Sampling

April 23, 2024

Sampling

  • Sampling the act of selecting a subset of individuals, items, or data points from a larger population to estimate characteristics or metrics of the entire population
  • Versus a census, which involves gathering information on every individual in the population
  • Why would you want to use a sample?

Today’s code is HERE

Sampling Activity

What proportion of all milk chocolate M&Ms are blue?

  • M&Ms has a precise distribution of colors that it produces in its factories
  • M&Ms are sorted into bags in factories in a fairly random process
  • Each bag represents a sample from the full population of M&Ms

Activity

  • Get in groups of 2. Each group will have 4-5 bags of M&Ms.
  • Keep the contents of each bag separate, and do not eat (yet!)
  • Open up your first bag of M&Ms: calculate the proportion of the M&Ms that are blue. Write this down. What is your best guess (your estimate) for the proportion of all M&Ms that are blue?

Activity

  • Do the same as above for the rest of your bags (you should have 4-5 estimates written down)
  • Draw a histogram of your estimates (by hand)
  • Add your estimates to this Google Sheet
  • Add your estimates to the class histogram on the whiteboard

Let’s Analyze the Data


We will use the googlesheets4 package to pull the data into R so be sure to install it.

#install.packages("googlesheets4")

library(tidyverse)
library(googlesheets4)

gs4_deauth() # to signify no authorization required

mnm_data <- read_sheet("https://docs.google.com/spreadsheets/d/136wGKZOnwOdo3O-4bUfF_TWKBCAWQUCyiVKSY4HyYAw/edit#gid=0")

glimpse(mnm_data)

Calculate Some Summary Stats


mnm_data |>
  summarize(
    mean_blue = mean(proportion_blue),
    median_blue = median(proportion_blue),
    sd_blue = sd(proportion_blue)
  )

Now Let’s Make a Histogram


ggplot(mnm_data, aes(x = proportion_blue)) +
  geom_histogram(fill = "steelblue") +
  labs(
    title = "Percentage of Blue M&Ms",
    x = "Proportion Blue",
    y = "Count"
  )

Discuss with Neighbor

  • What is the histogram/distribution showing?
  • Based on the histogram on the board, what is your answer to the question of what proportion of all milk chocolate M&Ms are blue? Why do you give that answer?
  • Why do some bag of M&Ms have proportions of blues that are higher and lower than the number you gave above?
  • How do our estimates relate to the actual percentage of blue M&Ms manufactured (ask Google or ChatGPT)

What did we just do?

  • We wanted to say something about the population of M&Ms
  • The parameter we care about is the proportion of M&Ms that are blue
  • It would be impossible to conduct a census and to calculate the parameter
  • We took a sample from the population and calculated a sample statistic
  • statistical inference: act of making a guess about a population using information from a sample

What have we just done?

  • We completed this task many times
  • This produced a sampling distribution of our estimates
  • There is a distribution of estimates because of sampling variability
  • Due to random chance, one estimate from one sample can differ from another
  • These are foundational ideas for statistical inference that we are going to keep building on

Zooming Out

Target Population

In data analysis, we are usually interested in saying something about a Target Population.

  • What proportion of adult Russians support the war in Ukraine?
    • Target population: adult Russians (age 18+)
  • How many US college students check social media during their classes?
    • Target population: US college students
  • What percentage of M&Ms are blue?
    • Target population: all of the M&Ms

Sample


In many instances, we have a Sample

  • We cannot talk to every Russian
  • We cannot talk to all college students
  • We cannot count all of the M&Ms

Parameters vs Statistics


  • The parameter is the value of a calculation for the entire target population
  • The statistic is what we calculate on our sample
    • We calculate a statistic in order to say something about the parameter

Inference


  • Inference–The act of “making a guess” about some unknown
  • Statistical inference–Making a good guess about a population from a sample
  • Causal inference–Did X cause Y? [topic for later classes]

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