Calculating Ratios and Percentages in Forensic Data

Hello again. In the previous lesson, you converted laboratory measurements into compatible metric units. That skill matters here because ratios are only meaningful when the quantities being compared use the same unit.

This lesson introduces two closely related ways to describe a small forensic dataset:

• a ratio, which compares two quantities;
• a percentage, which states how large one part is out of a chosen whole.

You will use fictional evidence counts and masses, keeping the calculation separate from any claim about what the evidence proves. Plan for roughly 40 minutes.

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Ratios: comparisons with an order

A ratio compares two quantities. It may be written with a colon, as a fraction, or in words:

$$18:12$$ $$\frac{18}{12}$$

“18 to 12”

The order is essential. A ratio of $18:12$ is not the same statement as $12:18$. Before calculating, write the comparison in words: “the ratio of red fibers to blue fibers” means red is first and blue is second.

Consider this small fictional dataset from a tape lift collected at a scene:

Fiber color	Number observed
Red	$18$
Blue	$12$
Total	$30$

The ratio of red fibers to blue fibers is:

$$18:12$$

Both numbers divide by $6$, so the simplified ratio is:

$$3:2$$

This means that for every $3$ red fibers observed, there were $2$ blue fibers observed. It does not mean there were only $3$ red and $2$ blue fibers. A simplified ratio preserves the relationship while removing a common multiplier.

A ratio can compare:

• part to part, such as red fibers to blue fibers: $18:12=3:2$;
• part to whole, such as red fibers to all observed fibers: $18:30=3:5$.

These answer different questions. Confusing them is one of the most common ratio errors.

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A percentage is always “part out of a whole”

The word percent means “per hundred.” A percentage therefore requires a reference whole.

The central relationship is:

$$\text{percentage}=\frac{\text{part}}{\text{whole}}\times 100\%$$

For the fiber dataset, red fibers are the part and all $30$ fibers are the whole:

$$\frac{18}{30}\times 100\%=60\%$$

So $60\%$ of the observed fibers are red. Similarly, blue fibers make up:

$$\frac{12}{30}\times 100\%=40\%$$

The image below shows this same idea using a ratio of $3:2$. There are $5$ equal total parts, so the two shares are $\frac{3}{5}$ and $\frac{2}{5}$, which become $60\%$ and $40\%$.

There is a useful connection between the two forms:

$$18:12=3:2$$

The ratio has $3+2=5$ total ratio parts. Therefore:

$$\text{red share}=\frac{3}{5}=60\%$$ $$\text{blue share}=\frac{2}{5}=40\%$$

This shortcut works when the ratio accounts for all categories in the chosen whole. If a dataset contains other categories, you cannot calculate a category’s percentage using only two selected parts.

For example, if the tape lift also contained $5$ green fibers, the red-to-blue ratio would still be $18:12=3:2$. But the percentage of red fibers in the full sample would change:

$$\frac{18}{18+12+5}\times 100\%=\frac{18}{35}\times 100\%\approx 51.4\%$$

The whole must match the question being asked.

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Working through a forensic-style dataset

Imagine a preliminary screening record for $20$ separate stain samples. In this fictional example, “positive” means the sample gave a positive screening result; it does not yet establish the substance conclusively.

Screening result	Number of samples
Positive	$7$
Negative	$13$
Total tested	$20$

Part-to-part ratio

The ratio of positive to negative screening results is:

$$7:13$$

There is no common whole-number factor other than $1$, so this is already simplified.

This comparison tells us how the two groups relate to one another. It is not itself a percentage.

Part-to-whole percentage

To find the percentage of samples with a positive result, use the total number tested as the whole:

$$\frac{7}{20}\times 100\%=35\%$$

Thus, $35\%$ of the tested samples had a positive screening result. The negative-result percentage is:

$$\frac{13}{20}\times 100\%=65\%$$

Because positive and negative are the only categories in this table, the percentages should add to $100\%$:

$$35\%+65\%=100\%$$

That is a valuable check. If category percentages are meant to partition one total but add up to $92\%$ or $108\%$, check the denominator, arithmetic, or rounding.

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Ratios involving measurements: convert first

Ratios of measured quantities need compatible units. This directly builds on the previous lesson.

Suppose two powder samples recovered from separate locations have masses:

Sample	Recorded mass
A	$0.75\text{ g}$
B	$680\text{ mg}$

It would be incorrect to write the ratio as $0.75:680$, because the first number is in grams and the second is in milligrams.

Convert sample A to milligrams:

$$0.75\text{ g}\times \frac{1000\text{ mg}}{1\text{ g}}=750\text{ mg}$$

Now the comparison is valid:

$$750\text{ mg}:680\text{ mg}$$

The common unit cancels conceptually, leaving:

$$750:680$$

Both values divide by $10$:

$$75:68$$

So the mass ratio of sample A to sample B is $75:68$.

You can also calculate each sample’s share of the combined recovered mass. The total mass is:

$$750\text{ mg}+680\text{ mg}=1430\text{ mg}$$

Sample A’s percentage of the combined mass is:

$$\frac{750}{1430}\times 100\%\approx 52.4\%$$

Sample B’s percentage is:

$$\frac{680}{1430}\times 100\%\approx 47.6\%$$

These add to $100\%$, apart from any small rounding difference.

The calculation describes only the distribution of recovered mass. It does not establish that the materials have the same composition, came from the same source, or have the same forensic significance. Those are separate scientific questions requiring suitable examinations.

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A reliable calculation routine

For every ratio or percentage question, use this sequence:

1. State the comparison in words.
   For example: “positive results to negative results” or “positive results as a percentage of all tested samples.”

2. Choose the correct whole.
   A percentage always has a denominator. Ask, “Out of what total?”

3. Make units compatible.
   Counts can usually be compared directly. Measurements such as grams and milligrams must first be converted into the same unit.

4. Calculate and simplify appropriately.
   Simplify ratios by dividing both sides by a common factor. Calculate percentages using part divided by whole, multiplied by $100\%$.

5. Check whether the result makes sense.
   A part that is less than its whole should produce a percentage below $100\%$. Mutually exclusive categories that include every case should total approximately $100\%$.

A calculator is useful when the division produces a long decimal. For example:

$$\frac{7}{19}\times 100\%\approx 36.842\%$$

Depending on the reporting context, this could be reported as $36.8\%$ or $37\%$. Do not imply unnecessary precision: a small dataset rarely justifies many decimal places.

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Key takeaways

A ratio compares two quantities in a stated order, while a percentage expresses one part relative to a chosen whole.

$$\text{percentage}=\frac{\text{part}}{\text{whole}}\times 100\%$$

Remember these distinctions:

• $7:13$ compares positive results to negative results.
• $\frac{7}{20}\times 100\%=35\%$ gives positive results out of all tested samples.
• Convert measurement units before forming a ratio.
• Category percentages should total $100\%$ only when the categories are mutually exclusive and cover the entire selected dataset.
• A numerical proportion describes the data; it does not, by itself, prove a forensic conclusion.

Next, you will build on these calculations by summarizing a small set of measurements with the mean, median, and range.