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The analysis of uncertainties (errors) in measurements and calculations is essential in the physics laboratory. For example, suppose you measure the length of a long rod by making three measurement x = x_{best }Â± âˆ†x, y = y_{best} Â± âˆ†y, and z = z_{best} Â± âˆ†z. Each of these measurements has its own uncertainty âˆ†x, âˆ†y, and âˆ†z respectively. What is the uncertainty in the length of the rod L = x + y + z? When we add the measurements do the uncertainties âˆ†x, âˆ†y, âˆ†z cancel, add, or remain the same? Likewise , suppose we measure the dimensions b = b_{best} Â± âˆ†b, h = h_{best} Â± âˆ†h, and w = w_{best} Â± âˆ†w of a block. Again, each of these measurements has its own uncertainty âˆ†b, âˆ†h, and âˆ†w respectively. What is the uncertainty in the volume of the block V = bhw? Do the uncertainties add, cancel, or remain the same when we calculate the volume? In order for us to determine what happens to the uncertainty (error) in the length of the rod or volume of the block we must analyze how the error (uncertainty) propagates when we do the calculation. In error analysis we refer to this as** error propagation**.

There is an error propagation formula that is used for calculating uncertainties when adding or subtracting measurements with uncertainties and a different error propagation formula for calculating uncertainties when multiplying or dividing measurements with uncertainties. Letâ€™s first look at the formula for adding or subtracting measurements with uncertainties.

**Adding or Subtracting Measurements with Uncertainties.**

Suppose you make two measurements,

x = x_{best } Â± âˆ†x

y = y_{best} Â± âˆ†y

What is the uncertainty in the quantity q = x + y or q = x â€“ y?

To obtain the uncertainty we will find the lowest and highest probable value of q = x + y. Note that we would like to state q in the standard form of q = q_{best} Â± âˆ†q where

q_{best} = x_{best} + y_{best}.

**(highest probable value of q = x + y): **

(x_{best}+ âˆ†x) + (ybest + âˆ†y) = (x_{best}+ y_{best}) + (âˆ†x +âˆ†y) = q_{best} + âˆ†q

**(lowest probable value of q = x + y):**

(x_{best}- âˆ†x) + (y_{best} - âˆ†y) = (x_{best}+ y_{best}) - (âˆ†x +âˆ†y) = q_{best} â€“ âˆ†q

Thus, we that

âˆ†q = âˆ†x + âˆ†y

is the uncertainty in q = x + y. A similar result applies if we needed to obtain the uncertainty in the difference q = x â€“ y. If we had added or subtracted more than two measurements x, y, ......, z each with its own uncertainty âˆ†x, âˆ†y, ......... , âˆ†z respectively , the result would be

âˆ†q = âˆ†x + âˆ†y + ......... + âˆ†z

Now, if the uncertainties âˆ†x, âˆ†y, ........., âˆ†z are random and independent, the result is

Ex. x = 3.52 cm Â± 0.05 cm

y = 2.35 cm Â± 0.04 cm

Calculate q = x + y We would like to state q in the standard form of q = q_{best} Â± âˆ†q

x_{best} = 3.52cm, âˆ†x = 0.05cm

y_{best} = 2.35cm, âˆ†y = 0.04cm

q_{best} = x_{best }+ y_{best} = 3.52cm + 2.35cm = 5.87cm

q = 5.87cm Â± 0.06cm

**Multiplying or Dividing Measurements with Uncertainties**

Suppose you make two measurements,

x = x_{best } Â± âˆ†x

y = y_{best } Â± âˆ†y

What is the uncertainty in the quantity q = xy or q = x/y?

To obtain the uncertainty we will find the highest and lowest probable value of q = xy. The result will be the same if we consider q = x/y. Again we would like to state q in the standard form of q = q_{best} Â± âˆ†q where now q_{best }= x_{best} y_{best}.

**.(highest probable value of q = xy): **

(x_{best}+ âˆ†x)(y_{best }+ âˆ†y) = x_{best}y_{best} + x_{best} âˆ†y +âˆ†x y_{best} + âˆ†x âˆ†y = q_{best} + âˆ†q

= x_{best}y_{best }+ (x_{best} âˆ†y + âˆ†x y_{best}) = q_{best }+ âˆ†q

**(lowest probable value of q = xy):**

(x_{best}- âˆ†x)(y_{best }- âˆ†y) = x_{best}y_{best} - x_{best }âˆ†y - âˆ†x y_{best} + âˆ†x âˆ†y = q_{best }â€“ âˆ†q

= x_{best}y_{best }â€“ (x_{best }âˆ†y + âˆ†x y_{best}) = q_{best} â€“ âˆ†q

Since the uncertainties âˆ†x and âˆ†y are assumed to be small, then the product âˆ†x âˆ†y â‰ˆ 0. Thus, we see that âˆ†q = x_{best }âˆ†y +âˆ†x y_{best} in either case. Dividing by xbestybest gives

Again, a similar result applies if we needed to obtain the uncertainty in the division of q = x/y. If we had multiplied or divided more than two measurements x, y, ......, z each with its own uncertainty âˆ†x, âˆ†y, ......... , âˆ†z respectively, the result would be

Now, if the uncertainties âˆ†x, âˆ†y, ........., âˆ†z are random and independent, the result is

Ex. x = 49.52cm Â± 0.08cm

y = 189.53cm Â± 0.05cm

Calculate q = xy We would like to state q in the standard form of q = q_{best} Â± âˆ†q

x_{best} = 49.52cm, âˆ†x = 0.08cm

y_{best} = 189.53cm, âˆ†y = 0.05cm

q_{best }= x_{best}y_{best }= (49.52cm)(189.53cm)=9.38553 x 10^{3} cm^{2}

âˆ†q = (1.63691 x 10^{-3})q_{best }= (1.63691 x 10^{-3}) (9.38553 x 10^{3} cm^{2})

âˆ†q = 15.3632cm^{2} â‰ˆ 20 cm^{2 }

q = 9390 cm^{2} Â± 20 cm^{2}

**Uncertainty for a Quantity Raised to a Power **

If a measurement x has uncertainty âˆ†x, then the uncertainty in q = x^{n}, is given by the expression

**Ex**. Let q = x^{3} where x = 5.75cm Â± 0.08 cm.

Calculate the uncertainty âˆ†q in the quantity q.

We would like to state q in the standard form of q = q_{best} Â± âˆ†q