Measurement and Uncertainty Flashcards
what is measurment
act of quantifying a characteristic element or object
the units used to quantify depend on what
the type of measurement being made
if it is a fundamental quantity, then the label is simply
meters or kilograms or seconds
if it is a derived quantity, then the label represents
the elements from which the measure is derived
-square meters or Pascal
if it is a derived quantity, then the label represents
the elements from which the measure is derived
- square meters or Pascal
every measurement has a degree of uncertainty, how is it determined?
by the scale used to make the measurement
- finer markings have less uncertainty
what is precision?
being able to make repeated measurements and get the same outcome each time
what is accuracy?
degree of closeness of the measurement to the actual measurement value
what is error and what is it used to do?
- the difference between the average value and the true value
- summarize all the flaws in the measurement
what is random error referred to as?
variability or random variation
what is random error?
error due to chance
random error does not have what?
direction
- average of all deviations from average value will be ~0
what happens to random error when more samples are taken and what is affected?
the reduction in error will be greater
- impacts precision
systematic error is referred to as what?
bias
what is systematic error?
error not due to chance alone
what does systematic error have?
net direction
- averaging over a large number of samples does not remove the error
how do you fix systematic errors and what is impacted?
by recognizing the source of the error
- recalibrating the equipment
- impacts accuracy and precision
which set is more precise?
A) 18.2 , 18.4 , 18.35
B) 17.9 , 18.3 , 18.85
C) 16.8 , 17.2 , 19.44
A
who is more accurate when measuring a book that has a true length of 17.0 cm?
Susan: 17.0 cm, 16.0 cm, 18.0 cm, 15.0 cm
Amy: 15.5 cm, 15.0 cm, 15.2 cm, 15.3 cm
Susan
who is more precise when measuring the same 17.0 cm book?
Susan: 17.0 cm, 16.0 cm, 18.0 cm, 15.0 cm
Amy: 15.5 cm, 15.0 cm, 15.2 cm, 15.3 cm
Amy
how do you decide what is accurate or precise?
Accuracy- consider the mean value and compare that to the expected (nominal value)
Precision- consider the standard deviation among the measurement (spread in the data)
summary for accuracy and precision
Accurate/Precise= on target #
Accurate/Not precise= near target #
Not accurate/precise= near e/o, not near target #
Not accurate/Not precise= random, not near e/o or target #
what to remember about accuracy/precision
- A measurement can be precise but not accurate
- A measurement can be neither accurate or precise
- A measurement can be accurate but not precise (increased variability)
- A measurement can be precise and accurate
if I measure a length of string and record that is it 10 cm long and my colleague measures the same string and finds that it is 3.93 inches, who is right?
Both are
- difference is the measurement system
how can we describe errors in measurement?
- random
- systemic (bias)
how is systemic bias split/what type of errors are systematic?
- offset errors
- scale factor errors
what is the difference between offset errors and scale factor errors?
offset- calibration error or no offset made
scale factor- errors proportional to “true” measurement
example of offset error-
imagine you have a home scale, and you want to measure 1 lb of hamburger to freeze from a large amount of hamburger you brought from the store. you haul out the scale and it is set to zero and you put the bowl that will hold the meat on the scale. what will be the offset error?
Any measurement you make will be off by the same amount- the weight of the bowl that holds the meat
consider that you are using a tape measure to measure some fabric and the tape measure has been stretched to 101% of its original size. what will be the scale factor error?
Any measurements that are made with this tape measure will be 101% of the actual measurement.
how do you calculate percent error?
% error= (observed result-expected result)/expected results x 100%
SI Prefixes
10^1 deca da
10^‐1 deci d
10^2 hecto h
10^‐2 centi c
10^3 kilo k
10^‐3 milli m
10^6 mega M
10^‐6 micro μ
10^9 giga G
10^‐9 nano n
10^12 tera T
10^‐12 pico p
non-SI units
time (minute (m), hour (h). day (d))
volume (liter (L or l))
mass (ton (t))
energy (electronvolt (eV))
non-SI unit relation to Si
time:
- 1 min= 60 s
- 1 h= 3600s
- 1 d= 86400s
volume:
- 1 L= 1 cm^3
mass:
- 1 t = 1000 kg
energy:
- 1 eV= 1.602 x 10^-19 J
base units
time (seconds- s)
length (meter- m)
mass (kilogram- kg)
what are derived units?
units that combine two base units
events that repeat
If there are events that occur repeatedly then it makes it easier to talk about the time it takes before the event starts again.
- This time between events is the period and is measured in units of time (s)
- Can also consider how many times an event happens in a unit of time (frequency)
Relation between period and frequency
Frequency (Hz)= 1/ Period (s)
- have inverse relationship
As freq increases, period decreases
As period decreases, freq increases
velocity
measure of distance over time (v (m/s)= d (m)/t (s))
acceleration and velocity relationship
if the velocity of an object changes then it is accelerating
how to calculate acceleration
acceleration (m/s^2)= change in velocity (m/s)/time (s)
change in velocity
velocity at the end of the measurement interval- velocity at the beginning of the measurement interval
what two measurements are needed to calculate area?
length and width
reported as m^2
what three measurements are needed to calculate volume?
length, width and height
reported as m^3
- need to count # of cubes if object is irregular shape
what is density and how do you calculate it?
measurement of mass and volume
d (kg/m^3) = mass (kg)/ volume (m^3)
density of water
1000 kg/m^3
density of air
1 kg/m^3
Accuracy in the lab/clinic
- measurement are expected to be no greater than 0.1%
Sig Fig reporting
3 sig figs
- # of leading 0 does not count!!
ex. 1.23, 0.340, 0.00000000631 are all 3 sig figs
how to get 3 sig figs
Round the values by looking at the 4th digit (ignore leading zeroes)
- If the 4th value is < 5 then truncate the number
- If the 4th value is > 4 then add 1 (round up) the third digit and truncate after the third digit
what are the two parts of a number when the calculator breaks the answer?
mantissa and exponent
what is the mantissa?
between 1 and 10
what is the exponent?
number 10 is raised to (10^-8, the exponent is -8)
what is scientific notation
a mean to express very large or small numbers in an efficient way
how to add/subtract exponents
if the exponent is the same, then add or subtract the mantissa and use the exponent
adding exponent example
2.0 x 10^3 + 1.2 x 10^2
- Convert 1.2 x 10^2 to 0.12 x 10^3 to make the exponent the same (larger exponent #)
- add the mantissas (2.0 and 0.12) = 2.12
- The final sum is 2.12 x 10^3 or 2120
subtracting exponent example
1.5 x 10^3 - 6.5 x 10^2
- Convert 1.5 x 10^3 to 15.0 x 10^2 to make the exponent the same (smaller exponent #)
- subtract the mantissas (15 and 6.5) = 8.5
- The final sum is 8.5 x 10^2 or 850
how to multiply/divide exponents
multiplication: mantissas are multiplied and exponents are added
division: mantissas are divided and exponents are subtracted
converting from standard notation to scientific notation
- Place a decimal after the first non‐zero digit
- Place the rest of the non‐zero digits after the decimal
- Add x 10^n
- Count the number of places the decimal was moved to get n
- If the standard # value is > 0 exponent is positive, if not exponent is negative
- Scientific notation: #(mantissa) x 10^n
what is an exponent
number of times a base is multiplied by itself
if the exponent is 0, the value is ALWAYS
1
what do exponents tell you about the number
10^0=1
10^positive #= # of zeros in result after mantissa
10^negative #= # of decimal places in result
Logs
logs are always base 10 unless told otherwise and is a rewrite of an exponent
log examples
log(1000)=4 is 10^4=1000
log base 2 of 16 = 4 is 2^4=16
exponent and log relation
log base b of a= c
b^c=a
ex. log base 5 of 25= 2
5^2=25
antilog
opposite of log
antilog equation
antilog base b of a= c
b^a=c
ex. antiog base 10 of 4 = 10,000
10^4=10,000
what is geometry
study of points, lines, angles, surfaces, and shapes
what are the two divisions of geometry
plane
solid
why does geometry matter
helps describe and understand how sound waves travel and how sound is afected as it encounts objects in space
angle
the space between the sides of a figure or the displacement of one side relative to the other
acute angle
displacement is <90 degrees
right angle
displacement is 90 degrees
obtuse angle
displacement is >90 degrees and <180 degrees
circle
every point along circumference is equidistant from the center
circumference
equivalent to the perimeter (C)
radius
line from the center of the circle to any point of the circle (R)
diameter
line from one side of the circle to the other side and passes through the center of the circle (D)
how to calculate circumference
c=2pi x Radius
c=pi x Diameter
what is the angle referred to as and in what
theta θ
degrees
Degrees
- Angles in the upper half of the circle are between 0 degrees and 180 degrees
- Angles in the lower half of the circle are between 0 degrees and 180 degrees
- These are conventions so it is possible to start anywhere.
The figure MUST be labelled
Radians
1 radian is equal to the angle that is created when the radius moves such that the arc on the circle is the same length as the radius
how many radians are in one complete circle?
6+
how to convert radians to degrees and vice versa
x(rad)= y(degree) x pi/180 degrees
x(degrees)= y(rad) x 180 degrees/pi
right triangle
one 90 degree angle
opposite side to right angle is hypotenuse
pythagorean theorem
a^2 +b^2 = c^2
where c is hypotenuse, b is adjacent, and a is opposite
trig functions
sinθ= opposite/hypotenuse
cosθ= adjacent/hypotenuse
tanθ= opposite/adjacent
cosecantθ= hypotenuse/oppositve=1/sinθ
Cartesian or rectangular coordinates
mapping system to identify point in 2D space
how do we identify a point in 2D space
provide an x‐ coordinate and y‐coordinate relative to some (0,0) location or center location or origin
what do the x and y coordinates do
capture how far to the right or left (x) the point is from the origin and how far up or down the point is from the origin
polar coordinates
each point in space is defined by the radius and the
angle (r, θ)
what is θ referred to as in polar coordinates
phase angle
trig in polar coordinates
sinθ= opposite/hypotenuse
sinθ= y/radius (r) or y= rsinθ
cosθ= adjacent/hypotenuse
cosθ= x/radius (r) or x= rcosθ
Cartesian and polar plots
same point in space can be labelled in (x,y) values and in (r, θ)
what two things can we do to describe a series of points in space over a time window
- draw a graph
- create a function that describes a value as a function of time
what is a function?
an equation that shows a relationship between values on one axis when the other axis is known; or the relationship between two sets of number
what is a graph?
a visual representation between two variables
what is interpolation?
determining the value of f(x) when x is not one of the numbers we used
what is extrapolation?
determining the value beyond what is graphed
straight line equation
y=mx+b
slope of line
m
change in y/change in x
rise/run
y-intercept
b
where the function crosses y-axis
linear function
no bends- straight lines only
no variables raised to a power >1
ordinal scale
- order is important
- no numerical quality assigned
ratio scale
- quantitative/numerical value
- absolute 0 (0-calorie condition)
- measurable distance
nominal scale
- nothing important about order
- # is typically assigned based on the order of registration
interval scale
- no true 0
- represents values below 0
- measures difference between values
Which of the following scales of measurement best describes values used to determine the number of shoe sizes for a specific footwear brand?
Interval
Which of the following represents an ordinal scale of measurement?
Rank of test scores
