UNITS AND MEASUREMENTS
PHYSICAL QUANTITY
TYPES OF PHYSICAL QUANTITY
 Fundamental quantities : The physical quantities which do not depend upon other physical quantities are called fundamental or base physical quantities. e.g. mass, length, time temperature electric current, luminous intensity and amount of substance.
 Derived quantities : The physical quantities which depend on fundamental quantities are called derived quantities
e.g. speed, acceleration, force, etc.
UNIT
 It should be suitable in size (suitable to use)
 It should be accurately defined (so that everybody understands the unit in same way)
 It should be easily reproducible.
 It should not change with time.
 It should not change with change in physical conditions i.e., temperature, pressure, moisture etc.
 It should be universally acceptable.
FUNDAMENTAL (OR BASE) AND DERIVED UNITS
SYSTEM OF UNITS
DIFFERENT TYPES OF SYSTEM OF UNITS
MERITS OF S.I. UNITS
 SI is a coherent system of units: This means that all derived units are obtained by multiplication and division without introducing any numerical factor.
 SI is a rational system of units: This is because it assigns only one unit to a particular physical quantity.
 SI is an absolute system of units: There is no gravitational unit in this system.
 SI system is applicable to all branches of science.
CONVENTIONS OF WRITING OF UNITS AND THEIR SYMBOLS
 Unit is never written with capital initial letter.
 For a unit named after scientist the symbol is a capital letter otherwise not.
 The unit or symbol is never written in plural form.
 Punctuations marks are not written after the symbol.
DEFINITIONS OF FUNDAMENTAL UNITS
 Metre : One metre is equal to 1650763.73 wavelength in vacuum of the radiation corresponding to transition between the levels 2p10 and 5d5 of the krypton – 86 atom
The distance travelled by light in vacuum in second is called 1 metre.
 Kilogram : The mass of cylinder (of height and diameter 39 cm) made of Platinumiridium alloy kept at International Bureau of weights and measures in paris is defined as 1kg.
 Second : It is the duration of 9,192,631,770 periods of radiation corresponding to the transition between the two hyperfine levels of the ground state of Caesium (133) atom.
 Ampere : It is the current which when flows through two infinitely long straight conductors of negligible crosssection placed at a distance of one metre in air or vacuum produces a force of 2 × 10–7 N/m between them.
 Candela : It is the luminous intensity in a perpendicular direction, of a surface of 1/600,000 square metre of a black body at the temperature of freezing platinum under a pressure of 1.013 × 105 N/m2.
 Kelvin : It is the 1/273.16 part of thermodynamic temperature of triple point of water.
 Mole : It is the amount of substance which contains as many elementary entities as there are in 0.012 kg of Carbon12.
S.I. PREFIXES
SOME IMPORTANT PRACTICAL UNITS
 For large distance (macrocosm)
1 A.U. = 1.496 × 1011m
1 ly = 9.46 × 1015m
1 parsec = 3.1 × 1016m
 For small distance (microcosm)
1 nanometre = 10–9m
1 angstorm = 10–10m
1 fermi = 10–15 m
 For small area
1 barn = 10–28m2  For heavy mass
1 ton = 1000kg
1 quintal = 100kg
1 slug = 14.57kg
1 C.S.L (chandrasekhar limit) = 1.4 times the mass of the sun  For small mass
1 amu = 1.67 x 10–27kg
1 pound = 453.6g = 0.4536 kg  For small time
1 shake = 10–8s  For large time
 For measuring pressure
DIMENSIONS
DIMENSIONAL FORMULA
DIMENSIONAL EQUATION
CLASSIFICATION OF PHYSICAL QUANTITIES (ON THE BASIS OF DIMENSIONS)
DIMENSIONAL FORMULA OF SOME IMPORTANT PHYSICAL QUANTITIES
 To find dimensions of a typical physical quantity which is involved in a number of formulae, try to use that formula which is easiest for you. For example if you want to find the dimensional formula of magnetic induction then you can use the following formulae
 If you have to find the dimensional formula of a combination of physical quantities, then instead of finding the dimensional formula of each, try to correlate the combination of physical quantities with a standard formula. For example, if you have to find the dimension of CV2, then try to use formula where E is energy of a capacitor.
 velocity of light in vacuum
 Dimensions of the following are same
 Dimensions of the following are same
 The dimension of RC = is same as that of time
 Dimensions of the following are same
 Dimensions of the following are same
 Dimensions of the following are same
 Dimensions of the following are same
 Dimensions of the following are same
 all trigonometric ratio is [M0L0T0]
 x in ex is [M0L0T0]
 ex is [M0L0T0]
 x in log x is [M0L0T0]
 log x is [M0L0T0]
DIMENSIONAL ANALYSIS AND ITS APPLICATIONS
 Conversion of one system of unit into another
 Checking the accuracy of various formulae
 Derivation of formula
LIMITATIONS OF DIMENSIONAL ANALYSIS
 No information about the dimensionless constant is obtained during dimensional analysis
 Formula cannot be found if a physical quantity is dependent on more than three physical quantities.
 Formula containing trigonometrical /exponential function cannot be found.
 If an equation is dimensionally correct it may or may not be absolutely correct.
SIGNIFICANT FIGURES
RULES TO DETERMINE THE NUMBERS OF SIGNIFICANT FIGURES
 All nonzero digits are significant. 235.75 has five significant figures.
 All zeroes between two nonzero digits are significant. 2016.008 has seven significant figures.
 All zeroes occurring between the decimal point and the nonzero digits are not significant provided there is only a zero to left of the decimal point. 0.00652 has three significant figures.
 All zeroes written to the right of a nonzero digit in a number written without a decimal point are not significant. This rule does not work if zero is a result of measurement. 54000 has two significant figures whereas 54000m has five significant figures.
 All zeroes occurring to the right of a nonzero digit in a number written with a decimal point are significant. 32.2000 has six significant figures.
 When a number is written in the exponential form, the exponential term does not contribute towards the significant figures. 2.465 × 105 has four significant figures.
 The significant figures depend upon the least count of the instrument.
 The number of significant figure does not depend on the units chosen.
ROUNDING OFF
 If digit to be dropped is less than 5 then preceding digit should be left unchanged.
 If digit to be dropped is more than 5 then one should raise preceding digit by one.
 If the digit to be dropped is 5 followed by a digit other than zero then the preceding digit is increased by one.
 If the digit to be dropped is 5 then the preceding digit is not changed if it is even.
 If digit to be dropped is 5 then the preceding digit is increased by one if it is odd.
ARITHMETICAL OPERATIONS WITH SIGNIFICANT FIGURES AND ROUNDING OFF
 For addition or subtraction, write the numbers one below the other with all the decimal points in one line. Now locate the first column from the left that has a doubtful digit. All digits right to this column are dropped from all the numbers and rounding is done to this column. Addition subtraction is then done.
Step2 : 23.6 + 8.7=32.3
 In multiplication and division of two or more quantities, the number of significant digits in the answer is equal to the number of significant digits in the quantity, which has minimum number of significant digits.
ACCURACY, PRECISION OF INSTRUMENTS AND ERRORS IN MEASUREMENTS
 Accuracy depends on the least count of the instrument used for measurement.
 In the addition and subtraction operation, the result contains the minimum number of decimal places of the figures being used
 In the multiplication and division operation, the result contains the minimum number of significant figures.
 Least count (L.C.) of vernier callipers = one MSD – one VSD
 Least count of screw gauge (or spherometer)
where pitch is the ratio of number of divisions moved on linear scale and number of rotations given to circular scale.  Pure number or unmeasured value do not have significant numbers
 Change in the position of decimal does not change the number of significant figures. Similarly the change in the units of measured value does not change the significant figures.
COMMON ERRORS IN MEASUREMENTS
EXPERIMENTAL ERRORS
 by the lack of perfection of observer
 if the measuring instrument is not perfectly sensitive.
 measuring instrument having a zero error.
 an instrument being incorrectly calibrated (such as slow runningstop clock)
 the observer persistently carrying out a mistimed action (e.g., in starting and stopping a clock)
METHODS OF EXPRESSING ERROR
TO FIND THE MAXIMUM ERROR IN COMPOUND QUANTITIES
We have to find the sum or difference of two values given as (a ± Δa) and (b ± Δb), we do it as follows
We add the fractional or percentage errors in case of finding product or quotient.
If x = an then
 More the accuracy, smaller is the error.
 Absolute error ΔX is always positive.
 ΔX has the same dimensions as that of X.
 If the least count of measuring instrument is not given and the measured value is given the least error in the measurement can be found by taking the last digit to be 1 and rest digit to be zero. For e.g. if the measured value of mass m = 2.03 kg then.
 If a number of physical quantities are involved in an expression then the one with higher power contributes more in errors and therefore should be measured more accurately.
 Relative error is a dimensionless quantity.
 We are always interested in calculating the maximum possible error.
Measurement and uncertainties
1.2.1 State the fundamental units in the SI system.
Many different types of measurements are made in physics. In order to provide a clear and concise set of data, a specific system of units is used across all sciences. This system is called the International System of Units (SI from the French “Système International d’unités”).
The SI system is composed of seven fundamental units:
Fundamental units:
Length: metre (m)
Mass: kilogram (kg)
Time: second (s)
Electrical current: ampere (A)
Temperature: kelvin (K)
Amount of substance: mole (mol)
Luminous intensity: candela (cd) (Not required for the IB!)
Quantity  Unit name  Unit symbol 
mass  kilogram  kg 
time  second  s 
length  meter  m 
temperature  kelvin  K 
Electric current  ampere  A 
Amount of substance  mole  mol 
Luminous intensity  candela  cd 
Note that the last unit, candela, is not used in the IB diploma program.
Metric multipliers: correspond to a power of ten, e.g. kilo (k) – 10³.
All available on the Physics Data Booklet! (page 5)
Scientific notation: convenient way of expressing numbers that are too small or too big.
Notation: m x 10^{n}, where 1 ≤ m < 10 and n is an integer (positive or negative).
Example: 213 000 000 = 2.13 x 10^{8}
Order of magnitude: approximation of a number to the nearest power of ten.
 Example: 0.0000945 ≅ 10^{4}
Significant figures (s.f.)
The number of digits that should be used to express a certain number, which shows how precise the information is. In any calculation or experimentation, the final answer should be expressed with the same number of s.f. as the value with least s.f. used.
Rules to count the number of significant figures:
Nonzero numbers are always significant. Example: 1234 – four s.f.
“Sandwiched” zeros are always significant. Example: 5403 – four s.f.
Zeros to the left are never significant. Example: 0.0004578 – four s.f.
Zeros to the right are only significant if there is a point. Example: 1403.00000 – nine s.f.
Rules to round a number:
If the number following the last significant digit is less than five, the digit remains equal, e.g. 678.4 (4 s.f.) rounded to 3 s.f. = 678.
If the number following the last significant digit is greater than five, the digit rounds up (i.e. +1), e.g. 678.6 (4 s.f.) rounded to 3 s.f. = 679.
If the number following the last significant digit is five and it is only followed by zeros:
The last significant digit (number before five) remains equal if it is even, e.g. 3.2500 (5 s.f.) rounded to 2 s.f. = 3.2.
The last significant digit (number before five) rounds up if it is odd, e.g. 3.3500 (5 s.f.) rounded to 2 s.f. = 3.4.
1.2.2 Distinguish between fundamental and derived units and give examples of derived units.
In order to express certain quantities we combine the SI base units to form new ones. For example, if we wanted to express a quantity of speed which is distance/time we write m/s (or, more correctly m s1). For some quantities, we combine the same unit twice or more, for example, to measure area which is length x width we write m2.
Certain combinations or SI units can be rather long and hard to read, for this reason, some of these combinations have been given a new unit and symbol in order to simplify the reading of data.
For example: power, which is the rate of using energy, is written as kg m2 s3. This combination is used so often that a new unit has been derived from it called the watt (symbol: W).
Below is a table containing some of the SI derived units you will often encounter:
SI derived unit  Symbol  SI base unit  Alternative unit 
newton  N  kg m s2  – 
joule  J  kg m2 s2  N m 
hertz  Hz  s1  – 
watt  W  kg m2 s3  J s1 
volt  V  kg m2 s3 A1  W A1 
ohm  Ω  kg m2 s3 A2  V A1 
pascal  Pa  kg m1 s2  N m2 
1.2.3 Convert between different units of quantities.
Often, we need to convert between different units. For example, if we were trying to calculate the cost of heating a litre of water we would need to convert between joules (J) and kilowatt hours (kW h), as the energy required to heat water is given in joules and the cost of the electricity used to heat the water is a certain price per kW h.
If we look at table 1.2.2, we can see that one watt is equal to a joule per second. This makes it easy to convert from joules to watt hours: there are 60 second in a minutes and 60 minutes in an hour, therefor, 1 W h = 60 x 60 J, and one kW h = 1 W h / 1000 (the k in kW h being a prefix standing for kilo which is 1000).
1.2.4 State units in the accepted SI format.
There are several ways to write most derived units. For example: meters per second can be written as m/s or m s1. It is important to note that only the latter, m s1, is accepted as a valid format. Therefor, you should always write meters per second (speed) as m s1 and meters per second per second (acceleration) as m s2. Note that this applies to all units, not just the two stated above.
1.2.5 State values in scientific notation and in multiples of units with appropriate prefixes.
When expressing large or small quantities we often use prefixes in front of the unit. For example, instead of writing 10000 V we write 10 kV, where k stands for kilo, which is 1000. We do the same for small quantities such as 1 mV which is equal to 0,001 V, m standing for milli meaning one thousandth (1/1000).
When expressing the units in words rather than symbols we say 10 kilowatts and 1 milliwatt.
A table of prefixes is given on page 2 of the physics data booklet.
1.2 Uncertainties and Errors
Errors (or uncertainties) in experimentation
All measurements are an estimate of the real value, since they are always subject to errors:
Systematic error: biases measurements in the same direction, e.g. always +0.1 cm.
Cause (e.g.): Not adequately calibrated equipment.
Cause (e.g.): Ignoring the effects of friction (given that it is constant).
Random error: biases measurements in all directions, yielding a wide spread of values.
Cause (e.g.): Using a stopwatch manually – some measurements (of time) will be above the real time and some measurements will be below the real time.
Cause (e.g.): Changing external circumstances, e.g. alternating atmospheric conditions.
Solution: Gathering a wide range of values and then taking the average.
1.2.6 Describe and give examples of random and systematic errors.
Random errors
A random error, is an error which affects a reading at random.
Sources of random errors include:
 The observer being less than perfect
 The readability of the equipment
 External effects on the observed item
Systematic errors
A systematic error, is an error which occurs at each reading.
Sources of systematic errors include:
 The observer being less than perfect in the same way every time
 An instrument with a zero offset error
 An instrument that is improperly calibrated
Accuracy and precision:
Accurate measurement: Low systematic error – average close to real value.
Precise measurement: Low random error – values close to each other.
1.2.7 Distinguish between precision and accuracy.
Precision
A measurement is said to be accurate if it has little systematic errors.
Accuracy
A measurement is said to be precise if it has little random errors.
A measurement can be of great precision but be inaccurate (for example, if the instrument used had a zero offset error).
Errors (or uncertainties) in measurements
Estimation of random errors in instruments: + uncertainty. The uncertainty should always have the same number of decimal places as the value measured, and normally only 1 s.f.
Analogical instrument (e.g. a ruler): Half the smallest possible width of graduation
Ruler 1:
Ruler 2: more precise than Ruler A.
Smallest width of graduation: 0.1 cm
Uncertainty: + 0.05 cm
Value:12.50 + 0.05 cm
Errors (or uncertainties) in calculations
Consider the following value: L1: 8.3 + 0.1 cm.
Absolute uncertainty (∆x): has the same units as the value, e.g. for L1: + 0.1 cm
Fractional uncertainty: division between the absolute uncertainty and the value itself, e.g. for L1: 0.1/8.3 = 0.012
Percentage uncertainty: the product of the fractional uncertainty by 100%, e.g. for L1: 0.012 x 100% = 1.2%
Now consider the following value as well: L2: 7.4 + 0.5 cm.
Propagation of uncertainties:
Addition or Subtraction: Addition or subtraction of the values and addition of the absolute uncertainties.
Example: L1 – L2: (8.3 + 0.1) – (7.4+ 0.5) = 0.9 + 0.6 cm.
Multiplication or division: Multiplication or division of the values and addition of the fractional uncertainties or percentage uncertainties.
Example: L1 x L2: (8.3 + 1.2%) x (7.4 + 6.8%) = 61 + 8.0% cm².
Power and roots: Value raised to a certain power and multiplication of the fractional uncertainty or percentage uncertainty by the value of the power.
Example: L1³: (8.3 + 1.2%)^{3} = 570 + 3.6% cm³.
Errors (or uncertainties) in graphs
Error box:
Uncertainties of one value in a graph is commonly represented by error bars.
Error of yvalue may differ from the error of xvalue.
Bestfit line: line that goes through all error bars (it does not have to be a straight line!)
Direct proportionality (in the form y = ax) only if bestfit line is a straight line that passes through the origin (0,0).
More than 2 points are needed to confirm a relationship between two variables (e.g. x and y)
Gradient: found by using two points at least half the line’s length away from each other: gradient = rise/run = ∆x/∆y
Uncertainty in the gradient:
∆gradient = gradientMAX – gradientMIN/2
Uncertainty in the yintercept:
∆y – intercept = y – interceptMAX – y – interceptMIN/2
1.2.8 Explain how the effects of random errors may be reduced.
The effect of random errors on a set of data can be reduced by repeating readings. On the other hand, because systematic errors occur at each reading, repeating readings does not reduce their affect on the data.
1.2.9 Calculate quantities and results of calculations to the appropriate number of significant figures.
The number of significant figures in a result should mirror the precision of the input data. That is to say, when dividing and multiplying, the number of significant figures must not exceed that of the least precise value.
Example:
Find the speed of a car that travels 11.21 meters in 1.23 seconds.
11.21 x 1.13 = 13.7883
The answer contains 6 significant figures. However, since the value for time (1.23 s) is only 3 s.f. we write the answer as 13.7 m s1.
The number of significant figures in any answer should reflect the number of significant figures in the given data.
1.2.10 State uncertainties as absolute, fractional and percentage uncertainties.
Absolute uncertainties
When marking the absolute uncertainty in a piece of data, we simply add ± 1 of the smallest significant figure.
Example:
13.21 m ± 0.01
0.002 g ± 0.001
1.2 s ± 0.1
12 V ± 1
Fractional uncertainties
To calculate the fractional uncertainty of a piece of data we simply divide the uncertainty by the value of the data.
Example:
1.2 s ± 0.1
Fractional uncertainty:
0.1 / 1.2 = 0.0625
Percentage uncertainties
To calculate the percentage uncertainty of a piece of data we simply multiply the fractional uncertainty by 100.
Example:
1.2 s ± 0.1
Percentage uncertainty:
0.1 / 1.2 x 100 = 6.25 %
1.2.11 Determine the uncertainties in results.
Simply displaying the uncertainty in data is not enough, we need to include it in any calculations we do with the data.
Addition and subtraction
When performing additions and subtractions we simply need to add together the absolute uncertainties.
Example:
Add the values 1.2 ± 0.1, 12.01 ± 0.01, 7.21 ± 0.01
1.2 + 12.01 + 7.21 = 20.42
0.1 + 0.01 + 0.01 = 0.12
20.42 ± 0.12
Multiplication, division and powers
When performing multiplications and divisions, or, dealing with powers, we simply add together the percentage uncertainties.
Example:
Multiply the values 1.2 ± 0.1, 12.01 ± 0.01
1.2 x 12.01 = 14
0.1 / 1.2 x 100 = 8.33 %
0.01 / 12.01 X 100 = 0.083%
8.33 + 0.083 = 8.413 %
14 ± 8.413 %
Other functions
For other functions, such as trigonometric ones, we calculate the mean, highest and lowest value to determine the uncertainty range. To do this, we calculate a result using the given values as normal, with added error margin and subtracted error margin. We then check the difference between the best value and the ones with added and subtracted error margin and use the largest difference as the error margin in the result.
Example:
Calculate the area of a field if it’s length is 12 ± 1 m and width is 7 ± 0.2 m.
Best value for area:
12 x 7 = 84 m2
Highest value for area:
13 x 7.2 = 93.6 m2
Lowest value for area:
11 x 6.8 = 74.8 m2
If we round the values we get an area of:
84 ± 10 m2
1.2.12 Identify uncertainties as error bars in graphs.
When representing data as a graph, we represent uncertainty in the data points by adding error bars. We can see the uncertainty range by checking the length of the error bars in each direction. Error bars can be seen in figure 1.2.1 below:
Figure 1.2.1 – A graph with error bars
1.2.13 State random uncertainty as an uncertainty range (±) and represent it graphically as an “error bar”.
In IB physics, error bars only need to be used when the uncertainty in one or both of the plotted quantities are significant. Error bars are not required for trigonometric and logarithmic functions.
To add error bars to a point on a graph, we simply take the uncertainty range (expressed as “± value” in the data) and draw lines of a corresponding size above and below or on each side of the point depending on the axis the value corresponds to.
Example:
Plot the following data onto a graph taking into account the uncertainty.
Time ± 0.2 s  Distance ± 2 m 
3.4  13 
5.1  36 
7  64 
Table 1.2.1 – Distance vs Time data
Figure 1.2.2 – Distance vs. time graph with error bars
In practice, plotting each point with its specific error bars can be time consuming as we would need to calculate the uncertainty range for each point. Therefor, we often skip certain points and only add error bars to specific ones. We can use the list of rules below to save time:
 Add error bars only to the first and last points
 Only add error bars to the point with the worst uncertainty
 Add error bars to all points but use the uncertainty of the worst point
 Only add error bars to the axis with the worst uncertainty
1.2.14 Determine the uncertainties in the gradient and intercepts of a straight line graph.
Gradient
To calculate the uncertainty in the gradient, we simply add error bars to the first and last point, and then draw a straight line passing through the lowest error bar of the one points and the highest in the other and vice versa. This gives two lines, one with the steepest possible gradient and one with the shallowest, we then calculate the gradient of each line and compare it to the best value. This is demonstrated in figure 1.2.3 below:
Figure 1.2.3 – Gradient uncertainty in a graph
Intercept
To calculate the uncertainty in the intercept, we do the same thing as when calculating the uncertainty in gradient. This time however, we check the lowest, highest and best value for the intercept. This is demonstrated in figure 1.2.4 below:
Figure 1.2.4 – Intercept uncertainty in a graph
Note that in the two figures above the error bars have been exaggerated to improve readability.
Mass Length Time Substance amount Electrical current Thermodynamic temperature Luminosity  Kilogram Meter Second Mole Ampere Kelvin Candela  kg m s mol A K cd 
Energy Force Frequency Pressure Power Voltage Resistance  Joule Newton Hertz Pascal Watt Volt Ohm  J N Hz Pa W V Ω  kg × m^2 / s^2 kg × m / s^2 1 / s kg / m / s^2 kg × m^2 / s^3 kg × m^2 / s^3 / A kg × m^2 / s^3 / A^2 
In an unidentified question, an answer is given in joules per second. The question requires a kilowatts per hour value. Which conversions should be made?
Answer: The joules per second value should be multiplied by 3600 to get watts per hour. This value can then be divided by 1000 to get kilowatts per hour.
Prefix exa  Symbol E  Scientific Notation 10^18 
Random
 Systematic

Uncertainties are presented as the tenth of the smallest decimal of a given number. For example:
13.21 m ± 0.01
0.002 g ± 0.001
1.2 s ± 0.1
12 V ± 1
These are called absolute uncertainties. Uncertainties can also be presented as fractional uncertainties, where the tenth of the smallest decimal is presented as a fraction of the original number. For example:
1.2 s ± 0.1
0.1 / 1.2 = 0.0625
Percentage uncertainties are fractional uncertainties converted to percentages:
1.2 s ± 0.1
0.1 / 1.2 x 100 = 6.25 %
Adding/subtracting numbers with uncertainties
 Multiplying/dividing numbers with uncertainties

 The value should be calculated with the highest possible components and lowest possible components then compared to find the uncertainty of the answer.
 For example: “Calculate the area of a field with a length of 12 ± 1 m and width of 7 ± 0.2 m.”
 The ‘best value’ would be to take 12 and 7 and calculate the area. The highest value uses 13 and 7.2 and the lowest value uses 11 and 6.8. When we work these areas out, we get 84, 93.6 and 74.8. When rounding to the appropriate number of significant figures, we get 84, 94 and 75. This translates to approximately 84 ± 10 m^2.
 The length of the error bars represents the uncertainty range for each value.
 The uncertainty range is the “± …” value (ie. 2 ± 0.1). This value can be directly drawn onto a graph at each point as an error bar, as shown in the image above.
 When drawing graphs for the IA, error bars only need to be included if the uncertainty is significant. If they are not included, the student should provide an explanation. If there is a large amount of data, only the largest data point, smallest data point and a few points inbetween require error bars.
 Error bars are not required for logarithmic or trigonometric functions.