topic 1.2 extended b - laboratory number crunching

Consider the following "experiment" where you construct a catapult which launches a dart at a target.The object is to hit the bulls eye.

Topic 1.2 Extended B - Laboratory number crunching

First Trial Second Trial Third Trial Fourth Trial

Low PrecisionHits not grouped

Low AccuracyAverage well below bulls eye

Low PrecisionHits not grouped

High AccuracyAverage right at bulls eye

High PrecisionHits grouped

Low AccuracyAverage well below bulls eye

High PrecisionHits grouped

High AccuracyAverage right at bulls eye

FYI: RANDOM ERROR is where accuracy varies in a random manner.

FYI: SYSTEMATIC ERROR is where accuracy varies in a predictable manner.

Having gathered your data (recorded to the most significant figures allowed by your measuring devices), you will want to analyze your data, and somehow decide on its validity.We have already talked of precision. For our meter stick we could measure to 3 significant figures with a precision of ± 0.05 mm:


0 1

1 cm 1 mm

I estimate the length to be 1.17 cm or 11.7 mm.Considering error, L = 11.7 mm ± .05 mm.

FYI: This is the precision in a SINGLE measurement.

We call the precision of 0.05 mm the absolute error.


There are three useful percent formulas:

Fractional Error =Absolute ErrorMeasured Value

The first formula compares absolute error to a measured value:

For example, since we measured L = 11.7 mm ± .05 mm, we have

·100%

Fractional Error = 0.43%

Fractional Error

FYI: FRACTIONAL ERROR is a way to evaluate the ACCURACY of a measurement.

Fractional Error =0.0511.7

·100%

FYI: Note that the SMALLER the MEASURED VALUE, the LARGER the FRACTIONAL ERROR.

FYI: FRACTIONAL ERROR between 1% and 5% are considered fairly accurate. Less than 1% is very accurate, and greater than 10% is considered "rough."



Percent Difference =DifferenceAverage

The second formula compares two measured values x1 and x2:

% Diff = x1 - x2

(x1 + x2)12

For example, suppose you measure the line of the previous example to be x1 = 11.7 mm the first time, and x2 = 11.8 mm the second time.

% Diff =11.7 - 11.8

(11.7 + 11.8)12

·100%

·100%

·100%

% Diff = 0.851%

Percent Difference

FYI: PERCENT DIFFERENCE is a way to evaluate the PRECISION in your measurements - provided they are expected to be the same.



Percent Error =DifferenceTrue Value

The third formula compares a measured value xmeas to an accepted or "true" value xtrue:

For example, suppose you measure the diameter and circumference of a circle, and calculate pi to be 3.11.

·100%

% Err = xmeas - xtrue

xtrue

·100%

% Error =3.11 - 3.14

3.14·100%

% Err = 0.955%

Percent Error

FYI: PERCENT ERROR is a way to evaluate the ACCURACY in your measurements - provided you know what they are supposed to be.

Sometimes you have lots of data that you need to compare to a known value:


For example, suppose you measure the freefall acceleration of a dropped ball.We will learn that all freely falling objects should have a CONSTANT acceleration.So we find the average, or mean value of the data.Average Value = Sum of Values

Number of Valuesx = xi

i=1

N

Average or Mean

Freefall Acceleration

0

2

10

18

20

0

5

10

15

20

25

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

Time (s)

Acce

lera

tion

(m/s

/s)

Bob's Data

a = 0 + 2 + 10 + 18 + 205

a = 10 m s-2

Suppose another student has this data for the same experiment:


0

5

10

15

20

25

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

Time (s)

Acc

eler

atio

n (m

/s/s

)

Abby's Data

Bob

a = 0 + 10 + 10 + 10 + 205

a = 10 m s-2 Abby

Who's lab results are best - Bob's or Abby's?

1N



0

2

10

18

20

0

5

10

15

20

25

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

Time (s)

Acce

lera

tion

(m/s

/s)

Bob's Data Freefall Acceleration

0

5

10

15

20

25

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

Time (s)

Acce

lera

tion

(m/s

/s)

Abby's Data

a a

At first glance we may say the results are identical: After all, both have the same average acceleration of 10 m/s2.But instinctively, you may feel that Abby's data is better, because, after all, three of the data points are exactly equal to the average acceleration.In technical terms, both data sets have the same ACCURACY, but Abby's data set has better PRECISION.

So how do we distinguish between the data sets (preferably with a single number)?One way is by examining the range of the data points.

Range = Max(xi) - Min(xi) Range

These data sets each have the same range (spread) of 20 m/s2 - 0 m/s2 = 20 m/s2.Thus range isn't the best indicator of a data set's precision.

Ran

ge

Ran

ge



0

2

10

18

20

0

5

10

15

20

25

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

Time (s)

Acce

lera

tion

(m/s

/s)


0

5

10

15

20

25

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

Time (s)

Acce

lera

tion

(m/s

/s)

Abby's Data

a a

Another indicator of precision is deviation from the mean:

ith deviation = (xi - x) Deviation from the Mean

- 10

- 80

810

- 10 0 0 0

10

FYI: The ith deviation only tells you how precise a SINGLE data point is. To find the PRECISION of the data SET, you might find the AVERAGE of all of the deviations. But the average in both cases is ZERO!

-10 + -8 + 0 + 8 + 105

= 0 m s-2

-10 + 0 + 0 + 0 + 105

= 0 m s-2



0

2

10

18

20

0

5

10

15

20

25

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

Time (s)

Acce

lera

tion

(m/s

/s)


0

5

10

15

20

25

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

Time (s)

Acce

lera

tion

(m/s

/s)

Abby's Data

a a

A way around this unfortunate circumstance is to take the AVERAGE OF THE ABSOLUTE VALUES of the deviations:

Average Absolute Deviation from the Mean

10

80

810

10 0 0 0

10

average absolute deviation = |xi - x|1N

10 + 8 + 0 + 8 + 105

= 7.2 m s-2

10 + 0 + 0 + 0 + 105

= 4 m s-2

Now we have an indicator (in the form of a single number) which tells us that Abby's data is better than Bob's.



0

2

10

18

20

0

5

10

15

20

25

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

Time (s)

Acce

lera

tion

(m/s

/s)


0

5

10

15

20

25

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

Time (s)

Acce

lera

tion

(m/s

/s)

Abby's Data

a a

Because absolute values are tedious (you have to take the opposite of negative differences), standard deviation is the accepted way to determine precision:

Standard Deviation

-10

-80

810

-10 0 0 0

10

(-102) + (-82) + 02 + 82 + 102

5-1

= 9.1 m s-2

(-10)2 + 02 + 02 + 02 + 102

5-1

= 7.1 m s-2

standard deviation = = (xi - x)21N-1

FYI: The standard deviation as the square root of the average of the squares of the deviations.

FYI: Squaring the deviations produces POSITIVE values, just like the absolute value.

FYI: For reasons we don't need to go into, we divide by ONE LESS THAN THE TOTAL NUMBER OF DATA POINTS when finding this average.

topic 1.2 extended b - laboratory number crunching

Documents