Significant Figures, Rounding Off, Accuracy, and Precision

Significant figures

Significant figures can be defined as digits in a number that is necessary to indicate the reliable value of measurement.

Reliable means those values that are within the measurement resolution or capacity.

For e.g, The length of an object is 14.3 mm measured by mm scale is not reliable. The last digit 3 is not reliable because the smallest scale of mm scale is 1mm and is not significant. Whereas, digits 1 and 4 are significant.

Suppose the mass of an object is measured to be 235.2045 kg. If the measurement resolution is 0.001 then, the last digit 5 is not significant.

Here is another example: The density of an object is measured to be 14.5 kg/m^3 with an uncertainty of \pm 0.4. The actual density can be in the range of 14.1 to 14.9 and is uncertain. But the value is reliable and is a significant figure.

Rules to identify significant digits

  1. The non-zero digits in a value are always significant if the value is within a measurement resolution.
    • The number 34.52 has three significant figures as all digit are non-zero.
    • The number 234.287 has ony four significant figures (2, 3, 4 and 2) if the measurement resolution is 0.1
  2. Any zeros between two significant digits are always significant.
    • The zeros in 134.0048 within a measurement resolution are signficant. So, it has 7 significant figures(1, 3, 4, 0, 0, 4, 8).
  3. The zeros after non-zero digit in decimal part (trailing zeros) are signficant within a measurement resolution.
    • The zeros in 23.400 are significant if measurement resolution is 0.001.
    • 18.00 has four significant figures 1, 8, 0 and 0.
  4. Exact numbers or constant numbers have infinite significant figures.
    • The constant \pi =3.14159265358979323… has all digits significant.
    • 1 kg or 1.0 kg or 1.00 kg or 1.000 kg, all digits are significant.

Rules to identify insignificant digits

  1. All zeros before the first non-zero digit (leading zeros) are not significant.
    • If a mass from a measurement gives 0.05 kg, then the leading zeros in 0.05 kg disapper in other unit. As 0.05 kg = 50 gm, the leading zeros are always insignificant.
  2. Trailing zeros in a whole number are not significant.
    • The distance value is 3400 m. The last two digits are not significant as they can be written as: 3400 m = 3.4 km.

Rounding off

Rounding off in physics generally means removing insignificant digits from the data. The data obtained from the experiment may contain many decimal place values, which may not be useful for us. So, we use rounding off to get our approximate result from decimal values.

Rules of Rounding off

  1. If the digits to be removed is greater than 5 (i.e 6, 7, 8, 9) then, we add 1 to previous digit.
    for e.g: NOTE: ( digit to be removed is underlined.)
    • Rounding off 345.238 = 345.24
    • Rounding off 2.800092 = 2.8001
    • Rounding off 42.65 = 42.7
  2. If the digits to be removed is less than 5 (i.e 1, 2, 3, 4) then, we leave the previous digit unchanged.
    for e.g NOTE: ( digits to be removed is underlined. )
    • Rounding off 345.234 = 345.23
    • Rounding off 2.800012 = 2.8000
    • Rounding off 42.63 = 42.6

Accuracy and Precision in measurement

Accuracy

Accuracy is the closeness of measured values to the exact value. for eg., Two students measured the density of water as 999 kg/m^3 and 1004 kg/m^3. The actual value of density of water at 4C^0 is 1000kg/m^3. Here 999 kg/m^3 is more close to actual value than 1004kg/m^3 hence more accurate.

The accuracy is determined by the significant figures in the measurement. The higher the number of significant figures, the higher is the accuracy.

Precision

Precision the closeness of measured values to each other. In the same example, students measured the density of water to be 999kg/m^3, 998kg/m^3, and 998.5kg/m^3 in three consecutive measurements. These measured values are precise as these values are close to each other. Not only precise, these values are accurate also as they are close to an exact value.

The values which are inaccurate can also be precise. In the same example of measurement, suppose, students measured 900kg/m^3, 901kg/m^3, and 989kg/m^3 in respective measurements. These values are not accurate but precise as they are close to each other.

The precision is determined by the least count of measuring instruments. The lower the least count of instruments, the higher is the precision of measurements.

The below figure shows an example of a dart game. The black dot represents the position of the dart hit on the board.

accuracy and precision
fig: accuracy and precision

Difference between accuracy and precision

AccuracyPrecision
Accuracy measures the degree of closeness of measured value to an exact value.Precision measures the degree of closeness of measured values to each other.
Accuracy is determined by the number of significant figures in the measured value.Precision is determined by the least count of measuring instruments from which measurement has been done.
Measurement cannot be accurate without being precise. Measurement can be precise without being accurate.

Some FAQ’s

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