**TL;DR:**Mathhammer does not accurately describe the most likely outcome for low burst weapons that deal random damage.

**Background:**

Since 8e introduced random damage for certain weapons, I have seen a number of posts claiming how unreliable certain weapons have become. Most people tend to fall back on mathhammer calculations when attempting to determine the effectiveness of a particular weapon, but I became suspicious that mathhammer was not accurately representing the actual outcomes on the game table.

**What is Mathhammer?**

In statistics terms, “mathhammer” calculates the average (arithmetic mean) damage of a particular shooting attack by multiplying the probability (P) each event in the shooting sequence. The general procedure is:

For a Firewarrior (BS 4+) shooting a pulse carbine (Assault 2, Str 5, AP 0) at a Guardsman (T3, Sv 5+), the calculation would be:

**What is the Problem?**

The issue with using the arithmetic mean only occurs when we introduce random damage (and/or random shots). The arithmetic mean requires us to calculate the “average” of a dice roll random roll. For D6 damage, the average is usually calculated as (1+6)/2 = 3.5. Furthermore, for weapons that deal bonus mortal wounds on a wound roll of 6+ (such as our rail weapons), we also have to calculate the average number of mortal wounds and add this to the standard damage. Additional wounds are calculated by:

To illustrate the problem with this approach, let us consider a scenario where a Hammerhead (BS 3+) fires a railgun (Heavy 1, Str 10, AP -4) at a Leman Russ (T8, Sv 3+). This results in the following calculation:

Mortal Damage: (1) x (4/6) x (1/6) x (2) = 0.89

Total Damage: 1.037 + 0.89 = 1.93

Now, let’s simulate 10,000 shooting phases* and look at the discreet probability distribution of the damage. This figure shows the probability distribution for dealing a specific amount of damage in 1 shooting round.

It is plainly visible from the figure that the probability distribution is heavily skewed. Not only that, but there is a 56% probability that the Hammerhead will deal 0 damage in a given shooting phase, which makes it suspicious that 1.93 damage is typical. In cases such as these, the mean is considered to be invalid for the purpose of estimating a “typical” outcome, as it is heavily influenced by outliers.

**If Not Mean, Then What?**

For skewed distributions, the median of the data is considered to be a more accurate representation of a “typical” outcome. Median is determined by sorting the data into increasing order (in this case, from 0 to 9) and then identifying the middle number of the data set. The median for this particular example is 0, which serves to emphasize the problem with the Railhead in its current configuration.

So let us look next at Longstrike to see what a more reasonable damage distribution looks like. Longstrike has better ballistic skill (BS 2+), and gains a +1 to his wound rolls which increases his probability of triggering mortal wounds from 1/6 to 2/6. Longstrike’s average damage output is 2.99, and his damage distribution is as follows:

Longstrike’s distribution is still quite skewed; however, there are two probability peaks, which gives us a median of 3. Since the median and mean are the same, Longstrike feels like he has the damage output that mathhammer says he should.

No discussion about the Hammerhead would be complete without someone bringing up the lascannon Predator, so let’s go ahead and look at it. The Predator has the same ballistic skill as the Hammerhead; however, the lascannon statline is a little different (Heavy 1, Str 9, Ap -3). The Predator has the option to bring 4 Lascannons, but let’s look at a single lascannon first. One lascannon has a mean damage output of 1.3 and a median damage output of 0. The distribution is as follows:

So far, the lascannon looks worse than the Railgun given that it has a 64% chance of dealing 0 damage. This doesn’t tell the whole story, so we need to look at the Predator with all 4 of its lascannons. A quad lascannon predator has a mean damage output of 5.19 and a median output of 5. Again, the mean and median line up to make the lascannon Predator feel more reliable, and even better, the distribution is closer to normal:

*Monte Carlo simulation was used to generate 10,000 shooting phases. Total damage was summed for each shooting phase and the frequency of damage was counted. This frequency was used to calculate the discreet probability of achieving each damage output.

**Conclusions:**

Weapons with random damage and a low number of shots are pretty bad at the moment, and using mathhammer to calculate average damage output dramatically overestimates the most likely outcome from a given round of shooting. In order to improve the reliability of damage output, weapons like the railgun either need to gain additional shots or they need special benefits (such as Longstrike’s +1 to wound) to shift the probability distribution. If we gave the Hammerhead the Grinding Advance rule that the Leman Russ got, or mean damage would be 3.56 and our median output would be 3. Distribution is as follows:

Let me know if there are other weapon distributions that you would like to see. I am currently working on an analysis of weapons that use a random a number of shots.