Definitions
Statistics people like to use all sorts of jargon, so here are a couple of terms. A parameter refers to data that is used to describe something of interest in the entire population, such as the average. However, if the data describes a sample from the population, it’s called a statistic.
For example, I have a large class of 300 students, and I want to find the average grade for this class. If I use the grades of all 300 students, then the average I get is called a parameter. If I use a sample of 50 students, then the average I get is called a statistic, which I use as an estimate of the parameter.
Differences
Think of parameters as the "true values" we wish we could know about an entire population. Unfortunately, measuring everyone is impossible, so we settle for statistics, which are the numbers we calculate from a sample, manageable group we actually study.
Since we can’t measure everyone, we calculate statistics from a sample to estimate those parameters. Statistics are our best guesses for the unknown truths.
Here’s the key differences between parameter and statistic:
Aspect | Population Parameter | Sample Statistic |
Definition | A numerical feature describing an entire population. | A numerical value derived from a sample within the population. |
Nature | Constant and unchanging | Can vary across different samples |
Value | Generally unknown for the full population | Known, as it is based on sample data |
Representation | Encompasses the full population | Reflects a subset of the population |
Usage | Primarily applied in theoretical and hypothesis-based work | Used for real-world analysis and research purposes |
Accuracy | More reliable due to representing the entire population | Can fluctuate depending on the sample’s characteristics |
Calculation | Cannot be precisely determined in most practical settings | Can be computed using data from the sample |
Purpose | Aims to describe the overall traits of the population | Aims to infer characteristics of the population based on sample data |
Example | The true average height of all adults in a given country | The average height calculated from a sample of adults within the same country |
Example
Imagine You are tasked with estimating the average height of adults in a region. The average height of all adults in the region (the population parameter) is unknown. You are given a dataset from a random sample of 100 adults, and you will use this sample to estimate the population average.
Data:
Suppose you are given the following heights of 100 randomly sampled adults (in cm):
[167,174,180,168,162,170,172,165,169,…,171] (100 values in total)
To calculate the sample mean height from the given 100 sample heights:
We suppose that after a complete survey of all adults in the region, you determine that the true population average height is 170 cm. So, the difference between the two values is 1,5 cm. This difference is due to sampling error, which occurs because we are only using a sample, not all entire population.
Assume that the standard deviation of the population is 8 cm. To calculate the 95% confidence interval for the population mean, we use the following formula:
Where:
is the Z-value for a 95% confidence level (which is 1.96).
σ is the population standard deviation (8 cm).
n is the sample size (100).
Now, plug in the values:
This means we are 95% confident that the true population average height lies between 166.93 cm and 170.07 cm, which include the population mean of 170 cm, so it is a very good estimate based on the sample data.
Conclusion
In summary, parameters and statistics play distinct yet interconnected roles in data analysis. A parameter describes the entire population, whereas a statistic represents a sample subset. While parameters are typically fixed and often unknown, statistics are variable and directly calculated from data. By distinguishing between these concepts and understanding their applications, researchers and analysts can ensure accurate interpretations and robust conclusions in their work.
