Statistical Analysis

3\.1 significant figures

fitness for purpose

degree to which data produced by a measurement enables a user to make correct decisions for a stated purpose

whether an analysis can answer our questions depends on both:

1. trueness of the analysis (accuracy)
2. its reproducibility (precision)

significant figures

the minimum number of digits needed to write a given value in scientific notation without loss of *precision*

zeros are significant when they occur:

1. in the middle of a number
2. at the end of a number on the RIGHT hand side of the DECIMAL point

minimum uncertainty in a measured quantity

±1 in the last significant digit (farthest to the right) bc the last σ digit in a measured quantity always has some associated uncertainty

there is uncertainty in any ____ quantity

MEASURED

3\.2 significant figures in arithmetic

addition and subtraction sig fig rules

1. if numbers to be added or subtracted have equal numbers of digits, the answer goes to the *same decimal place* as any of the individual numbers
2. if numbers being added or subtracted do not end at the same decimal place, we are limited by the least certain one (the decimal that goes the LEAST far out)

multiplication and division sig fig rules

limited to the number of digits contained in the number with the fewest sig figs

base 10 log

*n* = 10^*a* is the same as log *n* = *a*

antilogarithm

the antilogarithm of *a* is *b* if 10^*a* = *b*

characteristic

the part of a logarithm to the left of the decimal point

mantissa

the part of a logarithm to the right of the decimal point

logarithm sig fig rules

the number of digits in the MANTISSA (right of the decimal point) of the log *a* should EQUAL the number of sig fig in *a*

conversion of logarithm into antilogarithm sig fig rules

the number of sig figs in the antilogarithm of *a* should EQUAL the number of digits in the MANTISSA of *a* (right of the decimal point)

3\.3 types of error

experimental error

the difference bwn the “true” value and the measured value of a quantity

how is experimental error classified

1. systematic
2. random
3. blunder

systematic error

(type of experimental error)

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aka DETERMINATE error

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error due to PROCEDURAL or instrumental factors that cause a measurement to be CONSISTENTLY too large/small. The error CAN be correct in principle BUT this is hard bc the error is REPRODUCIBLE if conducted again the exactly the same manner

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arises from flaw in equipment or design of experiment

random error

(type of experimental error)

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aka INDETERMINATE error

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a type of error which can either be positive or negative and cannot be eliminated based on the ultimate limitations on a physical measurement

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arises from uncontrolled/uncontrollable variables in the measurement

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has an EQUAL chance of being positive OR negative and is always present + cannot be corrected

blunder

(type of experimental error)

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aka GROSS errors

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accidental error so large that it invalidates a measurement typically caused by human failure or instrument malfunction

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extreme instances of random or systematic error due to accidental but significant departures from procedure

type of error and how to discover it: pH meter has been standardized incorrectly

SYSTEMATIC error

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Suppose you think that the pH of the buffer used to standardize the meter is 7.00 but it is really 7.08. Then all your pH readings will be 0.08 units too low. When you read a pH of 5.60 the actual pH of the sample is 5.68. 

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This systematic error could be discovered by using a second buffer of known pH to test the meter.

type of error and how to discover it: uncalibrated buret

SYSTEMATIC error

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The manufacturer’s tolerance for a Class A 50mL buret is ±0.05mL. When you think you have delivered 29.43mL the real volume could be anywhere from 29.38 to 29.48mL and still be within tolerance.

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One way to correct for an error of this type is to construct a correction curve, such as that in Figure 2-23, by the procedure on page 48. To do this, deliver distilled water from the buret into a flask and weigh it. Determine the volume of water from its mass using Table 2-7. Figure 2-23 tells us to apply a correction factor of -0.03mL to the measured value of 29.43mL. The actual volume delivered is 29.43-0.03=29.40mL

type of error and how to discover it: reading a scale

RANDOM error

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Different people reading the volume in a buret report a range of values representing their subjective interpolation between the markings. One person reading the same buret several times might report several different readings.

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Improved technique can reduce random errors, but random errors cannot be completely eliminated.

type of error and how to discover it: electrical noise in an instrument

RANDOM error

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Another random error results from electrical noise in an instrument. Positive and negative fluctuations occur with approximately equal frequency.

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Improved technique can reduce random errors, but random errors cannot be completely eliminated.

type of error and how to discover it: calculation error

BLUNDER

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Record such incidents in your notebook. Blunders are unrecoverable errors due to procedural, instrumental, or clerical mistakes that may be so serious that you must reject the data or redo the whole experiment.

type of error and how to discover it: overshooting a titration endpoint

BLUNDER

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Record such incidents in your notebook. Blunders are unrecoverable errors due to procedural, instrumental, or clerical mistakes that may be so serious that you must reject the data or redo the whole experiment.

type of error and how to discover it: dropping, discarding, or contaminating a sample

BLUNDER

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Record such incidents in your notebook. Blunders are unrecoverable errors due to procedural, instrumental, or clerical mistakes that may be so serious that you must reject the data or redo the whole experiment.

type of error and how to discover it: instrument failure

BLUNDER

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Record such incidents in your notebook. Blunders are unrecoverable errors due to procedural, instrumental, or clerical mistakes that may be so serious that you must reject the data or redo the whole experiment.

precision

describes the REPRODUCIBILITY of a result

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how well replicate measurements agree with one another

uncertainty

the variability within a set of measurements

accuracy

describes how close a measured value is to the “true” value

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a measure of how close a measured value is to the “true” value

absolute uncertainty

an expression of the margin of uncertainty associated with a measurement

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could also refer to the difference bwn a measured value and the “true” value

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ex: if the estimated uncertainty in reading a calibrated buret is ±0.02mL, we say that ±0.02mL is the absolute uncertainty associated with the reading

relative uncertainty

uncertainty of a quantity divided by the value of the quantity

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compares the size of the absolute uncertainty with the size of its associated measurement

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usually expressed as a percentage of the measured quantity

calculating relative uncertainty

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ex: if the buret reading is 12.35 and the absolute uncertainty is ±0.02mL

relative uncertainty = absolute uncertainty / magnitude of measurement

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ex: rel unc = 0.02mL / 12.35mL = 0.002 = 0.2%

3\.4 propagation of uncertainty from random error

how can we measure/express uncertainty

standard deviation

standard deviation of the mean

confidence interval

uncertainty in addition and subtraction

the uncertainty in the answer is obtained form the ABSOLUTE uncertainties of INDIVIDUAL terms:

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**Absolute uncertainty:** e4 = √( (e1)^2 + (e2)^2 + (e3)^2 ), e is the experimental uncertainty

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**Relative uncertainty:** = e4 / ∑(e1 + e2 + e3)

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\*NOTE: result e4 is actually expressed as ±e4 bc √ can have both + and - answers

uncertainty in multiplication and division

FIRST: convert ALL uncertainties into PERCENT uncertainties (convert to percentages) to calculate uncertainty of the product or quotient

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SO CONFUSED (P57)

mixed operations

CONFUSED

absolute uncertainty and sig fig rules

the FIRST digit of the ABSOLUTE uncertainty is the LAST SIGNIFICANT digit in the answer

relative uncertainty in y = x^*a*

relative uncertainty in y (%ey) is *a* × the relative uncertainty in x (%ex)

(1) relative and (2) absolute uncertainty in y = x^*a* if y is the base 10 logarithm of x

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(1) relative and (2) absolute uncertainty in y = x^*a* if y is the natural logarithm of x

3\.5 propagation of uncertainty from systematic error

matrix

everything else in a sample BESIDES the analyte

4\.1 gaussian distribution

gaussian distribution

* theoretical bell-shaped distribution of measurements when all error is random
* center of the curve is the mean µ
* width is characterized by the stdev σ
* NORMALIZED gaussian distribution is called the *normal error curve*

arithmetic mean

x bar

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aka AVERAGE

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the sum of a set of results divided by the number of values in the set

calculating arithmetic mean

x bar = ∑ x(i) / n

standard deviation

s

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a statistical measuring how closely data are clustered abt the mean value

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smaller stdev = more closely data is clustered abt the mean

calculating standard deviation

s = √ (∑(xi - xbar)^2 / (n-1))

1. µ vs xbar
2. σ vs s

µ = true population mean (for an infinite set of data)

xbar = sample mean (for a finite set of data)

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σ = true population stdev (for an infinite set of data)

s = sample stdev (for a finite set of data)

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as the number of measurements under constant conditions increases, xbar approaches µ and s approaches σ IF there is NO systematic error

degrees of freedom

the quantity n-1

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the number of observations minus the number of parameters estimated from the observations

variance

the square of the stdev

relative standard deviation

aka *coefficient of variation*

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the stdev expressed as a percentage of the MEAN value

calculating relative standard deviation

rel stdev = 100 × s/xbar

excel functions for calculating xbar and s

xbar = average(insert data set)

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s = stdev.***s***(insert data set) ← NOT stdev or stdev.p

express mean and standard deviation in the form:

xbar ± s (n = ___), n is number of observations made

mean and standard deviation sig fig rules

experimental results are expressed in the form **xbar ± s (n = ___)**

(n is the number of data points)

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if mean xbar has *a* number of sig figs, the stdev s + xbar must go to however many sig figs *a* is, even if xbar is written with more than *a* number of sig figs and s has less than *a* number of sig figs

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ex: 823 ± 30 (n=4) means that the mean has only 2 sig figs (bc 823±30 results in answers up to 2 sig figs)

ALTERNATIVELY can write 823±30(n=4) as 8.2±0.3 × 10^2(n=4) and this still would go to 2 sig figs

formula for a Gaussian curve

(in image)

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* e ≈ 2.71828 (base of natural log)
* approximate µ by xbar for finite set of data
* approximate σ by s for finite set of data

Z score

number of stdevs away from the mean

calculating Z (to express deviations of the mean value in multiples, Z, of the standard deviation)

Z = (x - µ) / σ ≈ (x - xbar) / s

* x = observed value
* xbar = sample mean
* s = sample stdev

probability of measuring z in a certain RANGE

= the AREA of that range

68-95-99.7 rule

* 68.3% of the area under the Gaussian curve is in the rango of µ - 1σ to µ + 1σ (1 SD from the mean)
* 95.5% of the area lies within the range µ - 2σ to µ + 2σ (2 SDs from the mean)
* 99.7% of the area lies within the range µ - 3σ to µ + 3σ (3 SDs from the mean)

standard deviation of the mean

standard deviation of the mean = µ sub x

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aka STANDARD ERROR of the mean

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the relationship bwn how many times you measure a quantity and the associated confidence with how confident you can be that your mean is close to the population mean

calculating standard deviation of the mean

the standard deviation of a set of measurements (s) divided by the square root of the number of measurements (n) in this set

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µ sub x = (s sub x) / √n

* s sub x = s: measures uncertainty in x, approaches a constant value as n approaches infinity
* µ sub x measures uncertainty in the mean xbar, approaches a constant value as n approaches infinity

4\.2 comparison of standard deviations with the F test

null hypothesis

the statement that 2 sets of data are drawn from populations with the SAME properties such as standard deviation σ (F test) or mean µ (t test) and that observed differences arise only from random variation in the measurements

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NOTE: we FAIL TO REJECT the null hypothesis UNLESS there is strong evidence that it is false

when do we reject the null hypothesis

if there is < 5% probability of observing the experimental results from 2 populations with the same population SD

F test

* for two variances (s1 and s2, where s1≥s2), the statistic F is defined as F = s1^2 / s2^2
* to decide where s1 is significantly greater than s2, we compare F with the critical values in a table based on a certain confidence level under the appropriate degrees of freedom (df = n-1)
* if the calculated value of F is GREATER than the value in the table, the difference is significant and we REJECT the null hypothesis
* when F(calculated) > F(table), there is LESS than a p=5% chance that the data came from populations with the SAME population standard deviation → reject H0 if p

4\.3 confidence intervals

calculating confidence intervals

CI = xbar ± t\*(s/√n) = xbar ± t(µ sub x)

* t is taken from the t table at a certain degrees of freedom (n-1) and at a certain confidence level (%)
* s/√n = µ sub x = standard uncertainty (aka standard error of the mean)

finding CIs with excel

enter cells for:

* mean = average(insert data set)
* stdev = stdev.s(insert data set)
* n (number of observations) = count(insert data set)
* df = n-1
* CI as a decimal
* t = t.inv.2t(1-\[CI\],\[df\]) ← 1-CI = probability
* CI = \[t\]\*\[stdev\]/sqrt\[n\]) ← gives the ±ts/√n value

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lower bound = xbar - CI

upper bound = xbar + CI

4\.4 comparison of means with student’s t

t test

statistical test used to decide whether the results of 2 experiments are within experimental uncertainty of each each other (the uncertainty must be specified to within a certain probability)

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decides whether there is a statistically significant difference bwn the two means of diff data sets

null hypothesis for t test

H0: the two sets of measurements come from populations with the SAME population mean

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reject H0 if there is less than a p=5% chance that the two sets of measurements come from populations with the same population mean

deciding whether there is a statistically significant difference bwn the two means of diff data sets: *comparing a measured mean with a known mean*

compute a 95% CI for YOUR answer and see if that range includes the known answer → if the known answer is NOT within your 95% CI, the results do NOT agree

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95% CI = xbar ± t\*(s/√n) → if the known answer is outside the 95% CI, there is less than a 5% chance that our result agrees with the known answer → the two means are statistically different

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NOTE: this only works IF the known result is clearly inside or outside 95% CI → in the case that the 95% CI is so close to including the known result, would need more measurements before concluding that our new result is not accurate

deciding whether there is a statistically significant difference bwn the two means of diff data sets: *comparing replicate measurements (2 means from the same data set that has been measured one way and then another way)*

use F test to compare the SDs with the F test → 2 cases can happen:

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F(calculated) = s1^2 / s2^2, df = n-1


1. if F(calculated) < F(table) → SD differences are NOT significant
2. if F(calculated) > F(table) → SD differences ARE significant

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***a) if the SDs are NOT significantly different → conduct a t test to see if the 2 mean values are significantly diff***

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t test for comparison of means: t = | xbar1 - xbar2 | / s(pooled) \* √((n1 × n2) / (n1 + n2))

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s(pooled) = √((∑(x1-xbar1)^2 + ∑(x2-xbar2)^2) / (n1+n2 - 2)) = √((s1^2 × (n1-1) + s2^2 × (n2-1)) / (n1 + n2 - 2))

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if the calculated t is GREATER than the tabulated t at 95% CI, the two results are considered to be significantly different: if t(calculated) > t(table) (95%), the difference is significant

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***b) if the SDs ARE significantly different → conduct a t test to see if the 2 mean values are significantly diff***

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t(calculated) = |xbar1 - xbar2| / √(s1^2/n1 + s2^2/n2) = |xbar1 - xbar2| / √(µ1^2 + µ2^2)



df ≠ n-1 → INSTEAD df = \[equation in image\] → round to the nearest integer

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if the calculated t is GREATER than the tabulated t at 95% CI, the two results are considered to be significantly different: if t(calculated) > t(table) (95%), the difference is significant

deciding whether there is a statistically significant difference bwn the two means of diff data sets: *comparing individual differences (2 means found from 2 different data sets with n observations in each set)*

to see if there is a significant difference bwn the methods, use a PAIRED *t* TEST

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1. compute the mean difference (d sub i) bwn the two results for each sample
2. compute the mean (d bar) of the n number of differences
3. compute the standard deviation (s sub d) of the n number of differences


1. s sub d = √(∑(d sub i - d bar)^2 / (n-1))
4. once you have the mean, SD, and number of samples *n*, computer t(calculated)


1. t(calculated) = |d bar| / (s sub d) × √n
5. if t(calculated) < t(table) (95%) within a certain df (df = n-1) there is MORE than a 5% chance that the two sets of results come from populations with the SAME mean → the results are NOT significantly different


1. if t(calculated) > t(table) (95%) → reject H0 bc there is LESS than a 5% chance that the two sets of results come from populations with the SAME mean

two-tailed vs one-tailed test

***TWO-tailed:*** we reject the null hypothesis if the certified value lies in the low-probability region on either side of the mean

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***ONE-tailed:*** use if comparing a mean value with a REGULATORY limit (interested ONLY in the probability on ONE side (exceeding) of this limit)

t(calculated) = |xbar - regulatory limit| / s × √n → if t(calculated) > t at regulatory limit, H0 is rejected and Ha is accepted

4\.6 grubbs test for an outlier

outlier

a piece of data that lies far from the other data in a set of measurements

grubbs test

statistical test used to decide whether to discard a data point that appears to be an outlier

calculating the Grubbs statistic

G(calculated) = |questionable value - xbar| / s

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if G(calculated) > G(table) → questionable point SHOULD be discarded with 95% (or CL according to table) confidence

4\.7 the method of least squares

calibration curve

aka STANDARD curve

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a graph showing the value of some property vs the concentration of analyte. When the corresponding property of an unknown is measured, its concentration can be determined from the graph

method of least squares

process of fitting a mathematical function to a set of measured points by minimizing the sum of the squares of the distances from the points to the curve

finding the equation of the line using the least squares method

***NOTE:*** the procedure we use assumes that:


1. the uncertainties in y values are substantially greater than uncertainties in x values ← this condition is often true in a calibration curve in which the experimental response (y values) is less certain the quantity of analyte (x values)
2. uncertainties (standard deviations) in all y values are similar

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draw the best fit line through the points by minimizing the VERTICAL deviations (bc of assumption 1)

* equation of straight line: y = mx + b → minimizing Δy bwn the point (x(actual), y(actual)) and the y(expected) calculated from y = mx + b
* vertical deviation for the point (xi, yi) is yi - y
* bc we wish to minimize the MAGNITUDE of the deviations irrespective of their signs, we SQUARE all the deviations so that we are only dealing with positive numbers
* (di)^2 = (yi - y)^2 = (yi - m\*xi - b)^2
* bc we minimize the squares of the deviations → this is called the method of *least squares*

determinant

a single numerical value which is used when calculating the inverse or when solving systems of linear equations

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a scalar (a number) that indicates how that matrix behaves. It can be calculated from the numbers in the matrix.

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the determinant from the 2x2 matrix |row:e f, row:g h| = eh - fg (multiply diagonally and subtract)

how to use determinants to calculate the best-fit line’s slope and y intercept in least-squares method

least squares best-fit line follows equation y = mx + b, m is slope, b is y-intercept

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(follow image for how to calculate m and b)

how to estimate uncertainty in the slope and intercept calculated from least squares method

uncertainties in m and b are related to uncertainty in measuring each value of y

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1. estimate the standard deviation that describes the population of y values → stdev = σ sub y


1. σ sub y ≈ s sub y = √(∑(di - dbar)^2 / df


1. di = yi - y = yi - (m\*xi + b)
2. dbar = 0 for the best straight line
3. df = n - 2
4. equation can be rewritten as σ sub y ≈ s sub y = √∑(di^2)/df
2. calculate standard uncertainty (u) of slope and intercept:


1. standard uncertainty of slope = u sub m = √ ((s sub y)^2 × n) / D) ← D is determinant
2. standard uncertainty of intercept = u sub b = √ ((s sub y)^2 × ∑(xi)^2) / D) ← D is determinant
3. u decreases as you measure more points
4. standard deviation s is approx. constant if you measure more points

slope and y intercept sig fig rules

the first decimal place of the uncertainty is the last significant figure of the slope or intercept

how to express uncertainty in slope and y int as a confidence interval

multiply the standard uncertainty by student’s *t* from the table for n-2 degrees of freedom

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slope = ±t × u(m) based on n - 2 df

4\.8 calibration curves

4\.8 calibration curve

standard solutions

solutions containing known concentrations of analyte

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a solution whose composition is known by virtue of the way that it was made from a reagent of known purity or by virtue of its reaction with a known quantity of a standard reagent

blank solutions

solutions containing all reagents and solvents used in the analysis but no deliberately added analyte

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blanks measure the response of the analytical procedure to impurities or interfering species

constructing a calibration curve

1. prepare known samples of analyte covering the range of concentrations expected for unknowns. Measure the response of the analytical procedure to these standards to generate data
2. subtract the avg absorbance of the *blank* solutions from each measured absorbance to obtain *corrected absorbance*. The blank measures the response of the procedure when no protein is present.
3. make a graph of corrected absorbance vs. quantity of protein. Inspect the graph to determine over what range the data are linear, whether there are any outliers, and that the y-uncertainty is approximately the same over the range.
4. use the least-squares procedure to find the best straight line through the linear portion of the data. Find the slope and intercept and their standard uncertainties


1. the equation of the linear calibration line is: corrected absorbance (y) = mx + b
2. y as the corrected absorbance = observed absorbance - blank absorbance
5. if you analyze an unknown at a future time, run a blank at that time. Subtract the new blank absorbance from the unknown absorbance to obtain corrected absorbance.