19  Population Distributions

In statistics, we use the phrase population to refer to the group of folks/organizations/countries we are interested in studying. The population does not necessarily imply that we are talking about all of the countries in the world or all of the citizens in a given jurisdiction. It really is just the group of interest.

For example, say that I am interested in studying Black Americans’ political attitudes on federal taxes. My population of interest would not include non-Black Americans. Another example would be that if I wanted to understand the voting intentions for American voters, my population does not include every American as there are a number of disenfranchised populations that are not allowed to vote. If I want to examine whether the EU decreased interstate conflict between those countries, I would not include countries outside of the EU in my analysis.

19.1 Population distribution

# Import needed functions
import plotly.express as px # for making plotly histograms
import pandas as pd # for manipulating pd.DataFrame objects
# Import needed objects
from inference_data import population_N, population_data # to access population distribution data
# Convert to ojs object
ojs_define(population_data = population_data)

# Print calculated percentage of example
print("Let's try visualizing this. Say that we have identified a population that we are interested in studying that has {:.0f}".format(population_N) + " people in it (N = {:.0f}".format(population_N) +")")

Let’s try visualizing this. Say that we have identified a population that we are interested in studying that has 1000000 people in it (N = 1000000)

Say we want to analyze three characteristics (variables) of observations in the population. Figure 19.1 displays the distribution of our population if we could collect all of the data. In practice, this is often theoretical (based on our expertise). Meaning, we rarely are able to collect data on our full population.

Plot.plot({
    grid: true,
    marks: [
        Plot.frame(),
        Plot.rectY(
            transpose(population_data),
            Plot.binX(
                {y: "count"},
                {x: "data.normal",
                    fill:"#D3D3D3",
                    strokeWidth: 0.5,
                    stroke: "#000000",
                    thresholds: 10}
            ))
    ]
})

(a) Normal distribution

Plot.plot({
  grid: true,
  marks: [
    Plot.frame(),
    Plot.rectY(transpose(population_data), Plot.binX(
      {y: "count"},
      {x: "data.poisson",
       fill: "#D3D3D3",
       strokeWidth: .5, // linewidth = 0.5
       stroke: "#000000", // equivalent of edgecolor="black"
       thresholds: 10 // equivalent of "bins=10"
      }
    ))
  ]
})

(b) Poisson distribution

Plot.plot({
  grid: true,
  marks: [
    Plot.frame(),
    Plot.rectY(transpose(population_data), Plot.binX(
      {y: "count"},
      {x: "data.binomial",
       fill: "#D3D3D3",
       strokeWidth: .5, // linewidth = 0.5
       stroke: "#000000", // equivalent of edgecolor="black"
       thresholds: 10 // equivalent of "bins=10"
      }
    ))
  ]
})

(c) Binomial distribution

Figure 19.1: Population Distributions

Just for our records, let’s calculate the mean and the standard deviation of each variable for our population. Again, we are rarely ever able to actually do this.

print(
    "Normal distribution variable mean: {:.0f}".format(population_data["data.normal"].mean()) + ", standard deviation: {:.0f}".format(population_data["data.normal"].std())
)

Normal distribution variable mean: 0, standard deviation: 1

print(
    "Poisson distribution variable mean: {:.0f}".format(population_data["data.poisson"].mean()) + ", standard deviation: {:.0f}".format(population_data["data.poisson"].std()),
)

Poisson distribution variable mean: 1, standard deviation: 1

print(
    "Binomial distribution variable mean: {:.0f}".format(population_data["data.binomial"].mean()) + ", standard deviation: {:.0f}".format(population_data["data.binomial"].std())
)

Binomial distribution variable mean: 0, standard deviation: 1