what is meant by accuracy and precision of a measurement define mean absolute error relative error and percentage error

Accuracy of a measurement means how close the measured value is to the true (actual) value of the quantity.

Precision of a measurement means how close repeated measurements of the same quantity are to one another (how consistent they are), regardless of the true value.

Accuracy vs precision (quick idea)

• A measurement is accurate if it is correct (near the true value).

• A measurement is precise if repeated readings are tightly clustered together, even if they are all a bit wrong in the same way.

• You can have:
  • High accuracy, high precision
  • High precision, low accuracy (consistently wrong)
  • Low precision, high accuracy on average
  • Low accuracy, low precision

Example:
If the true length of a rod is 50.0 cm and your readings are 49.9 cm, 50.0 cm, 50.1 cm, they are both accurate (near 50.0) and precise (close to each other).

Mean absolute error

Suppose you measure a quantity AAA a number of times and get values A1,A2,…,ANA_1,A_2,\dots,A_NA1​,A2​,…,AN​. Let the mean (average) of these measurements be AmeanA_{\text{mean}}Amean​.

1. Absolute error of one measurement
   For the iii-th measurement:

ΔAi=∣Ai−Amean∣\Delta A_i=|A_i-A_{\text{mean}}|ΔAi​=∣Ai​−Amean​∣

(Magnitude of the difference between that measurement and the mean value.)

2. Mean absolute error
   This is the average of all absolute errors:

ΔAmean=1N∑i=1N∣ΔAi∣\Delta A_{\text{mean}}=\frac{1}{N}\sum_{i=1}^{N}|\Delta A_i|ΔAmean​=N1​i=1∑N​∣ΔAi​∣

It tells you, on average, how far your measurements deviate from the mean value.

Relative error

Relative error compares the size of the error with the size of the measured quantity.

• Defined as:

Relative error=ΔAmeanAmean\text{Relative error}=\frac{\Delta A_{\text{mean}}}{A_{\text{mean}}}Relative error=Amean​ΔAmean​​

where ΔAmean\Delta A_{\text{mean}}ΔAmean​ is the mean absolute error and AmeanA_{\text{mean}}Amean​ is the mean measured value.

• It is a dimensionless quantity (no unit), and shows how large the error is in proportion to the measurement.

Percentage error

Percentage error is simply the relative error expressed as a percentage.

• Defined as:

Percentage error=(ΔAmeanAmean)×100\text{Percentage error}=\left(\frac{\Delta A_{\text{mean}}}{A_{\text{mean}}}\right)\times 100Percentage error=(Amean​ΔAmean​​)×100

This tells you the error as “so many percent of the measured value.”

Example:
If Amean=10.0A_{\text{mean}}=10.0Amean​=10.0 and ΔAmean=0.2\Delta A_{\text{mean}}=0.2ΔAmean​=0.2, then

• Relative error =0.2/10.0=0.02=0.2/10.0=0.02=0.2/10.0=0.02.

• Percentage error =0.02×100=2%=0.02\times 100=2%=0.02×100=2%.

TL;DR:

• Accuracy: closeness to true value.

• Precision: closeness of repeated readings to each other.

• Mean absolute error: average of the magnitudes of individual deviations from the mean.

• Relative error: mean absolute error divided by mean value.

• Percentage error: relative error × 100.

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