Q.A student measures the thickness of a human hair by looking at it through a microscope of magnification 100. He makes 20 observations and finds that the average width of the hair in the field of view of the microscope is $3.5\ \text{mm}$. What is the estimate on the thickness of hair?

The key idea is that the microscope magnifies the true thickness by a factor of 100, so the measured average width must be divided by the magnification to recover the actual thickness.

Step 1: The average observed width in the magnified image is $3.5\ \text{mm}$.

Step 2: Magnification $M = 100$ means the image is 100 times larger than the object. …

The key idea here is that the microscope magnifies the image — what the student sees in the field of view is 100 times larger than the real hair. So to get the actual thickness, you simply reverse the magnification: divide the measured average by 100.

This is not a statistics-heavy problem despite the 20 observations. The 20 readings help reduce random error in the observed width, but the conversion from observed to real is a straightforward scaling. The average observed width is already given as $3.5\ \text{mm}$, so the estimate of the true thickness is:

That’s the whole calculation. But let’s walk through it step by step to make the reasoning crystal clear.

1. Understand what magnification means.

   A microscope with magnification 100 makes the object appear 100 times larger in the field of view. So if the real hair has thickness $t$, the image you see has thickness $100 \times t$. The student measures this image thickness.

2. The student’s measurement is of the image, not the object.

   He sees an average width of $3.5\ \text{mm}$ in the microscope. That $3.5\ \text{mm}$ is the size of the magnified hair. The real hair is 100 times smaller.

3. Reverse the magnification.

   To find the real thickness $t$, divide the observed width by the magnification factor:

4. Why 20 observations matter (but don’t change the formula). …

Method: Direct Measurement with Magnification Correction

This is a single-measurement scaling problem — the microscope magnifies the actual thickness, so we simply divide the observed width by the magnification factor.

Steps

1. Identify the given data

   • Magnification of microscope: $M = 100$
   • Average observed width in the field of view: $W_{\text{obs}} = 3.5\ \text{mm}$
   • Number of observations: $20$ (this tells us the average is reliable, but doesn't change the calculation)

2. Apply the magnification relation

   • The microscope magnifies the actual thickness by a factor of $100$.
   • Therefore:

3. Calculate

4. Express in appropriate units
   • $0.035\ \text{mm} = 3.5 \times 10^{-2}\ \text{mm}$ …

Common Mistakes in Measurement Error Estimation (Microscope Magnification Problem)

This problem tests your understanding of magnification, significant figures, and error propagation — three concepts that frequently trip students up.

Mistake 1: Forgetting to Divide by Magnification

The error:

Students directly report $3.5\ \text{mm}$ as the hair thickness, forgetting that the microscope magnifies the image by 100 times.

Why it’s wrong:

The measured $3.5\ \text{mm}$ is the image size in the field of view. The actual hair thickness is:

How to avoid:

Always ask: “Is this the real size or the magnified size?” Write the formula before plugging numbers:

Mistake 2: Ignoring Significant Figures / Precision

The error:

Reporting the answer as $0.035\ \text{mm}$ without considering that the original measurement $3.5\ \text{mm}$ has only 2 significant figures.

Why it’s wrong:

The division by 100 (an exact number) does not increase precision. The result should also have 2 significant figures:

How to avoid:

• Count significant figures in the least precise measurement (here, $3.5$ has 2 sig figs).
• The final answer must have the same number of sig figs.
• Use scientific notation to avoid ambiguity: $3.5 \times 10^{-2}\ \text{mm}$.

Mistake 3: Confusing Average Width with Error in Measurement

The error:

Students think “20 observations” implies they must calculate a standard deviation or error bar, and then report something like $0.035 \pm 0.001\ \text{mm}$.

Why it’s wrong:

The problem only gives the average width — no individual readings, no spread. Without the range or standard deviation, you cannot estimate random error. The question asks for the estimate of thickness, not an uncertainty interval.

How to avoid:

• Read carefully: “estimate” here means the best value (the mean after correction).
• Only compute error if the problem provides individual measurements or a stated uncertainty.
• If asked for “error estimation,” you need the spread — not just the number of observations.

Mistake 4: Misinterpreting “Magnification of 100”

The error:

Thinking magnification means the hair appears 100 times thinner (i.e., dividing the wrong way).

Why it’s wrong:

Magnification $M = 100$ means the image is 100 times larger than the object. So:

How to avoid:

• Remember: Magnification > 1 makes things look bigger. …