Q.The mean length of an object is 5 cm. Which of the following measurements is most accurate?

Concept: Accuracy and Absolute Error

Accuracy measures how close a measurement is to the true value. The most accurate measurement has the smallest absolute error.

Solution:

The true (mean) length is $5$ cm. Calculate the absolute error for each option:

| Measurement | Absolute Error $= |x - 5|$ |

|-------------|---------------------|

| (A) 4.9 cm | $|4.9 - 5| = 0.1$ cm |

| (B) 4.805 cm | $|4.805 - 5| = 0.195$ cm |

| (C) 5.25 cm | $|5.25 - 5| = 0.25$ cm | …

Understanding Accuracy vs. Precision

When we talk about accuracy in measurement, we're asking: how close is this value to the true or accepted value? This is fundamentally different from precision, which tells us how finely we can measure (how many decimal places) or how reproducible our measurements are.

A measurement can be very precise (many decimal places) but wildly inaccurate if it's far from the true value. Conversely, a measurement with fewer decimal places can be highly accurate if it's very close to what we're trying to measure.

In this problem, the mean length of 5 cm represents our reference or true value. We need to find which measurement deviates least from this reference.

Step-by-Step Analysis

1. Identify the reference value

   The true or mean length is $5 \text{ cm}$.

2. Calculate the absolute deviation for each measurement

   Accuracy is determined by $|\text{measured value} - \text{true value}|$.

   Measurement	Deviation from 5 cm	Absolute Deviation
   (A) 4.9 cm	$4.9 - 5 = -0.1$	$0.1 \text{ cm}$
   (B) 4.805 cm	$4.805 - 5 = -0.195$	$0.195 \text{ cm}$
   (C) 5.25 cm	$5.25 - 5 = 0.25$	$0.25 \text{ cm}$
   (D) 5.4 cm	$5.4 - 5 = 0.4$	$0.4 \text{ cm}$

3. Compare the deviations …

Concept: Accuracy and Absolute Error

Accuracy of a measurement refers to how close the measured value is to the true value. The most accurate measurement is the one with the smallest absolute error.

Absolute Error = | Measured Value − True Value |

Method: Absolute Error Comparison

Steps

1. Identify the true value

   True mean length = 5 cm

2. Calculate absolute error for each option

   • (A) $|4.9 - 5| = 0.1$ cm
   • (B) $|4.805 - 5| = 0.195$ cm
   • (C) $|5.25 - 5| = 0.25$ cm
   • (D) $|5.4 - 5| = 0.4$ cm

3. Compare the errors

   Smallest absolute error = 0.1 cm (Option A)

4. Select the measurement with the smallest error …

Common Mistakes & How to Avoid Them

Mistake 1: Confusing "Accuracy" with "Precision"

The error: Students often pick the measurement with the most decimal places (4.805 cm) thinking more digits = more accurate.

Why it's wrong: Accuracy is about closeness to the true value (5 cm), not the number of decimal places. 4.805 cm is far from 5 cm (error = 0.195 cm), while 4.9 cm is closer (error = 0.1 cm).

How to avoid: Always calculate the absolute error:

• $|4.9 - 5| = 0.1$ cm
• $|4.805 - 5| = 0.195$ cm
• $|5.25 - 5| = 0.25$ cm
• $|5.4 - 5| = 0.4$ cm

The smallest error = most accurate. Answer: (A) 4.9 cm

Mistake 2: Thinking "Closest to the Mean" Means the Same as the Mean

The error: Some students pick 5.25 cm because it's "close to 5" without comparing all options.

Why it's wrong: 5.25 cm has an error of 0.25 cm — larger than 4.9 cm's error of 0.1 cm. "Close" is relative; you must compare all errors.

How to avoid: List all errors side-by-side:

| Measurement | Error = |Value − 5| |

|-------------|----------------|

| 4.9 cm | 0.1 cm ✓ |

| 4.805 cm | 0.195 cm |

| 5.25 cm | 0.25 cm |

| 5.4 cm | 0.4 cm |

The smallest absolute difference wins.

Mistake 3: Misreading the Question as "Most Precise"

The error: Students see "most accurate" but mentally substitute "most precise" (most decimal places).

Why it's wrong: In exam terminology:

• Accuracy = closeness to true value (use absolute error)
• Precision = consistency / number of decimal places

How to avoid: Underline the keyword "accurate" in the question. If it said "precise," then 4.805 cm would be correct. Here it says "accurate," so use error calculation.

Mistake 4: Forgetting to Use Absolute Value …