Error, Accuracy, Precision, and

Standard DeviationNotes

Errors

Two types of errors: random and systematic

• Random errors: uncontrollable events, like air currents, temperature variations, and electrical variations.

The error can be minimized by taking a large number of measurements (100 or more).

Random errors are recognized by the fact that the values are BOTH above and below the true value

• Systematic errors: controllable events.

Find the error, fix the error and repeat the experiment.

Types of systematic errors:Equipment : the equipment is worn, out of

calibration or broken, fix the equipment and repeat the measurements.

Technique: some part of the procedure is incorrect. Examples: not looking at eye level at a graduated cylinder or balance, cars are not released at the same time.

Bias: eliminating a number because you do not like it. Unless a known error has occurred (can eliminate then) you cannot throw out a value because it is different.

• Systematic errors are recognized because all of the values are EITHER above or below the true value.

• All of the numbers we measure need to be evaluated and accuracy, precision and standard deviation are the tools we use to do this.

Accuracy

• How close a value is to the true or accepted value (an average can be compared to the accepted value)

• Only one measurement is necessary for calculating an accuracy but many numbers is preferred and the accuracy of the average is then taken.

Accuracy: % error = True value – experimental value

x100True value

* The experimental value can be the average

Desired value is zero.

Precision

• How close a set of values are to each other.

• Requires at least 2 values; more are better.

% difference = High – Low x 100Average

* Desired value is zero.

Acceptable ranges are arbitrary but for Physics we will use

0-1% Excellent1-7% Good7-15% Fair15 and up Redo (Unacceptable)

• What is the precision is high, how can it be fixed?

• Look to see if there is an outlier in the set and statistically try to eliminate it.

Standard Deviation (S)(sample standard deviation)

• Population Standard Deviation (σx on the calculator)The standard deviation of the entire population of data

• Sample Standard Deviation (Sx on the calculator)The standard deviation of a small sample of the whole population – this is all that we are able to collect.

√ Σ(x-ave)2

n-1

√ - the square root of the entire thingΣ – sum ofx – a valueave – average of all valuesn – the number of values

√ Σ(x-ave)2

n-1

Take the value, subtract the average and square this number. (Do this for all values.)

Add all of these together. Subtract one from number of values.Divide your sum by this difference. Take the square root of the whole

thing.

Example Values

2.542.552.562.572.58

Ave = 12.80/5= 2.560

Example Values value – average

difference2 2.54 -.02 0.00042.55 -.01 0.00012.56 0 02.57 .01 0.00012.58 .02 0.0004

Ave = 12.80/5= 2.560

Example Values value – average

difference squared2.54 -.02 0.00042.55 -.01 0.00012.56 0 02.57 .01 0.00012.58 .02 0.0004

12.80/5 0.0010Average = 2.560 n-1 = 5-1 = 4

0.0010/4 = 0.00025√ 0.00025 = 0.016

Standard deviation values are hard to interpret (2.560 + 0.016)

Hard to say from the numbers whether they are good or not.

Therefore, we use Relative Standard Deviation.

• Relative Standard Deviation = s/average x 100

0.016 x100 = 0.63%2.560

Easier to interpret: 2.560 + 0.63% very close

If you have a value that does not fit the set, you must statistically show if it is an outlier.

Two methods to do so are: 1. 2 standard deviations2. q test

If have a set of values, is 2.79 an outlier?

2.542.552.562.792.572.58

2 Standard Deviations

2.542.552.562.792.572.58

Average = 2.598Sx = 0.092.598 + .18 = 2.78 2.598 -.18 = 2.42Range of values 2.42 to 2.78The value 2.79 would be an outlier because it

is beyond 2 standard deviations from the average.

Q test

Questionable value – closest value numerically

Range of all values

= q value

Compare the results to the Q values, if your questionable value is larger than the 95% confidence Q value, then it is an outlier.

2.79 – 2.58 = 0.842.79 – 2.54

Number of values 95% confidence Q value

3 0.943

4 0.754

5 0.640

6 0.564

7 0.510

8 0.469

9 0.438

10 0.412

Using the Calculator for Standard Deviation

Plug the values into the calculatorHit STAT buttonSelect 1: EditEnter the list of data Hit STAT buttonSelect CALC menuSelect 1: 1-Var Stats Hit Enter

Avg 2.56 (¯x)Sum 12.8 (Σx)Sx = 0.0158 = 0.016 (sf of standard deviation values is first non- zero digit unless it is a one then keep 2 digits)