Continuous Distributions
A continuous distribution describes measurements that can take any value within a range, rather than separate, countable outcomes. Height, weight and time are common examples: a person could be 170 cm, 170.5 cm or 170.52 cm tall, with no gaps between possible values.
Because of this, continuous distributions are drawn as a smooth curve rather than separate bars. The probability of any single exact value is effectively zero, so probability is instead measured across a range, as the area under the curve.
The Normal Distribution
The normal distribution is the bell-shaped curve that appears in the dice example. Most values sit near the middle, and values become less common the further they are from it, in the same way on both sides. Many real measurements follow this pattern, such as people's heights.
A normal distribution is described by just two numbers: the mean, which marks the centre of the curve, and the standard deviation, which describes how spread out the values are.
It also follows a simple rule. About 68% of values fall within one standard deviation of the mean, and about 95% fall within two. For example, if the average height is 170 cm with a standard deviation of 10 cm, about 68% of people are between 160 and 180 cm tall, and about 95% are between 150 and 190 cm.
Other types of continuous distributions
Not all data follow a normal distribution.
Income is a common example: most people earn a moderate amount, but a small number earn a very large amount, which stretches the curve out to the right. This is called a skewed distribution.
Some common distributions:
A uniform distribution is flat, because every outcome is equally likely. Rolling a single die is the simplest example: each number from 1 to 6 has the same chance of appearing.
A skewed distribution has a long tail stretching to one side. Right-skewed data have most values at the low end and a few very large values, as with income or house prices. Left-skewed data are the reverse, with most values at the high end and a few very low ones, as with scores on an easy exam where most students do well.
A bimodal distribution has two peaks, which usually means two different groups are mixed together. The heights of a combined group of adults and children would show one peak for each group.
An exponential distribution starts high and drops off steeply, so small values are very common and large values are increasingly rare. It often describes waiting times. For example, the time until the next customer walks into a shop is usually short, occasionally longer, and only rarely very long.
Why are distribution shapes important?
In data science, the shape of a distribution affects almost every decision that follows.
First, it determines the right summary.
For normally distributed data, the mean describes a typical value well, but for skewed data such as income, a few very large values pull the mean upwards, so the median is more accurate.
Second, it determines which methods are appropriate.
Many common techniques, such as the t-test and linear regression, assume the data or the errors are roughly normal. If that assumption is wrong, the results can be misleading, and the analyst may need to transform the data or choose a different method.
Third, the shape can reveal something about the data itself.
A bimodal distribution, for example, suggests that two different groups are mixed together, such as new and returning customers, and analysing them as one group would hide the difference between them.
Finally, the shape matters for prediction and risk. A distribution with a long tail means extreme values are more likely than a normal distribution would suggest, which is important when estimating anything from insurance losses to delivery delays. In other words, checking the shape of the data is one of the first steps of any analysis, because it tells the analyst which tools can be trusted.
Connect with Me!
LinkedIn: caroline-rennier
Email: caroline@rennier.com