Clinical Chemistry (CHE221) Professor Hicks Week 1 Statistics Made Slightly Less Boring and Introduction to Spectrophotometry 3 Accuracy vs Precision Precision is the consistency of a measurement made in different trials Accuracy is the agreement of a measure value with an accepted value accurate and precise not accurate but precise 4 not accurate not precise
Mean (x) Mean is just what we used to call the average before college Add em up and divide by how many you added If we repeated identical measurements the mean is our best guess at the true value How close the mean is to the accepted value reflects the accuracy of the measurement Accuracy How close is the center of the circle to the center of the target? Closer to center More accurate XX X no matter how small the circle gets (precise) if it is not near center it is not accurate More accurate 6 Standard deviation (s) Standard deviation is a measure of how widely values vary from the mean In other words a measure of the precision Smaller standard deviation means better precision s = x xi n 1 2 You should recognize the formula when you see it You can always use Excel or a calculator to calculate it
35 30 25 20 15 10 5 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 Precision radius standard deviation precision Smaller radius smaller standard deviation more precise XX X no matter how small the circle gets (precise) if it is not near center it is not accurate 8 Histograms Special kind of graph Frequency vs outcome Outcomes -what happened, - on x axis Frequency - how often an outcome happened, - on y axis Frequency 70 60 50 40 30 20 10 0 Histogram of Exam Grades frequency = 10 people (got a grade between 90-100) 10 20 30 40 50 60 70 80 90 100 Exam Grade outcome = a grade between 90-100 Bins in Histograms Must decide when making a histogram what range of outcomes will be grouped together # students that got 70-75, 75-80, or # students that got 70-80, 80-90 etc same data different bin sizes bin size = 5 points Frequency Frequency Histogram of Exam Grades Exam Grade bin size = 10 points Histogram of Exam Grades 70 60 50 40 30 20 10 0 10 20 30 40 50 60 70 80 90 100 Exam Grade
90 91 92 93 94 95 96 97 98 99 100 bell-shaped curves Many histograms have a profile that looks like a bell-shaped curve as smaller bin sizes large number samples Called Normal distributions Distribution of what? Why? Histogram of Blood Glucose Levels of College Students 0 Sometimes distributions reflect the Blood Glucose (mg/dl) differences in a population - Different people have different blood This bell-shaped curve reflects sugar levels differences between individuals Frequency 80 60 40 20 Sometimes distributions reflect random experimental error - If you analyzed the same sample over and over the result would probably be at least a little different for each trial - This would reflect the quality of the measurement not differences in the sample (you used the same sample!) Frequency Histogram of Blood Glucose Levels measured on a single sample 10 9 8 7 6 5 4 3 2 1 0 Blood Glucose (mg/dl) This bell-shaped curve reflects experimental error Confidence 100 throws total 68 are in the circle #1 double the radius 95.5 are in the circle #2 triple the radius 99 are in the circle #3 3 2 1 circles 1,2, and 3 have radii of 1 standard deviation 2 standard deviations 3 standard deviations if this dart thrower makes another throw: we can be 68.3% confident it will be within 1 standard deviation of the center 95.5% confident it will be within 2 standard deviations of the center 99.7% confident it will be within 3 standard deviations of the center this assumes the data is Normal (histogram looks like a bell curve)
Standard error (of the mean) standard error = s n Used to get the range of values with a certain confidence for a value that has been obtained by averaging Based on the assumption that the data is accurate, this range is the range of values that the true value is within Example If a patients blood glucose was measured many times and the average was found to be 90 mg/dl, which is in the reference range of 70-110 mg/dl, but the standard error for this average was 20 mg/dl we would be only 68% certain it actually was in the reference range. 68% confidence Mean +/- s n How does averaging improve data? as number of measurements (n) gets larger range at any confidence gets smaller s standard error = n mean s n this range is 68.3% confidence 1.96 2.0 s larger n mean this range is 95.5% confidence smaller range n 1.96s mean n this range is 95% confidence 3s mean n this range is 99.7% confidence Coefficient of variation (CV) The range at any confidence will be different for every type measurement Typical ranges glucose 70-110 mg/dl HDL cholesterol 29-60 mg/dl (males) CV expresses error as % of the mean so 5% for glucose or cholesterol reflects similar quality measurements CV reflects accuracy and precision- Do you see why? CV = standard deviation mean 100% or s CV 100% x
Wavelength ( ) take a snapshot of any wave wavelength distance between high points this waves wavelength is ½ the wavelength of the wave above Different colors of light are described by their wavelength 32 Only some light is visible to our eye Many types of light we cannot see Visible light has wavelengths of 400-700 nm aka visible spectrum 33 Spectrophotometry Shining light through samples How much light gets through? Often used to determine concentrations Most common light used is visible + UV light, infrared (IR) less common
Spectrophotometer Prism, color filter or diffraction grating detector I/I o = transmittance T (fraction light that got through) I and I o measured separately I o (reference/background) measured first - solvent alone I is the amount of light that gets through when the sample + solvent is in the light beam Why does some light get absorbed (not get through)? All matter has electrons Electrons are in orbitals with different energies When matter absorbs energy (light, heat) electrons go up to higher energy orbitals = green photon = 550 nm Energy = hc heat or sometimes light released Transmittance (T) I/I o called Transmittance 0 1 The fraction of the light that got through the sample Sometime expressed as % Transmittance 0 100% Affected by: - identity of substances - concentration of dissolved substances - thickness of the path the light must travel
Transmittance and concentration I o = all the light I = the light that got through of 8 photons total 4 photons absorbed 4 photons transmitted % transmittance = 4/8 100% = 50% = absorbing molecule light rays = Doubling concentration halves light that gets through (again) If % transmittance was proportional to concentration then 2 concentration 50% T 0% transmission not true! transmission is halved, again to 25% 8 photons total (I o ) 6 photons absorbed 2 photons transmitted (I) transmittance = 2/8 x 100% 25% Beer s Law (or here s what concentration is proportional to) A = bc A = Absorbance = log(i/i o ) = is the Molar Absorptivity aka Molar Extinction Coefficient b is the pathlength that the light travels through the sample c is the concentration, usually in moles/liter (M) sometimes written A= lc ( b the pathlength = l)
Beer s Law A = bc Which Kool-aid looks sweeter? How do we know? When a solution is darker colored its absorbance (A) is larger If the containers are the same size then b the pathlength is the same If they are both koolaid is the same The darker one must have larger concentration (c) of red food coloring (and sugar) Extreme pathlengths (NY to Paris in cm) A = bc If pathlength b gets longer absorbance gets larger Fiber optic cables have very long pathlengths Impurities present at very low concentrations can lead to large absorbance's Absorption Spectrum Graph of absorbance vs wavelength Every wavelength of light has different absorbance for every substance Beer s law true for each color (wavelength) light The spectrum is a unique property of a substance - like a fingerprint
Common misconceptions Transmittance and absorbance do not add up to 1.0 or 100% Absorbances can be larger than 1.0 though 0.1 1.0 is a desirable range to measure Negative absorbance's indicate an experimental problem such as - Error in measuring background I o - Sample fluorescing (releasing light) Selecting a cuvette cuvettes (sample tubes) can be made of several different materials - glass - several different types plastic - quartz (most expensive) The cuvette materials themselves all have different UV absorption spectra all will give similar valid results 400-700 nm because they all transmit close to 100% of visible light - If a material is transparent and uncolored to your eye it transmits all visible light plastic and glass can distort data severely in UV range 200-400 nm Square cuvette Round cuvettes Measuring A in the UV range 200-400 nm Before making A measurements of samples: 1) Measure absorption spectrum of the cuvette filled with water using air as the background 2) This A should be less than 0.30 over range of wavelengths to be measured When measuring samples place reference solvent and sample in cuvettes determined to have the same absorption spectrum in steps 1 and 2 above If you do not have two identical cuvettes measure background then clean/dry the cuvette then measure sample in same cuvette useful range glass and most plastics quartz and some plastics 400 nm-700 nm 200 nm-700 nm
What are the functions of the parts? Source Monochromator Slit before sample Slit after sample Spectrophotometer Detector Prism, filter or diffraction grating detector Photomultiplier Tubes Light detectors Convert light into an electrical signal Based on Einstein s Photoelectric Effect In a spectrophotometer they are used to measure only 1 wavelength (color) of light at a time The wavelength(s) that reaches it is selected by the slit if the monochromator is a prism or grating Monochromators Separate light into narrow range of wavelength (colors) Different types 1) Prism 2) Diffraction grating 3) Light filters Prism, filters or diffraction grating detector
Photodiode Arrays Light detectors Convert light into an electrical signal Measure wide range of wavelengths (colors) of light at a time Allow a complete spectrum to be collected without scanning Spectrophotometer with a photodiode array detector Light is not separated into different wavelengths until after it goes through sample Complete spectrum is collected instantly without scanning through different wavelengths No slit needed between source and cuvette Monochromator positioned after the light has passed through sample Aperture or slit Photodiode array Detector source Bandpass of an Instrument The range of wavelengths that are measured together as a group Specified in units of wavelength (nm) Determined by the range of wavelengths that reach the photomultiplier through the slits, or Range of wavelengths that strike the photodiode array on a single pixel
Bandpass of an Instrument Example: An instrument with a bandpass of 20 nanometers that has a photomultiplier tube as a detector allows the light from 510 nm to 530 nm to pass through the sample and reach the photomultiplier tube when the instrument is set at 520 nm Example: An instrument with a bandpass of 20 nanometers that has a photodiode array detector allows the light from 510 nm to 530 nm to reach a single pixel on the detector and records the light intensity at that pixel as the intensity at 520 nm Single Beam vs Double Beam Spectrophotometers Single beam most basic design measures I and I o at different times Double Beam Instrument has a chopper that splits the light into two equal portions that travel through a sample and background cuvettes as the instrument scans Equivalent to two spectrophotometers: one for I and one for I o Allows spectra to be collected without needing a computer to save each absorbance at a different wavelength Absorbance's add up When solutions with different components are mixed the resulting absorbance is the sum of the absorbance's (after dilution is accounted for) Scanning a background is equivalent to subtracting the absorbance of the cuvette and solvent if its absorbance were measured against air
Troubleshooting absorbance measurements Most common problems: Light source burns out Light path is blocked Background measured incorrectly - Different cuvette or solvent for background - Cuvette s absorbance too large at the wavelength measurement is being made Determining concentration from A using calibration curves Calibration curves are constructed by measuring absorbance (A) for a series of samples where concentration (c) is known Graphs of Absorbance vs concentration Beer s law can be used at any wavelength How to pick best wavelength? Larger values larger Absorbances most sensitive low concentrations Smaller values smaller Absorbances can produce measurable A even at high concentrations or long pathlengths Absorbance at 500 nm Concentration (millimoles/liter) Determining concentration from A using calibration curves Measure absorbance of a sample of unknown concentration Find A value on y axis of calibration curve Trace across and down to the corresponding concentration Example measured unknown A say A= 0.70 find 0.70 on y-axis read c off x axis c = 18 mm concentration unknown 0.70 Using calibration curves is a graphical way to determine concentration from a samples measured absorbance Absorbance at 500 nm Concentration (millimoles/liter) 18 mm
Beer s Law instead of using a calibration curve concentration can be calculated mathematically from measured A values using and b A = bc A = absorbance = log(i/i o ) = is the molar absorptivity aka molar extinction coefficient ranges from ~ 0-100,000 M -1 cm -1 b is the pathlength that the light travels through the sample most often 1.0 cm c is the concentration, usually in moles/liter (M) Sometimes rearranged c = A b Mathematical Analysis of Absorbance Data Usually measure A and want to calculate c b is constant (thickness of the cuvette) Beer s Law rearranged Many protocols suggest c = A b A = 0.10 1.0 is most accurate straight line predicted by Beer s law b/c all instruments data agrees with Beers law A < 2 obey Beer s Law at low A values Method only accurate if Beer s Law is obeyed Absorbance at 500 nm deviations from Beers law fall below prediction for A>2 Concentration (millimoles/liter) Measuring concentration of samples with large absorbance values Directly measuring high A values is inaccurate Usually it is an underestimate of the samples true absorbance Samples should be diluted to lower A to the range 0.10 1.0 A of diluted sample is measured Concentration of diluted sample is determined from either a Graphical or Mathematical analysis Concentration of undiluted sample is calculated from dilution formula V dil M conc V conc = M dil V dil M conc = V M dil conc called the Dilution Factor
1.0 Limits on the Graphical Determination of Concentration from Beers Law Calibration Curves Data can deviate from Beers Law significantly when absorbances are outside the instruments measurable range A calibration curve relates concentration to absorbance with no assumption of any theory (Beer s Law) 7.0 6.0 5.0 4.0 3.0 2.0 hard to tell where it hit when the graph is almost horizontal graph is curved but concentration can be determined from A 0 10 20 30 40 50??? The limitation on the range of A that can be related to concentrations is in reading the calibration curve not the range that Beers Law is obeyed uncertainty in concentration gets big! in other words poor precision This curve applies to this instrument and should not be applied to other instruments Example: A middle aged male boogeyman was not feeling well and his primary care doctor suspected Malingerer s Syndrome, a disease characterized by levels of Fetid Hormone in the sweat higher than 100 mm. A sweat sample collected from the patient had an absorbance of 3.2 at 500 nm. The technician performed a dilution of the patients sample by taking 1.0 ml and diluting it up to a total volume of 10.0 ml and found the absorbance of this diluted sample was 0.50. Calculate the concentration of Fetid Hormone in this patients sweat using the Beer s Law calibration curve for Fetid Hormone at 500 nm shown below and the dilution formula. Absorbance at 500 nm Concentration (millimoles/liter) Fetid Hormone (mm)
Example: A standard sample of fetid hormone with a concentration of 5 mm was found to have an absorbance of 0.1321 at 500 nm. Assuming that Beer s law is obeyed calculate the concentration of the previous sample mathematically from Beers law. Alternate method of calculating concentrations mathematically from Beer s Law Often applied in clinical chemistry protocols Based on assumption Beer s law is obeyed ratios created A proportional to C A (standard) = A (patient sample) C (standard) C (patient sample) rearranged A (patient sample) C (patient sample) = C (standard) A (standard) very similar to the dilution factor