Having independent and identically distributed (IID) data is a common assumption for statistical procedures and hypothesis tests. But what does that mouthful of words actually mean? That’s the topic of this post! And, I’ll provide helpful tips for determining whether your data are IID.
Let’s break the components down one-by-one.
We talk about independent and identically distributed variables in the context of samples. Samples are drawn from a population sequentially. And, IID relates to the values of a characteristic for the objects that you are sequentially sampling.
Values for a characteristic is easy. That’s just the variable you are measuring. For example, imagine you measure the IQ for each person in your study. Let’s examine the properties of independence and identically distribution in depth!
In the context of sampling, events are independent when observing the current item doesn’t influence or provide insight about the value of the next item you measure, or any other items in the sample. There is no connection between the observations.
The classic example of independent events is flipping a coin. As you flip the coin, one result does not influence or predict the next outcome at all. Even if you get five heads in a row, the next coin flip still has a 50 percent chance of being heads.
You can apply the same thinking to other characteristics. For example, in our IQ study, if we measure an individual’s IQ, it shouldn’t provide any information about the next subject we assess. If we’re selecting subjects randomly, that should be true. However, if we’re not using random selection, it might not be accurate.
Imagine we assess one person’s IQ, and then measure their sibling because it is convenient. A correlation between their scores is likely. By measuring the first person, we gain some insight into the second person. Hence, they are not independent observations.
Independence relates to how you define your population and the process by which you obtain your sample. It pretty much boils down to random sampling and not using a convenience sample. The best practice is to define your population and then draw a random sample from that population.
Most hypothesis tests assume that observations are independent. Violating that assumption can cause the results to be untrustworthy. However, a few tests work with dependent samples, such as paired t-tests.
Related post: Independent and Dependent Samples
Identically distributed relates to the probability distribution that describes the characteristic you are measuring. Specifically, one probability distribution should adequately model all values you observe in a sample. Consequently, a dataset should not contain trends because they indicate that one probability distribution does not describe all the data.
Probability distributions define both discrete and continuous variables. Let’s look at what this entails for both types.
Discrete events have a few specific outcomes. For each event, the sum probability of all possible outcomes equals one. Flipping a coin is the traditional example, but I’m going to use rolling a six on a die. I’ve found that the equal probability aspect of heads and tails gets mixed into the definition too often. Identically distributed does not require equal probabilities.
Analysts model rolling a six versus not rolling a six using the binomial distribution because they are binary data (6 or not 6). The probability of rolling a six is 16.6%. This probability should not change over your data collection run. If it stays consistent while you’re collecting data, the values are identically distributed.
Continuous events have an infinite number of possible outcomes. A probability distribution for continuous data defines the probabilities of these outcomes. The most common is the normal distribution. For continuous distributions, you often need to ensure there are no trends in multiple parameters. For example, the normal distribution has two parameters, the mean and standard deviation. There should not be a trend in either measure as you collect your sample.
For example, with our fictitious IQ study, IQ scores follow a normal distribution. As we collect our scores, we should ensure that our scores don’t tend to increase or decrease because that affects the mean. Additionally, we need to be sure that the spread of the IQ scores remains constant.
Why Are Identically Distributed Data Important?
For both discrete and continuous data, there should be no trends. One probability distribution can describe your sample.
Identically distributed data are vital for most hypothesis tests because they indicate you are assessing a stable phenomenon. For example, if you measure the strength of a product and the mean strength increases as you collect more samples, it’s hard to draw conclusions. What is the mean strength of the product? It depends on when you measure it! Is it stronger than a particular value? That also depends on when you measure it!
Of course, if your analysis compares groups, the groups can have different means, proportions, or other properties, but each group must be identically distributed.
Finally, you can assess measures that have trends over time, but you’ll need to use an analysis designed for it, such as time series analysis.
Assessing IID in Your Dataset
How do you know whether your data are independent and identically distributed? Here are some tips!
For independence, consider how you collected your data. Did you use random sampling, or did you obtain a sample of convenience? If you used readily available subjects, do you believe sequential observations are related or influence each other, such as siblings’ IQ scores?
Understanding your data collection process and the subject area can help you determine whether your observations are independent. Random sampling is great way to help ensure independent observations!
For the identically distributed portion, determine whether there are any trends in the data. Graphs can help you with this aspect. Graph your data in the order that you measured each item and look for patterns.
Fortunately, there’s a special type of graph that quality analysts use to assess this aspect—control charts! Unfortunately, analysts outside that field don’t use these charts often enough.
Control charts are specifically designed to track characteristics overtime, including proportions, means, and variability. These charts indicate whether your sample has problematic trends or patterns suggesting your data don’t follow a single probability distribution. For more information, read my post about using control charts with hypothesis tests.
Hopefully, you understand both the independent and identically distributed portions of the IID assumption. Also, keep in mind that there are statistical tests and procedures specifically designed for data that do not satisfy these assumptions!