−μ

  • There are as many deviations as there are data points (n)

  • We can measure the average or standard deviation of a variable from its mean

  • Before we get there...

Variance (Population)

  • The population variance (σ2) of a population distribution measures the average of the squared deviations from the population mean (μ)

σ2=1NN∑i=1(xi−μ)2

  • Why do we square deviations?

  • What are these units?

Standard Deviation (Population)

  • Square root the variance to get the population standard deviation (σ), the average deviation from the population mean (in same units as x)

σ=√σ2=√1NN∑i=1(xi−μ)2

Variance (Sample)

  • The sample variance (s2) of a sample distribution measures the average of the squared deviations from the sample mean (ˉx)

σ2=1n−1n∑i=1(xi−ˉx)2

  • Why do we divide by n−1?

Standard Deviation (Sample)

  • Square root the sample variance to get the sample standard deviation (s), the average deviation from the sample mean (in same units as x)

s=√s2=√1n−1n∑i=1(xi−ˉx)2

Sample Standard Deviation: Example

Example: Calculate the sample standard deviation for the following series:

{2,4,6,8,10}

Sample Standard Deviation: Example

Example: Calculate the sample standard deviation for the following series:

{2,4,6,8,10}

sd(c(2,4,6,8,10))
## [1] 3.162278

The Steps to Calculate sd(), Coded I

#  first let's save our data in a tibble
sd_example<-tibble(x=c(2,4,6,8,10))
# first find the mean (just so we know)
sd_example %>%
  summarize(mean(x))
## # A tibble: 1 x 1
##   `mean(x)`
##       <dbl>
## 1         6
# now let's make some more columns:
sd_example <- sd_example %>%
  mutate(deviations = x-mean(x), # take deviations from mean
         deviations_sq = deviations^2) # square them

The Steps to Calculate sd(), Coded II

sd_example # see what we made
## # A tibble: 5 x 3
##       x deviations deviations_sq
##   <dbl>      <dbl>         <dbl>
## 1     2         -4            16
## 2     4         -2             4
## 3     6          0             0
## 4     8          2             4
## 5    10          4            16

The Steps to Calculate sd(), Coded III

sd_example %>%
  # sum the squared deviations
  summarize(sum_sq_devs = sum(deviations_sq), 
            # divide by n-1 to get variance
            variance = sum_sq_devs/(n()-1), 
            # square root to get sd
            std_dev = sqrt(variance))
## # A tibble: 1 x 3
##   sum_sq_devs variance std_dev
##         <dbl>    <dbl>   <dbl>
## 1          40       10    3.16

Sample Standard Deviation: You Try

You Try: Calculate the sample standard deviation for the following series:

{1,3,5,7}

Sample Standard Deviation: You Try

You Try: Calculate the sample standard deviation for the following series:

{1,3,5,7}

sd(c(1,3,5,7))
## [1] 2.581989

Descriptive Statistics: Populations vs. Samples

Population parameters

  • Population size: N

  • Mean: μ

  • Variance: σ2=1NN∑i=1(xi−μ)2

  • Standard deviation: σ=√σ2

Sample statistics

  • Population size: n

  • Mean: ˉx

  • Variance: s2=1n−1n∑i=1(xi−ˉx)2

  • Standard deviation: s=√s2

Review From 1.2: Two Big Problems with Data

  • We want to use econometrics to identify causal relationships and make inferences about them

  1. Problem for identification: endogeneity

  2. Problem for inference: randomness

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