# German Gun Dispersion and Accuracy

By Christian Ankerstjerne

To aid gun crews in determining the chance of hitting a target, German manuals and firing tables included gun accuracy tables. To conserve ammunition, tank gun crews were generally only allowed to open fire if they had at least a 20 percent chance of hitting their target. The methods used to calculate these tables used a combination of empirical dispersion data and theoretical accuracy formulas. When using German gun accuracy tables, it is important to understand these methods to properly interpret what they mean.

## Gun Dispersion Data

### 50 percent dispersion

A key concept in German army gun dispersion was the term 50 percent dispersion

(50 %ige Streuung), which was measured in meters. Three different dispersion numbers were used:

- Width dispersion, indicating target width.
- Height dispersion, indicating target height. This dispersion number was only measured for direct fire weapons, and was greater than the width dispersion.
- Length dispersion, indicating the length along the flight path. For direct fire weapons, such as rifles and tank guns, this number was significantly larger than the width and height dispersions. For indirect fire guns, such as artillery, the length dispersion was larger than the width dispersion at close ranges, but smaller at long ranges. For an example, see the 15 cm Nebelwerfer 41 accuracy numbers.

These numbers were based on firing range tests. In German war-time manuals, there are two different descriptions of how these numbers were measured:

- In the 1934 shooting regulation for rifles, light machine guns, and pistols (H Dv 470/20), the numbers indicate the width, height, and length within which half the hits fell.
- In the 1944 training manual for armor gunnery (H Dv 470/20), the numbers indicate a quarter of the width and height within which all the hits fell, measured from the center, as seen in the illustration.

While the first method measures the true half of the hits, the second method uses a statistical approach. While this allows for easier accuracy calculations, it assumes that all guns have the same distribution of hit inaccuracy.

In the tables below are examples of the 50 percent dispersion numbers for German tank guns:

### Dispersion examples

#### 5 cm Kw K 39 (L/60)

Distance | 5 cm Pzgr 39 | 5 cm Pzgr 40/1 | 5 cm Sprgr 38 | ||||||
---|---|---|---|---|---|---|---|---|---|

Width | Height | Length | Width | Height | Length | Width | Height | Length | |

200 | 0.06 | 0.06 | 35 | 0.10 | 0.11 | 105 | 0.0 | 0.1 | 19 |

400 | 0.12 | 0.12 | 35 | 0.22 | 0.23 | 105 | 0.1 | 0.1 | 18 |

600 | 0.18 | 0.18 | 30 | 0.35 | 0.37 | 95 | 0.1 | 0.2 | 18 |

800 | 0.24 | 0.26 | 30 | 0.49 | 0.51 | 95 | 0.2 | 0.3 | 17 |

1000 | 0.30 | 0.33 | 3 | - | - | - | 0.3 | 0.4 | 17 |

1200 | 0.37 | 0.42 | 28 | - | - | - | 0.4 | 0.5 | 16 |

1400 | 0.45 | 0.52 | 28 | - | - | - | 0.6 | 0.7 | 16 |

1600 | - | - | - | - | - | - | 0.8 | 0.8 | 16 |

1800 | - | - | - | - | - | - | 1.0 | 1.0 | 15 |

2000 | - | - | - | - | - | - | 1.2 | 1.2 | 15 |

2200 | - | - | - | - | - | - | 1.4 | 1.4 | 15 |

2400 | - | - | - | - | - | - | 1.6 | 1.7 | 15 |

2600 | - | - | - | - | - | - | 1.8 | 2.0 | 15 |

2800 | - | - | - | - | - | - | 2.0 | 2.3 | 15 |

3000 | - | - | - | - | - | - | 2.2 | 2.6 | 15 |

#### 7,5 cm Kw K (L/24)

Distance | 7,5 cm Gr 38 Hl/C | 7,5 cm Sprgr 34 | ||||
---|---|---|---|---|---|---|

Width | Height | Length | Width | Height | Length | |

200 | 0.1 | 0.1 | 19 | 0 | 0 | 23 |

400 | 0.2 | 0.2 | 19 | 0 | 0 | 23 |

600 | 0.3 | 0.4 | 18 | 0 | 0 | 23 |

800 | 0.5 | 0.5 | 18 | 0 | 1 | 24 |

1000 | 0.6 | 0.7 | 18 | 0 | 1 | 24 |

1200 | 0.8 | 0.9 | 18 | 0 | 1 | 40 |

1400 | 1.0 | 1.2 | 18 | 1 | 1 | 48 |

1600 | - | - | - | 1 | 2 | 56 |

1800 | - | - | - | 1 | 2 | 65 |

2000 | - | - | - | 1 | 2 | 74 |

2200 | - | - | - | 1 | 3 | 84 |

2400 | - | - | - | 1 | 3 | 93 |

2600 | - | - | - | 1 | 4 | 104 |

2800 | - | - | - | 1 | 4 | 114 |

3000 | - | - | - | 2 | 4 | 125 |

#### 7,5 cm Kw K 40 (L/48)

Distance | 7,5 cm Pzgr 39 | 7,5 cm Pzgr 40 | 7,5 cm Gr 38 Hl/C | 7,5 cm Sprgr 34 | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Width | Height | Length | Width | Height | Length | Width | Height | Length | Width | Height | Length | |

200 | 0.1 | 0.1 | 49 | 0.1 | 0.1 | 85 | 0.1 | 0.1 | 19 | 0.1 | 0.1 | 67 |

400 | 0.2 | 0.2 | 50 | 0.2 | 0.2 | 85 | 0.2 | 0.2 | 19 | 0.2 | 0.3 | 65 |

600 | 0.3 | 0.3 | 52 | 0.3 | 0.3 | 85 | 0.3 | 0.4 | 18 | 0.3 | 0.6 | 64 |

800 | 0.4 | 0.4 | 53 | 0.4 | 0.5 | 85 | 0.5 | 0.5 | 18 | 0.5 | 0.9 | 62 |

1000 | 0.5 | 0.6 | 54 | 0.6 | 0.7 | 85 | 0.6 | 0.7 | 18 | 0.7 | 1.1 | 61 |

1200 | 0.7 | 0.7 | 55 | 0.7 | 0.9 | 80 | 0.8 | 0.9 | 18 | 0.9 | 1.4 | 60 |

1400 | 0.8 | 0.9 | 57 | 1.0 | 1.1 | 80 | 1.0 | 1.2 | 18 | 1.0 | 1.7 | 58 |

1600 | 1.0 | 1.1 | 58 | 1.1 | 1.3 | 80 | - | - | - | 1.2 | 2.0 | 57 |

1800 | 1.1 | 1.3 | 59 | 1.3 | 1.5 | 80 | - | - | - | 1.4 | 2.4 | 56 |

2000 | 1.3 | 1.6 | 61 | 1.5 | 1.8 | 80 | - | - | - | 1.6 | 2.7 | 54 |

2200 | 1.5 | 1.9 | 62 | - | - | - | - | - | - | 1.8 | 3.1 | 53 |

2400 | 1.7 | 2.2 | 63 | - | - | - | - | - | - | 2.0 | 3.5 | 52 |

2600 | 1.9 | 2.6 | 65 | - | - | - | - | - | - | 2.2 | 3.9 | 51 |

2800 | 2.1 | 2.9 | 66 | - | - | - | - | - | - | 2.4 | 4.3 | 50 |

3000 | 2.3 | 3.3 | 67 | - | - | - | - | - | - | 2.5 | 4.7 | 49 |

#### 8,8 cm Kw K 36 (L/56)

Distance | 8,8 cm Pzgr 39 | 8,8 cm Sprgr L/4,5 | ||||
---|---|---|---|---|---|---|

Width | Height | Length | Width | Height | Length | |

200 | 0.1 | 0.1 | 43 | 0.1 | 0.1 | 70 |

400 | 0.1 | 0.2 | 43 | 0.1 | 0.2 | 65 |

600 | 0.2 | 0.2 | 40 | 0.2 | 0.3 | 60 |

800 | 0.2 | 0.3 | 40 | 0.2 | 0.4 | 55 |

1000 | 0.2 | 0.4 | 40 | 0.3 | 0.5 | 55 |

1200 | 0.3 | 0.5 | 38 | 0.4 | 0.6 | 55 |

1400 | 0.3 | 0.5 | 38 | 0.5 | 0.7 | 55 |

1600 | 0.4 | 0.6 | 39 | 0.6 | 0.9 | 55 |

1800 | 0.4 | 0.8 | 39 | 0.7 | 1.1 | 60 |

2000 | 0.5 | 0.9 | 39 | 0.8 | 1.3 | 60 |

2200 | 0.6 | 1.0 | 40 | 0.9 | 1.5 | 60 |

2400 | 0.7 | 1.2 | 40 | 0.9 | 1.7 | 60 |

2600 | 0.8 | 1.3 | 40 | 1.0 | 1.9 | 60 |

2800 | 0.9 | 1.5 | 40 | 1.1 | 2.1 | 60 |

3000 | 1.0 | 1.7 | 40 | 1.1 | 2.1 | 60 |

## Hit Probability

In addition to dispersion, German firing tables also included calculations on the chance to hit a target. The standard firing table target was 2.5 meters wide and 2 meters tall. For other target sizes, a formula was given to calculate the chance to hit the target.

### Probability calculation

The calculation of the probability of hitting a given target was done by first calcularing the ratio between the size of a target and the 50 percent dispersion numbers at the target distance. The resulting ratios were then looked up in a table to find corresponding pre-calculated probabilities. The probabilities were then multiplied to get a single hit probability.

`Width probability factor = Target width ÷ Width dispersion`

`Height probability factor = Target height ÷ Height dispersion`

`Hit probability = Width probability × Height probability`

To take into account that accuracy on the battlefield would not be comparable to that on a firing range, for example due to poor maintenance and errors by the gunner, a modified formula was used to simulate combat accuracy. This was done by multiplying the firing table dispersions by two. There does not appear to have been any studies done to specifically verify whether this double dispersal formula reflected actual battlefield performance. A 1943 study did point out, however, that gunners required further training for long-range engagements.

#### Probability factor table

Probability factor | Hit probability |
---|---|

0.10 | 5% |

0.20 | 11% |

0.30 | 16% |

0.40 | 21% |

0.50 | 26% |

0.60 | 31% |

0.70 | 36% |

0.80 | 41% |

0.90 | 46% |

1.00 | 50% |

1.10 | 54% |

1.20 | 58% |

1.30 | 62% |

1.40 | 65% |

1.50 | 69% |

1.60 | 72% |

1.70 | 75% |

1.80 | 78% |

1.90 | 80% |

2.00 | 82% |

2.10 | 84% |

2.20 | 86% |

2.30 | 88% |

2.40 | 89% |

2.50 | 91% |

2.60 | 92% |

2.70 | 93% |

2.80 | 94% |

2.90 | 95% |

3.00 | 96% |

3.10 | 96% |

3.20 | 97% |

3.30 | 97% |

3.40 | 98% |

3.50 | 98% |

3.60 | 98% |

3.70 | 99% |

3.80 | 99% |

3.90 | 99% |

4.00 | 99% |

4.10 | 99% |

4.20 | 100% |

### Hit probability examples

A 5 cm Kw K 39 (L/60) fires against a target 2 meters wide and 1 meter tall at a range of 600 meters using a 5 cm Pzgr 39.

- Standard calculation
`Width probability factor = Target width ÷ Width dispersion = 2 ÷ 0.35 = 5.71`

`Height probability factor = Target height ÷ Height dispersion = 1 ÷ 0.37 = 2.7`

`Hit probability = Height probability × Width probability = 100% × 93% = 93%`

- Combat simulation
`Width probability factor = Target width ÷ Width dispersion = 2 ÷ (0.35 × 2) = 2.85`

`Height probability factor = Target height ÷ Height dispersion = 1 ÷ (0.37 × 2) = 1.35`

`Hit probability = Height probability × Width probability = 98% × 65% = 64%`

A 7,5 cm Kw K 40 (L/48) fires against a target 2.5 meters wide and 2 meter tall at a range of 1200 meters using a 7,5 cm Pzgr 39.

- Standard calculation
`Width probability factor = Target width ÷ Width dispersion = 2.5 ÷ 0.7 = 3.57`

`Height probability factor = Target height ÷ Height dispersion = 2 ÷ 0.7 = 2.85`

`Hit probability = Height probability × Width probability = 98% × 95% = 93%`

- Combat simulation
`Width probability factor = Target width ÷ Width dispersion = 2.5 ÷ (0.7 × 2) = 1.79`

`Height probability factor = Target height ÷ Height dispersion = 2 ÷ (0.7 × 2) = 1.43`

`Hit probability = Height probability × Width probability = 78% × 65% = 51%`

## Sources

*Schießvorschrift für Gewehr (Karabiner), leichtes Maschinengewehr, Pistole und Bestimmungen für das Werfen scharfer Handgranaten*. Berlin : Reichwehrministerium, 1934. 162 p. H Dv 240. NARA T283 R136.*Schußtafel für die 7,5 cm Kampfwagenkanone 40 (7,5 cm Kw K 40) und 7,5 cm Sturmkanone 40 (7,5 cm Stu K 40) und die 7,5 cm Panzerjägerkanone 40 (7,5 cm Pak 40)*. 1943. 74 p. H Dv 119/324. NARA T283 R12.*Ausbildungsvorschrift für die Panzertruppe : Panzerschießvorschrift (Schießvorschrift für Panzerkampfwagen, Sturmgeschütze und Panzerspähwagen)*. Generalinspekteur der Panzertruppen, 1944. 136 p. H Dv 470/20. NARA T283 R136.- CRANZ, Dr. Carl.
*Lehrbuch der Ballistik : Erster Band - Äussere Ballistik*. Berlin : Verlag von Julius Springer, 1925. 711 p.