A distribution of scores where one tail extends out further than the other.

A unimodal distribution is a distribution that has one clear peak. The values increase first, rising to a single highest point where they then start to decrease. A unimodal distribution can either be symmetrical or nonsymmetrical.

A symmetrical distribution is one where the mean, mode, and median are all equal. In such a distribution, the intervals of gains or losses exhibit the same frequency. For instance, if we have a symmetrical distribution with a mean return of zero, then the frequency of losses between -2% and -1% will be equal to the frequency of gains between 1% and 2%.

A distribution that deviates from the symmetrical distribution is said to be nonsymmetrical, and that’s how we end up having positive skewness and negative skewness.

Unimodal Distribution – Positively Skewed

This is the tendency of a given frequency curve leaning towards the left. Conversely, the ‘tail’ extends to the right. In this distribution, candidates should note the following:

1. The mean is bigger than both the median and the mean.
2. The median always occurs between the mode and the mean.
3. The observations below the mean are more than those above it.

A good example of a positively skewed distribution would be the age distribution in a developing country.

Unimodal Distribution – Negatively Skewed

This is a frequency curve where the long tail extends to the left. It has the following characteristics:

1. The mode is usually bigger than both the median and the mean.
2. The median is found between the mean and the mode.
3. The observations above the mean are usually more than those below it.

A good example of a negatively skewed distribution would be the age distribution in a developed country.

Project forecasting is not a straightforward process – it’s a scientific and statistical exercise that deals with numerous interconnected variables. The value of these variables can have significant impact on your final prediction. Knowing the frequency distribution of these values enables you to make data-driven decisions.

The type of frequency distribution is especially important when making predictions with the Monte Carlo simulation. In this article, we’ll explain the frequency distribution shapes that you will encounter most frequently.

Normal Distribution

The normal distribution, also known as a Gaussian distribution or “bell curve” is the most common frequency distribution. This distribution is symmetrical, with most values falling towards the centre and long tails to the left and right. It is a continuous distribution, with no gaps between values.

Normal distributions are found everywhere, for both natural and man-made phenomena. This could include time taken to complete a task, IQ test results, or the heights of a group of people. In project management, when performing estimations while you have no further information about the type of frequency distribution, it is usually best to assume a normal distribution.

Skewed Distribution

When a normal curve slopes to the left or right, it is known as a skewed distribution. The location of the long tail – not the peak – is what gives this frequency distribution shape its name. A long tail on the right is referred to as right-skewed or positively skewed, while a long tail on the left is referred to as left-skewed or negatively skewed.

Positively skewed distributions are common in situations where there is a fixed lower boundary. For example, delivery of a component – if most deliveries happen within 3 days, the minimum value is 0, but the long tail could stretch far to the right if some deliveries are late.

Negatively skewed distributions are less common in general, but still appear when fixed or near-fixed upper boundaries are in play. For example, a company that guarantees all orders will be delivered within 1 week will most likely see some faster deliveries, but most values clustering close to the 1 week point.

One important fact about skewed distributions is that, unlike a bell curve, the mode, median and mean are not the same value. The long tail skews the mean and median in the direction of the tail. There is a very easy way to calculate the different average values using a histogram diagram. If you rely on average values to make quick predictions, pay attention to which average you use!

Bimodal/Multimodal Distribution

All of the frequency distribution types that we’ve looked at so far have been unimodal – values cluster around a single peak. A bimodal distribution occurs when two unimodal distributions are in the group being measured. When more than two peaks occur, its known as a multimodal distribution.

This distribution shape happens frequently when the measured data can be split into two or more groups. One example would be the throughput of all of your team’s tasks. If your team are using Classes of Service to tackle emergency tasks faster than regular tasks, you will most probably see a bimodal distribution.

If you spot a bimodal frequency distribution, it’s worth checking if you can split the measured data into sub-groups to see the shape for each group.

Uniform Distribution

In a uniform or rectangular distribution, every variable value between a maximum and minimum has the same chance of occurring. The probability of rolling a certain number on a dice or picking a certain card from the pack is described by this frequency distribution shape.

This frequency distribution appears at the start of every project. A uniform distribution assumes that all samples from its population are equally probable. When rolling a die, all numbers on the die have an equal chance of coming up on each throw. Let’s say you have nineteen samples from a uniformly distributed population. In a uniform distribution there is a very high probability that the next sample will be between the min and the max of the previous samples. That means that you have a fairly good understanding of the range of your uniform distribution after having collected only twenty data points.

Logarithmic/Pareto

Some data sets have nearly all their frequency values clustered to one side of the graph. This frequency distribution shape is known as logarithmic. A common example of this in real life is found in distributions of wealth and income, with large numbers of people at the bottom but extreme outliers extending the tail to the right.

This distribution type is often known as a Pareto distribution, named after famous Italian economist and sociologist Vilfredo Pareto. You’ve almost certainly heard of his 80-20 rule. For example, 80% of the wealth of a society is held by 20% of society, 80% of revenue comes from 20% of clients and 80% of productivity comes from 20% of your team.

While the percentages are not always 80-20, this pattern appears mainly in financial estimation models.

PERT/Triangular

The PERT and triangular frequency distribution types are both modelled from the same 3 values – a minimum, a maximum and a mode. This distribution type is especially useful when only a small amount of past performance data is available. It uses only three values as the inputs – a, m and b.

While the triangular distribution is a simple shape made using straight lines between each of the 3 values, the PERT distribution assumes that the long tail values appear less frequently. The frequency distribution shape generated from these three values is then used to estimate likely completion times.

Understanding the frequency distribution of your data is important for both input and output of your forecasts. Realistic outputs are simply impossible without accurate inputs. Calculations that rely on subjective estimates are risky – we recommend to always draw from your past performance data.

What is the frequency distribution of your data? Have you used histogram diagrams to analyse it? Do you use histograms to make your estimations? Tell us about your experience in the comments!

What is the tail of a skewed distribution?

What does it mean when one of the scores falls in the tail of the distribution?

What is a skewed distribution called?

In which type of distribution does the tail extend to the left?