Summarizing Data with Mean, Median, and Range

Hello again. Previously, you calculated ratios and percentages from forensic-style data. Those calculations describe parts of a whole. This lesson shifts to a different question: What single summary best describes a set of measurements, and how much do the measurements vary?

You will learn to summarize a small numerical dataset with the mean, median, and range. These are useful when recording measurements such as particle sizes, stain dimensions, or instrument readings. They describe the data; on their own, they do not establish the source or significance of an item of evidence.

Plan for about 40 minutes, including a short video and worked examples.

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Three ways to describe a dataset

Suppose a laboratory examiner records several measurements of the same kind, all in the same unit. Three summaries are especially useful:

Summary	What it tells you	Main calculation
Mean	The average, using every value	Total of values divided by number of values
Median	The central value after ordering the data	Middle value, or average of the two middle values
Range	The total spread from smallest to largest value	Maximum minus minimum

The mean and median are both measures of center: they describe a typical location in the data. The range is a measure of spread: it describes how far apart the extreme values are.

Before using any of them, check two things:

1. The values must be numerical measurements or counts, not labels. Case numbers, evidence-item IDs, and collection dates are labels, not values to average.
2. Measurements must use a common unit. You could summarize five lengths in millimeters, or convert them all to millimeters first; do not mix millimeters and centimeters in one calculation.

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A complete worked example

Imagine that an examiner measures the visible lengths of five fibers in a simulated comparison exercise. The values are recorded in millimeters:

Fiber	Measured length
A	$2.1\text{ mm}$
B	$1.7\text{ mm}$
C	$2.3\text{ mm}$
D	$1.9\text{ mm}$
E	$2.0\text{ mm}$

The measurement order does not matter for the mean or range. It does matter for finding the median, so it is usually efficient to place the data in order first:

$$1.7,1.9,2.0,2.1,2.3$$

Mean: the average of every measurement

The mean is calculated with:

$$\text{mean}=\frac{\text{sum of all values}}{\text{number of values}}$$

For the five fiber lengths:

$$\text{mean}=\frac{2.1+1.7+2.3+1.9+2.0}{5}$$ $$\text{mean}=\frac{10.0}{5}=2.0\text{ mm}$$

The mean length is $2.0\text{ mm}$. Notice that the mean can be a value not found in the original dataset. It is a calculated summary, not necessarily an observed measurement.

Median: the center after sorting

The ordered values are:

$$1.7,1.9,2.0,2.1,2.3$$

There are five values, an odd number. The third value sits in the middle, with two values on either side:

$$\text{median}=2.0\text{ mm}$$

A median is only meaningful after the data are ordered. Choosing the middle value from the original recording order would be unreliable.

Range: the full spread

The largest value is $2.3\text{ mm}$ and the smallest is $1.7\text{ mm}$:

$$\text{range}=2.3\text{ mm}-1.7\text{ mm}=0.6\text{ mm}$$

So this dataset can be summarized as:

• Mean: $2.0\text{ mm}$
• Median: $2.0\text{ mm}$
• Range: $0.6\text{ mm}$

The mean and median happen to be equal here, but they do not have to be.

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The median when there is an even number of values

Now consider six simulated particle-diameter measurements, in micrometers:

$$12,15,13,14,16,20$$

First, order them:

$$12,13,14,15,16,20$$

Because there are six values, there is no one number in the exact center. The two central positions are the third and fourth values, $14$ and $15$. The median is the average of those two values:

$$\text{median}=\frac{14+15}{2}=14.5\text{ micrometers}$$

The median does not need to be one of the observed values. It represents the point halfway between the two central observations.

The image below uses the same method with six temperature values: after the outer pairs are removed, the two central values are averaged.

For the particle measurements, the other summaries are:

$$\text{mean}=\frac{12+13+14+15+16+20}{6}=\frac{90}{6}=15\text{ micrometers}$$ $$\text{range}=20-12=8\text{ micrometers}$$

The full summary is therefore:

Measure	Result
Mean	$15\text{ micrometers}$
Median	$14.5\text{ micrometers}$
Range	$8\text{ micrometers}$

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Why report more than one summary?

Each summary notices a different feature of the data.

The mean uses every measurement, so it changes when any value changes. The median depends on position after sorting, so it is less affected by one especially high or low value. The range focuses entirely on the smallest and largest measurements.

Consider the earlier six particle values:

$$12,13,14,15,16,20$$

Their mean is $15$, median is $14.5$, and range is $8$.

Now imagine that the final reading was $40$, not $20$:

$$12,13,14,15,16,40$$

The median remains:

$$\frac{14+15}{2}=14.5$$

But the mean becomes:

$$\frac{12+13+14+15+16+40}{6}=\frac{110}{6}\approx 18.3$$

And the range becomes:

$$40-12=28$$

One unusually large value has pulled the mean upward and made the range much wider, while the median has not changed. This does not mean the value $40$ should automatically be discarded. In laboratory work, an unusual value might be a genuine observation, an instrument issue, a transcription error, or the result of measuring a different kind of material. The appropriate response is to check the record and measurement process, not to remove a value simply because it is inconvenient.

Reporting all three summaries gives a more informative description:

The six recorded diameters had a mean of approximately $18.3$ micrometers, a median of $14.5$ micrometers, and a range of $28$ micrometers.

That statement accurately describes the measurements while avoiding unsupported claims about what caused the variation.

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A dependable calculation routine

For a small dataset, use this routine:

1. Identify the values and units.
   Confirm that every entry represents the same quantity in the same unit.

2. Write the values in ascending order.
   This immediately reveals the minimum, maximum, and the median position.

3. Calculate the mean.
   Add every value, then divide by the number of values. Count carefully; the denominator is the number of observations, not the sum.

4. Find the median.
   With an odd number of values, select the one middle value. With an even number, average the two middle values.

5. Calculate the range.
   Subtract the minimum from the maximum:

   $$\text{range}=\text{maximum}-\text{minimum}$$

6. Report units and sensible precision.
   If lengths were measured in millimeters, the mean, median, and range should all be reported in millimeters. Do not report more decimal places than the original measurements justify.

A compact calculation record might look like this:

Item	Working
Ordered values	$1.7,1.9,2.0,2.1,2.3\text{ mm}$
Mean	$10.0÷5=2.0\text{ mm}$
Median	Third value $=2.0\text{ mm}$
Range	$2.3-1.7=0.6\text{ mm}$

This format makes the summary easy for another person to check.

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Key takeaways

The mean, median, and range summarize different aspects of numerical data:

• The mean is the total of all values divided by the number of values.
• The median is the middle value in ordered data; for an even number of values, it is the average of the two middle values.
• The range is the largest value minus the smallest value.
• Keep measurement units consistent and include them in your final result.
• A large gap between the mean and median, or a large range, may flag variation worth checking. It does not by itself prove an error or a forensic conclusion.

Next, you will use these summaries while interpreting a table or graph of forensic measurements.