Finding Measures of Central Tendency for Weighted Data
In the previous sections dealing with measures of central tendency, you learned the fundamentals of calculating the mean, median and mode as well as how to handle grouped data. We also walked you through the basics of quartiles. Here, we’ll expand upon these topics and show you how to handle weighted data as well as the basics of range.
We’ve all been there: that dreaded question at the end of a test that is impossible to answer and yet is worth so much more than all the questions preceding it. While it may seem like simply another tool designed to make us struggle, making some questions worth more tends to reflect a higher level of complexity.
This is one of the most basic forms of weighting. A weight is assigned to a value to make it worth more or less than other values, hence the term “weight.” Weights are commonly assigned to values in order to give them more or less of an impact on a variable or data set. In statistics, samples or groups within a sample are typically assigned weights in order to correct for errors and variation occurring naturally from sampling methods.
Another basic form of weighting is arriving at a weighted average, discussed in the following paragraphs.
Recall that the mean is one measure of central tendency that attempts to give the most typical value of a data set. There are three types of Pythagorean Means:
Here, we’ll deal only with the arithmetic mean, which is simply the mean as you’ve learned it probably dozens of times. The formula for the arithmetic mean of a population and a sample is summarized in the table below.
The formula for the arithmetic weighted mean, on the other hand, is
The arithmetic mean can also be considered to be a weighted average, where all values have the equal weight of 1. Take the following numbers as an example.
Where we would calculate the weighted mean as
Which boils down to the formula for the sample mean,
Where the weighted mean diverges from the sample mean is that the weighted mean assigns unequal weights to each value. Meaning, there is at least one weight that’s different from 1. Take the table below as an example.
|Value||Weight||Value * Weight|
Where the weighted mean is calculated as,
Giving us the weighted mean of about 22.