Descriptive Statistics

• Definition 1: A statistic is a quantity computed from a sample that can be used for a statistical purpose (inference if you’ve heard the term, otherwise you can loosely view inference as decision making).
• Definition 2: A descriptive statistic is a statistic that summarizes or conveys information about some aspect of a sample of data

Popular types of information to convey using descriptive statistics are:

1. Central tendency - mean or median value
2. Variability - standard deviation, range, interquartile range
3. Frequencies or relative frequencies - counts in a category or range, proportions, rates
4. Relationships - correlation coefficient, difference in means or proportions, odds ratio/relative risk

The goal of any descriptive statistic is to accurately convey important information without needing to showcase the entire set of data. For example, a correlation coefficient can provide a good summary of the relationship in a scatter plot (though not in all cases!)

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Practice

Descriptive statistics are an essential tool when working data; however, there are often many different descriptive statistics that could be used to convey information. It is important to understand the types of choices that can be made and their implications.

Scenario 1: Suppose you’d like to describe how financially well off the residents of a particular neighborhood are. Additionally, suppose you have the power to ascertain any information about each resident that you think might be useful. What you share to most accurately convey this aspect of the neighborhood?

• Question: Each of the following are descriptive statistics one might consider in this scenario. Identify one or more limitations of each (assuming your goal is to provide accurate information), then choose the 1 or 2 descriptive statistics that you feel are most effective.
  1. The aggregate income, which is the sum of the annual incomes of all residents of the neighborhood
  2. The average individual income, which is the aggregate income divided by the number of individual residents
  3. The average household income, which is the aggregate income divided by the number of households (which each may contain one or more residents)
  4. The total wealth, which is the sum of the net worth (assets - debts) for every resident
  5. The average individual wealth, which is the total wealth divided by the number of individual residents
  6. The average household wealth, which is the total wealth divided by the number of households

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• Remarks: Question 1 can be broken down into 3 different choices:
  • choice of the variable/attribute to focus on
  • use of raw/crude numbers vs. rates/scaled numbers
  • choice of denominator when opting for a rate or scaled number

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Scenario 2: Suppose a new disease has emerged in your community and you are trying to decide how worried you should be. Further, suppose you have the ability ascertain any descriptive statistic related to the impacts of this disease without any barriers or challenges; however, you may only rely upon one descriptive statistic.

• Question:
  • Part A: What variable/attribute would you focus on? Some examples might include infections, or deaths, or hospitalizations, etc.
  • Part B: Would you use a raw/crude number or a rate/scaled number?
  • Part C: What descriptive statistic would you ultimately rely upon?

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• Remarks: In reality you aren’t limited in how many descriptive statistics you can share in the context of a particular argument. However, more is not better.
  • The Gish gallop is an argumentation strategy that attempts to overwhelm someone by providing an an excessive amount of information with no regard for quality or accuracy
  • Brandolini’s law is the idea that the amount of effort required to refute a faulty claim is orders of magnitude greater than the effort needed to make the faulty claim.
  • An awareness of these ideas can help shape the way you approach argumentation/debate.