Statistical Parameter: A True Measure of Population Value

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A parameter is a value that conveys information about an entire group, also known as a statistical population. Parameters describe or summarize aspects of a statistical population such as average, standard deviation, or total population count. Parameters describe true values calculated from the entire of a specific group.

Parameters are sometimes considered unobservable and require estimations to be calculated from smaller parts (samples) of the larger population—estimators. An example of an estimator would be the Exponential Moving Average (EMA) of a stock.

This estimator is calculated from the entire pricing history (population) of the security but isn’t itself a static value because it reflects only a sampling of the entire pricing history (generally 12, 26, 50, or 200 pricing observations.) An estimator has no business being equal to the value from which it is derived—it is synonymous with a sample statistic.

Types of Parameters

Statistical Parameters can be grouped into one of two broad categories: measures of central tendency and measures of variation. The first describes observed values as a function of their middlemost value; the most common car color, the average fuel economy of mid-range SUVs, or the most common salary among all employees including upper management, middle management, owners, full-time employees, and temporary contractors.

Measures of Central Tendency

Measures of central tendency strive to describe a population or sample population by identifying the central position of observed values. For example, the average previous closing price among assets in an index fund would be a measure of central tendency. Below are the three most common measures of central tendency that are commonly used as statistical parameters:

  • Mean: The sum of all observed values divided by the number of observations.
  • Median: The middle observed value of all values arranged in order of magnitude.
  • Mode: The most frequently observed value among all observed values.

Measures of Variation

Measures of variation describe how far, wide, or deep a population of observed values varies from their central values. For example, the average of three securities priced at $100, $150, and $200 is $150—fairly representative. However, the average of three securities priced at $0.015, $0.045, and $1,500 is $500—not very representative. Measures of variation convey how variable a collection of observed values are, on average, from their central value (usually the mean.) Below are common measures of variation:

Range: The difference between the largest and smallest observed values.

Standard Deviation: Average amount that observed value varies from the mean.

Variance: Average squared amount that observed values vary from the mean.

Population Parameters vs. Sample Estimators

Parameters are referred to as true values in that they represent the totality of an observed value of a population. The average return of every asset in a portfolio, the average height of all employees of a Fortune 500 company, the average salary of every CEO in the United States; are all values that represent a total population.

When data representing an entire population isn’t available we rely on statistical sampling to estimate population values. The average return of half the assets in a portfolio—used to estimate the average return on the other half; the average height of the accounting department of a single Fortune 500 company—used to predict the average height of every employee at all Fortune 500 companies; the average salary of 10 CEOs—used to predict the salary of all CEOs; these are the applications of estimators.

Access to entire data for entire populations is often not available or feasible to collect. Sample statistics serve as estimators for population values while still considered a statistic for the sample population. In other words, the average value of a population is a parameter; the average value of a sample population is a statistic of that sample population, but an estimator for the population value. It’s a spaghetti bowl of vocabulary but also used to draw important distinctions.

References

  1. Coolidge, Frederick. Statistics: A Gentle Introduction. 4th ed., SAGE Publications, Inc, 2020.
Zack West
Entrepreneur, programmer, designer, and lifelong learner. Can be found taking notes from Mother Nature when not hammering away at the keyboard.