1. Significant Figures
The significant figures of a (measured or calculated) quantity are the meaningful digits in it. There are conventions which you should learn and follow for how to express numbers so as to properly indicate their significant figures.
Any digit that is not zero is significant. Thus 459 has three significant figures and has four significant figures.
Zeros between non zero digits are significant. Thus 2006 has four significant figures.
Zeros to the left of the first non zero digit are not significant. Thus has only two significant figures. This is more easily seen if it is written in a scientific notation, i.e., 7.4×10-5.
For numbers with decimal points, zeros to the right of a non zero digit are significant. Thus 3.00 has three significant figures and has two significant figures. For this reason it is important to keep the trailing zeros to indicate the actual number of significant figures.
For numbers without decimal points, trailing zeros may or may not be significant. Thus, 400 indicates only one significant figure. To indicate that the trailing zeros are significant a decimal point must be added. For example, 400. has three significant figures, and 4× has one significant figure.
Exact numbers have an infinite number of significant digits. For example, if there are two oranges on a table, then the number of oranges is Defined numbers are also like this. For example, the number of centimeters per inch (2.54) has an infinite number of significant digits, as does the speed of light ( m/s).

2. Error means the uncertainty (limitation) in the measurements=
Common Mistakes
(2.80) (4.5039) =

3. Classification of Error: Systematic errors and Random errors
Systematic errors are errors which tend to shift all measurement data in a systematic way so their mean value is displaced. This may be due to such things as incorrect calibration of equipment.
Random errors are errors which fluctuate from one measurement to the next. They yield results distributed about some mean value. They can occur for a variety of reasons.
Random errors displace measurements in an arbitrary direction whereas systematic errors displace measurements in a single direction.
Real Value
Measured
Real Value
Measured
Systematic errors: Data Shift
Random errors: Data dispersion (fluctuation)
In most case, we may not know the Real Value! Do your best to avoid Systematic errors that course Data Shift .
But you should always give your Random errors in your specific Lab. That is how big your data fluctuates.

4. Expressing Experimental Error and Uncertainty: Percent ErrorXT XE X1 X2
Real Value
Measured
Measured
The absolute difference (absolute error):
Absolute difference/ error = |XE  XT|.
XE : experimental measured value
XT : theoretical (true) value
Percent Difference
It is not always possible to find theoretical value for a physical quantity to be measured. In such circumstance we resort to comparison of results obtained from two equally dependable measurements. The comparison is expressed as a percent difference which is given by
The fractional error (relative error) is given by
In your Lab reports, one may use 𝑋 (average value) as 𝑋 𝐸 .

5. Percent error: ------ accuracy of the measurements
Ex: You are given a cube of pure copper.
You measure the sides of the cube to find the volume and weigh it to find its mass. When you calculate the density using your measurements, you get 8.78 grams/cm3.
Copper’s accepted density is 8.96 g/cm3.
What is your percent error?
experimental value = 8.78 g/cm3 accepted/ theoretical value = 8.96 g/cm3
Absolute error = |XE  XT|= 8.78−8.96 =0.18 g/cm3
Percent error = =0.02=2%
The accuracy of the mean value experimental measurements should be expressed in terms of the percent error or percent difference.
Example:
Percent error may not show the data dispersion very well
Real Value
Measured
One Special Case:
Theoretical value = 8.96 g/cm3
Average experimental value = 8.96 g/cm3
Percent error = 0 %
Example: Percent Difference
Car M costs $50,000 and car L costs $40,000.
We wish to compare these costs. With respect to car L.
𝑋 1 =$50,000
𝑋 2 =$40,000
Percent Difference = 10,000 45,000 =22.2%
Need better way to describe the Random errors distribution
Two Key Words: Accuracy and precision

6. Accuracy and precision
Accuracy is how close a measured value is to the actual (true) value
(aCcuracy is Correct).
Accuracy depends on the instrument we are measuring with. 
Precision is how close the measured values are to each other.
(pRecision is Repeating).

7. Accuracy and precision Accuracy ----- Percent error
Real Value
Measured
Real Value
Measured
Low Accuracy
Low Precision
Low Accuracy
High Precision
Real Value
Measured
Real Value
Measured
High Accuracy
Low Precision
High Accuracy
High Precision
Accuracy Percent error
Precision ---- standard deviation

8. Question: What is the experiment data pattern looks like?
mean value 𝑋
Measure quantity X, Measured Data: 𝑋 1 , 𝑋 2 …, 𝑋 𝑁
N: Suppose an experiment were repeated N times
Question: What is the experiment data pattern looks like?
Answer: Most measured data of Physical quantities have distribution that is called Normal distribution (or Gaussian distribution).
If one made one more measurement of quantity X then it would have some 68% probability of lying within
x0: most probable value (mean value, or expectation)
s: called the standard deviation
𝜎 2 : called variance.
Now we can express the experimental value Xv as
Note that this means that about 32% of all experiments will disagree with the accepted value by more than one standard deviation!

9. measure of the dispersion of the data about the mean.
Normal distribution
If one made one more measurement of quantity X then it would have some 68% probability of lying within
𝜇: mean value
s: standard deviation
𝑋 𝜈 =52 ±6 (Measurement with higher precision)
𝑋 𝜈 =52 ±12 (Measurement with lower precision)
standard deviation
measure of the dispersion of the data about the mean.

10. Calculation of Deviation from the mean ( 𝒅 )
Measure quantity X, Measured Data: 𝑋 1 , 𝑋 2 …, 𝑋 𝑁
Average/mean deviation:
N: Suppose an experiment were repeated N times
𝑋 : mean value
𝒅 : Average/mean deviation
s: standard deviation
The experimental value Xv of a measured quantity is given in the form
Deviation from the mean
Average/mean deviation relate Accuracy
uncorrected sample standard deviation, or sometimes the standard deviation of the sample (considered as the entire population)

11. Calculation of standard deviation s
Measure quantity X, Measured Data: 𝑋 1 , 𝑋 2 …, 𝑋 𝑁
Now we can express the experimental value Xv as
N: Suppose an experiment were repeated N times
If one made one more measurement of x then (this is also a property of a Gaussian distribution) it would have some 68% probability of lying within
𝑋 : mean value
s: standard deviation
Deviation from the mean
Standard deviation
Note that this means that about 32% of all experiments will disagree with the accepted value by more than one standard deviation!
𝜎 𝑋 = 𝑘=1 𝑁 𝑋 𝑘 − 𝑋 𝑁 = 𝑘=1 𝑁 𝑑 𝑘 2 𝑁
standard deviation  
(consider the entire
population)
Sometimes our data is only a sample of the whole population. Researchers may not able to draw many samples from the population of interest. Therefore, it is essential for them to be able to determine the probability that their sample measures are a reliable representation of the full population, so that they can make predictions about the population. The Example: Sam has 20 rose bushes, but only counted the flowers on 6 of them! But when we use the sample as an estimate of the whole population, the Standard Deviation formula changes to Sample Standard Deviation.
N: The number of values (population).
Sample/true standard deviation
𝜎 𝑋 = 𝑘=1 𝑁 𝑋 𝑘 − 𝑋 𝑁−1 = 𝑘=1 𝑁 𝑑 𝑘 2 𝑁−1
N: The number of values (the sample size. a random sample drawn from some large parent population).

12. Example. Standard deviation
Population standard deviation of grades of eight students
Suppose that the entire population of interest was eight students in a particular class. The grades of a class of eight students (that is, a statistical population) are the following eight values:
Sample standard deviation of grades of eight students
Suppose that the entire population of interest was eight students in a particular class. Now take 4 students’ grades as the sample. The grades of the four students (that is, the sample) are the following eight values: 2, 5, 7, 9.
𝑋 = =5.75
𝑑 1 = 2−5.75 =3.75
𝑑 2 = 5−5.75 =0.75
𝑑 1 = 7−5.75 =2.75
𝑑 1 = 9−5.75 =4.75
The variance
𝑑 𝑑 𝑑 𝑑 −1 = =15
Sample standard deviation
15 =3.9
First, calculate the deviations of each data point from the mean, and square the result of each:
Deep question: Why there is a N and N1 difference?
the population standard deviation is equal to the square root of the variance:
4 = 2
The reason why we divide by N to get the best estimate of the mean and only by N-1

13. Use "sample standard deviation".
In The Physics Lab, when do we use standard deviation; when do we use sample standard deviation?
N: Suppose an experiment were repeated N times
If N less than 10,
Use "sample standard deviation".

14. Standard error of the mean Standard error of the mean
Example
This shows four samples of increasing size. Note how the standard error reduces with increasing sample size.
Standard error of the mean
If you measure a sample from a wider population, then the average (or mean) of the sample will be an approximation of the population mean. But how accurate is this?
Standard error of the mean

15. Propagation of Errors (optional)

16. Pre-lab Questions
1. (a) Distinguish among the types and causes of experimental errors.
(b) How can one reduce each type of experimental error?
2. Explain the difference between measurement precision and accuracy. How are these
related to the types of error you have discussed in question (1)?
3. What determines the number of significant figures used in reporting measured values?
4. For a series of experimentally measured values give the definition of
(a) average or mean value
(b) the deviation from the mean
(c) average deviation
(d) standard deviation
5. Explain how one can express experimental result using the quantities discussed in (4).
6. (Optional) Explain how one can account for uncertainty in calculation of area from
measurements of length and width of a rectangular block. [Let dl and dw be
uncertainties in length and width measurement respectively. Express dA in terms of
the two uncertainties ]

17. Lab1 Report Cover Page Index Name … 1.5 Data and Data Analysis. 1. 2.3. 4. 5.