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GATE 2024 IN Q.37: solution and answer

GATE 2024 · INQ.37 · 2 marks · MCQ

The measured values from a flow instrument, whose range is between 0 and 2 flow units, are shown in the histogram. The systematic error (bias) and the maximum error (in flow units), respectively are

GATE 2024 IN question 37: Histogram on a dotted grid.Tap to open full size
Figure from the GATE 2024 IN paper. It is part of the question.
Figure description

Histogram on a dotted grid. Horizontal axis: Q (flow units), with ticks labelled 0, 0.25, 0.35, 0.36, 0.37, 0.38, 0.39 and 2, and axis-break symbols between 0 and 0.25, between 0.25 and 0.35, and between 0.39 and 2. Vertical axis: 'Number of occurrence of a particular reading', from 0 to 12 in steps of 1. A vertical dashed line at Q = 0.25 is labelled 'True Value'. Narrow bars, grouped under a brace labelled 'Measured Values': Q = 0.35 has height 1, Q = 0.36 has height 4, Q = 0.37 has height 7, Q = 0.38 has height 5, Q = 0.39 has height 2 (19 readings in total).

  1. (A)

    0.12 and 0.14

  2. (B)

    0.01 and 0.10

  3. (C)

    0.10 and 0.14

  4. (D)

    0.04 and 0.12

Answer and solution

ANSWER

The systematic error (bias) is about 0.12 flow units (mean reading 0.3716 minus the true value 0.25). The maximum error is 0.14 flow units (the farthest reading, 0.39, minus 0.25). This is option (A).

Official answer: ASource: Official answer key, GATE 2024 IN (IISc Bengaluru)

  1. Read the histogram. The true value is . The readings are (1 time), (4 times), (7 times), (5 times) and (2 times), so the number of readings is .

  2. The systematic error (bias) is the difference between the mean of the readings and the true value. Random scatter averages out over many readings, but a bias does not, so the mean is where the bias shows up.

  3. Mean reading:

  4. Bias: . As a check, the most frequent reading (the mode, ) gives the same value: .

  5. The maximum error is the largest deviation of any single reading from the true value. The two extreme readings are (error ) and (error ), so the maximum error is .

  6. So the bias is and the maximum error is flow units, which is option (A).

KEY IDEA

Always measure errors from the true value. The bias is the mean of the readings minus the true value. The maximum error is the largest single-reading deviation from the true value. How far the readings spread around their own mean is precision, not accuracy.

WATCH OUT FOR

The usual mistake is measuring from the wrong reference. The spread of the readings, 0.39 - 0.35 = 0.04, describes precision, not maximum error, and taking it for the maximum error leads to option (D). Using the reading nearest the true value (0.35 - 0.25 = 0.10) as the bias leads to option (C). A third slip is dividing by the 0 to 2 full-scale range, which gives a percentage of full scale when the question asks for flow units.

From the GATE 2024 IN paper. Every question of this paper · Download the full paper · All 11 Errors & standards questions, with notes

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