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Estimating Uncertainty from Repeated Measurements (Biology Edition)

You just measured the same thing several times… and got slightly different numbers.

That’s not you “doing it wrong.” That’s reality being a little wiggly.

In this lesson, you’ll learn how to turn repeated measurements into a clear, beginner-friendly uncertainty statement like:

mass ≈ 1.26 g ± 0.02 g

…and how your instrument’s resolution (the smallest step it can show) controls the digits you should report.

The scenario: weighing a microcentrifuge tube + sample

Say you’re weighing a small biology sample on a balance that reads to 0.01 g.

That means the display jumps like:

• 1.25 g → 1.26 g → 1.27 g …

So the resolution is 0.01 g.

Now you weigh the same tube (same setup) 6 times:

These tiny changes come from normal factors: slight repositioning, air currents, small vibration, or just the balance “settling.”

Step 1: Find a “center” value (a typical value)

A simple center is the average (mean).

Add them and divide by 6:

$\text{mean} = \frac{1.25+1.26+1.27+1.26+1.24+1.26}{6}$

The sum is:

$7.54$

So:

$\text{mean} = \frac{7.54}{6} \approx 1.2567\text{ g}$

Now pause: your balance reads to 0.01 g, so reporting 1.2567 g pretends you can see beyond the screen.

So we round the mean to the instrument’s decimal place:

• Mean (rounded to 0.01 g) = 1.26 g

Step 2: Describe variability with the range

The range is:

$\text{range} = \text{max} - \text{min}$

Here:

• max = 1.27 g
• min = 1.24 g

So:

$\text{range} = 1.27 - 1.24 = 0.03\text{ g}$

Range is quick and intuitive: “My repeats span 0.03 g.”

Step 3: Make a simple “typical spread”

Range is the full width. Often you want a typical wiggle around the center.

A beginner-friendly option is half the range:

$\text{typical spread} \approx \frac{\text{range}}{2} = \frac{0.03}{2} = 0.015\text{ g}$

But again: your balance can’t show 0.001 g steps, so 0.015 g is too fussy.

Round the spread to match the resolution (0.01 g):

• typical spread ≈ 0.02 g

(Why 0.02 and not 0.01? Because 0.015 rounds to 0.02 at the 0.01 g level.)

Step 4: Write a clear uncertainty statement

Now combine the pieces:

• Typical value: 1.26 g
• Typical spread: 0.02 g

A clear, readable report is:

Mass ≈ 1.26 g ± 0.02 g

This tells a reader:

• “Most repeats land around 1.26 g.”
• “It typically wiggles by about 0.02 g.”

And notice the formatting rule you just followed naturally:

The value and the uncertainty should match decimals

• 1.26 g has two decimals
• ± 0.02 g has two decimals

That’s good scientific hygiene.

How instrument resolution connects to uncertainty (the digits story)

If your balance only reads 0.01 g, then:

• Reporting 1.2567 g implies magical extra precision.
• Even if you repeat many times, you can’t honestly claim digits the instrument cannot display.

Also, the resolution creates a built-in “rounding fuzz.” If the true mass sits between display steps, the balance must round.

So resolution affects:

1. Reported digits (don’t go beyond 0.01 g)
2. Minimum believable uncertainty (it’s hard to justify uncertainty much smaller than a few last digits)

Your repeats help estimate the real-world wiggle, which can be larger than the resolution if technique or environment adds extra variation.

Two kinds of changes: random sampling wiggle vs procedural drift

Repeated measurements can vary for two main reasons:

1) Sampling fluctuation (random wiggle)

This is the “normal bounce” around a stable value.

• Up, down, up, down…
• No consistent direction

Your six masses above look like this: mostly around 1.26, with a low (1.24) and a high (1.27).

2) Procedural drift (a trend over time)

Drift is when measurements creep in one direction as time passes.

That might happen if:

• the balance warms up
• evaporation slowly reduces mass
• the sample settles
• a pipette tip slowly leaks

Here’s a trend over time example (same kind of balance, same 0.01 g resolution), shown as an ASCII table:

Time order:     1     2     3     4     5     6
Mass (g):     1.25  1.25  1.26  1.27  1.28  1.29

Even though each step is small, the pattern is consistent: it keeps increasing.

That’s a clue you shouldn’t summarize it as “random uncertainty” only—because the process is changing.

A practical takeaway:

• Random wiggle → summarizing with “typical ± spread” makes sense.
• Drift → you may need to fix the procedure (stabilize temperature, reduce evaporation, recalibrate) or report that the value changed over time.

Putting it all together (the mindset)

When you repeat a measurement, you’re doing two helpful things:

1. Finding a typical value (a sensible center)
2. Quantifying the wiggle (uncertainty) in a way others can trust

For our biology weighing example:

• Repeats give a mean near 1.26 g
• The spread suggests typical variation about ± 0.02 g
• The balance resolution (0.01 g) tells you which digits are fair to report

Quick motivational takeaway

Uncertainty isn’t “bad news”—it’s a confidence label.

When you can say 1.26 g ± 0.02 g, you’re not just giving a number… you’re telling the truth about how the measurement behaves in the real world.