Here are the steps to calculate variance and standard deviation from a data set: 1. Find the mean (average) of all the measurements. Add up all values and divide by the total number of measurements. 2. For each measurement, calculate the difference between that measurement and the mean. Square each difference. 3. Add up all the squared differences and divide by the total number of measurements minus 1. This gives the variance. 4. Take the square root of the variance. This gives the standard deviation. The standard deviation indicates how far the measurements spread out from the mean. A low standard deviation means the values are close to the mean, while a high standard deviation means they are more spread out.

1. Physical Quantities
and Measurements
Uncertainties and Deviations

2. Measuring Instruments
• Accuracy is how close a measurement is to the expected or correct value
(true value) of a physical quantity. It is synonymous to validity.
• Precision of a measurement system refers to how close or consistent the
agreement is between repeated measurements (which are repeated under the
same conditions). It is synonymous to reliability.
• Measurements can be both accurate and precise, accurate but not precise,
precise but not accurate, or neither

3. Significant Figures
• These are digits in a reading that specify the precision of the instrument.
RULES
Rule Example
1. All non – zero digits are significant 11 has 2 significant figures.
98762 has 5 significant figures.
2. Zeroes placed before other digits are not significant 0.012 has 2 significant figures
3. Zeroes placed between other digits are significant 2009 has 4 significant figures
4. Zeroes placed after other digits but behind a decimal point are
significant
1.80 has 3 significant figures

4. 5. When the zeroes in a number are placeholders, write
the number in scientific notation to indicate the number
of significant figures
In the number 2000, it is not clear if the
zeroes are significant or not. To show the
number of significant figures, wrote 2000
in scientific notation:
2 x 103 mean 1 significant figure
only
2.00 x 103 mean 3 significant
figures
Rounding off
A number 1.876 rounded off to three significant digits is 1.88 while the number
1.872 would be 1.87. The rule is that if the insignificant digit (underlined) is more than
5, the preceding digit is raised by 1, and is left unchanged if the former is less than 5.
If the number is 2.845, the insignificant digit is 5. In this case, the convention is
that if the preceding digit is even, the insignificant digit is simply dropped and, if it is
odd, the preceding digit is raised by 1. Following this, 2.845 is rounded off to 2.84
where as 2.815 is rounded off to 2.82.

5. • All measurements are made with the help of instruments.
• Physical quantities obtained from experimental observation always have
some uncertainty. Measurements can never be made perfect. Precision of a
number is often indicated by following it with ± symbol and a second
number indicating the maximum error likely.
• Measured value = (true value + uncertainty) units
Mini Activity:
What is the length (measured
value)of the fish?
________________________

6. Accurate? Precise? or What?
• A group of boys were
playing darts
Mini Activity:

7. Zeus pulls of his
stunt first
Accurate? Precise?
or What?

8. Hades pulls of his
stunt first
Accurate? Precise?
or What?

9. Hephaestus pulls
of his stunt first
Accurate? Precise?
or What?

10. Poseidon pulls of
his stunt first
Accurate? Precise?
or What?

11. Physical Quantities
and Measurements
Errors

12. Errors
• The uncertainty in the measurement of a physical quantity is
called error.
• It is the difference between the true value and the measured value
of the physical quantity.
• Errors may be classified into the following types:
• Systemic Errors
• Random Errors

13. SYSTEMATIC ERROR RANDOM ERROR
Characteristics Constant magnitude; always positive (measurement
greater than actual size) or always negative
(measurement less than actual value)
Magnitude is not constant; can be positive or negative
Source The instrument used:
- Zero error in a micrometer screw gauge
- Change in length of a steel rule or vernier
calipers depending on the surrounding
temperature
The observer:
- Reaction time of the observer who uses a
stopwatch
The physical condition of the surroundings:
- Assumed value of the acceleration due to
gravity maybe higher or lower than the actual
value at the location of the experiment
The main source is the observer:
- Wrong reading of the scale of the scale of the
instrument
- Wrong count of the number of oscillations of
pendulum
- Parallax error: error in a reading due to the wrong
positioning of the eye when reading a scale
- Error due to changing temperature of the
surroundings during the experiment where the
temperature is assumed to be constant
Methods of
reducing or
eliminating
error
- To eliminate zero errors, note the zero error,
and subtract the error from the readings
- Check a measurement by using a different
instrument, for example, use another stopwatch
to check the time measured.
- To reduce random errors, make repeated readings
and calculate the average reading. Discard any
reading that differs significantly from the rest
before finding the average.

14. Take your pulse rate following the steps.
Answer what is being asked below.
• Place your pointer and middle fingers on the inside of your opposite wrist just below
the thumb.
• Don’t use your thumb to check your pulse, as the artery in your thumb can make it
harder to count accurately.
• Once you can feel your pulse, count how many beats you feel in one full minute using a
stop watch.
• Do this 5 times. The first pulse rate is to be labelled ME (since you are the one who got
the results). For the other 4 results, let someone in your household (parents, siblings,
househelp) get your pulse rate following the steps above.
• Please note a normal pulse rate is 80 beats per minute.
Mini Activity:

15. Where there instances that there is/are random errors? Why did it
happen? How are these obstacles reduced or eliminated?
_____________________________________________________
_____________________________________________________
_____________________________________________________
_____________________________________________________
_____________________________________________________
_____________________________________________________
_____________________________________________________
TRIALS LABEL PULSE PER
MINUTE
1st ME
2nd
3rd
4th
5th
Where there instances that there is/are systematic errors? Why did it
happen? How are these obstacles reduced or eliminated?
_____________________________________________________
_____________________________________________________
_____________________________________________________
_____________________________________________________
_____________________________________________________
_____________________________________________________
_____________________________________________________

16. Variance
• It is a way to estimate errors from multiple measurements of a physical
quantity. It is by measuring how wide the points in a data set are spread
about the mean or average.

17. STEPSINCALCULATINGTHE
VARIANCEOFASETOF
MEASUREMENTS

18. Using the sets of pulse rates gathered.
• Determine the following:
• Mean
• Variance
• Standard Deviation
Mini Activity: