The NORMAL DISTRIBUTION (sometimes referred to as the Gaussian distribution) is a continuous probability distribution that can be found in many places: height, weight, IQ scores, test scores, errors, reaction times, etc. Understanding that a variable is normally distributed allows you to:
- predict the likelihood (i.e., probability) of observing certain values
- apply various statistical techniques that assume normality
- establish confidence intervals and conduct hypothesis tests
Characteristics of the Normal Distribution
There are several key characteristics of the normal distribution:
- mean, median, and mode are equal and located at the center of the distribution
- the distribution is symmetric about the mean (i.e., the left half of the distribution is a mirror image of the right half); “Scores above and below the mean are equally likely to occur so that half of the probability under the curve (0.5) lies above the mean and half (0.5) below” (Meier, Brudney, and Bohte, 2011, p. 132)
- the distribution resembles a bell-shaped curve (i.e., highest at the mean and tapers off towards the tails)
- the standard deviation determines the SPREAD of the distribution (i.e., its height and width): a smaller standard deviation results in a steeper curve, while a larger standard deviation results in a flatter curve
- the 68-95-99 RULE can be used to summarize the distribution and calculate probabilities of event occurrence:
– approximately 68% of the data falls within ±1 standard deviation of the mean
– approximately 95% of the data falls within ±2 standard deviations of the mean
– approximately 99% of the data falls within ±3 standard deviations of the mean - there is always a chance that values will fall outside ±3 standard deviations of the mean, but the probability of occurrence is less than 1%
- the tails of the distribution never touch the horizontal axis: the probability of an outlier occurring may be unlikely, but it is always possible; thus, the upper and lower tails approach, but never reach, 0%
Why the Normal Distribution is Common in Nature: The Central Limit Theorem
The CENTRAL LIMIT THEOREM states that the distribution of sample means for INDEPENDENT, IDENTICALLY DISTRIBUTED (IID) random variables will approximate a normal distribution, even when the variables themselves are not normally distributed, assuming the sample is large enough. Thus, as long as you have a sufficiently large random sample, we can make inferences about the population parameters (what we are interested in) from sample statistics (what we often are working with).
What Does “IID” Mean?
Variables are considered independent if they are mutually exclusive. Variables are considered identically distributed if they have the same probability distribution (i.e., normal, Poisson, etc.)
Do Outliers Matter?
In a normal distribution based on a large number of observations, it is unlikely that outliers will skew results. If you are working with data involving fewer observations, outliers are more likely to skew results; in these situations, you should identify, invest, and decide how to handle outliers.
Example of a Normal Distribution: IQ Tests
Because the IQ test has been given millions of times, IQ scores represent a normal probability distribution. On the IQ test, the mean, median, and mode are equal and fall in the middle of the distribution (100). The standard deviation on the IQ test is 15; applying the 68-95-99 rule, we can say with reasonable certainty:
- 68% of the population will score between 85 and 115, or ±1 standard deviation from the mean
- 95% of the population will score between 70 and 130, or ±2 standard deviations from the mean
- 99% of the population will score between 55 and 145, or ±3 standard deviations from the mean
Rarely will you encounter such a perfect normal probability distribution as the IQ test, but we can calculate z-scores to standardize (i.e., “normalize”) values for distributions that aren’t as normal as the IQ distribution.