Error, Uncertainty, and Statistics

Introduction to Error, Uncertainty, and Statistics

Introduction

• Every measurement has some degree of uncertainty.
• Experimental error is the difference between the true value and the measured value.
• Three types of experimental error:
  • Systematic
  • Random
  • Gross (blunders)

Systematic Error

• Systematic error (determinate error) arises from a flaw in equipment or experiment design.
• If an experiment is conducted again in the exact same manner then the error is reproducible.
• Can be discovered and corrected (in theory)
• Ways to Detect:
  • Analyze a known sample (certified reference material).
  • Use a different method to measure the same quantity.
  • Different labs analyze identical samples (“round robin”).
  • Analyze a blank sample. If a nonzero result is observed, the method has systematic error.
• Ways to Correct:
  • Calibrate glassware and instruments.
  • Use standard addition or internal standards to correct for matrix effects.

Random Error

• Random error (indeterminate error) arises from uncontrolled variables in measurement.
• There is an equal chance of the error being positive or negative.
• It is always present and cannot be eliminated.
• Might be reduced with better technique
• Examples of Random Error:
  • Subjective reading of a scale (varies with individual).
  • Electrical noise in an instrument.

Gross Error (Blunders)

• Gross error (blunder) is due to accidental but significant departures from procedure.
• Caused by procedural, instrumental, or clerical mistakes (unrecoverable).
• Should be recorded in the lab notebook.
• May be so serious that data are rejected or the experiment is redone
• Examples of Blunders:
  • Calculation errors.
  • Overshooting a titration endpoint.
  • Dropping, discarding, or contaminating a sample.
  • Instrument failure.

Precision, Uncertainty, and Accuracy

• Precision: describes the reproducibility of a result
  • If measured several times and values agree = precise.
  • If measured several times and values vary widely = not precise.
• Uncertainty: the variability within a set of measurements.
• Accuracy: how close a measured value is to the “true” value.

Absolute and Relative Uncertainty

• Absolute uncertainty: expresses the margin of uncertainty associated with any measurement (with units).
  • Typically written as a $±$ value.
  • Example: A buret reading of $12.35 ± 0.02$ mL means the true value could be in the range $12.33$ mL to $12.37$ mL.
• Relative uncertainty: compares the size of the absolute uncertainty with the size of its associated measurement.

Propagation of Uncertainty

• General formulas for how uncertainty propagates through calculations:
  • Addition/Subtraction: If $y = x<em>1 + x</em>2 - x<em>3$, then $e</em>y = \sqrt{e<em>{x</em>1}^2 + e<em>{x</em>2}^2 + e<em>{x</em>3}^2}$
  • Multiplication/Division: If $y = \frac{x<em>1 \cdot x</em>2}{x<em>3}$, then $\%e</em>y = \sqrt{\%e<em>{x</em>1}^2 + \%e<em>{x</em>2}^2 + \%e<em>{x</em>3}^2}$
  • Logarithm: If $y = \log(x)$, then $e<em>y = \frac{1}{\ln(10)} \cdot \frac{e</em>x}{x} ≈ 0.434 \frac{e_x}{x}$
  • Natural Logarithm: If $y = \ln(x)$, then $e<em>y = \frac{e</em>x}{x}$
  • Exponential: If $y = 10^x$, then $e<em>y = (\ln(10)) \cdot e</em>x \cdot 10^x ≈ 2.303 \cdot e_x \cdot 10^x$
  • If $y = e^x$, then $e<em>y = e</em>x$
  • Exponents: If $y = x^a$, then $\%e<em>y = a \cdot \%e</em>x$
• $x$ represents a variable and $a$ represents a constant that has no uncertainty.
• $e<em>x$ is the absolute error in $x$ and $\%e</em>x$ is 100 × the relative error.

Error in Arithmetic

• Addition and subtraction: use absolute uncertainty of the individual terms (include units).
• Multiplication and division: use percent relative uncertainty.
• Mixed operations: follow proper algebraic rules for mathematical manipulation.

Error in Addition and Subtraction

• Use absolute uncertainty of the individual terms (include units).
• Example: $1.76 m (±0.03) + 1.89 m (±0.02) – 0.59 m (±0.02) = 3.06 m (±0.041)$
• $e = \sqrt{(0.03)^2 + (0.02)^2 + (0.02)^2} = 0.041$

Example Problem 1

• Initial reading: $0.05 (±0.02)$ mL
• Final reading: $17.88 (±0.02)$ mL
• Uncertainty in the volume delivered?
• $17.88 (±0.02) mL – 0.05 (±0.02) mL = 17.83 (±e) mL$
• $e = \sqrt{(0.02)^2 + (0.02)^2} = 0.028$
• The volume delivered is the difference (the volume delivered by a buret is the difference between final and initial readings).
• Regardless of the initial and final readings, if the uncertainty in each reading is ±0.02 mL, the uncertainty in volume delivered is ±0.03 mL.

Error in Multiplication and Division

• Use percent relative uncertainty
• To convert relative uncertainty to absolute uncertainty; multiply the relative uncertainty by the value.
• Example: $\frac{0.494 (±0.004) \cdot 5.00 (±0.01)}{100.00 (±0.08)} = \frac{0.494 (±0.81\%) \cdot 5.00 (±0.20\%)}{100.00 (±0.080\%)}$
• $\%e = \sqrt{(0.81)^2 + (0.20)^2 + (0.080)^2} = 0.84\%$
• $0.0247 M ± 0.84\% (relative uncertainty)$
• $0.84\% \cdot 0.02470 M = ± 0.00021 M$
• $0.0247 M ± 0.00021 M (absolute uncertainty)$

Example Problem 2

• A $0.250 M NH<em>3$ solution is prepared by diluting $8.46 (±0.04)$ mL of $28.0 (±0.5) wt\% NH</em>3$ up to $500.0 (±0.2)$ mL. [density = $0.899 (±0.003) g/mL$].
• Find the uncertainty in $0.250 M$.
• The molecular mass of $NH_3$, $17.031 g/mol$, has negligible uncertainty relative to other uncertainties in this problem.
• To find the uncertainty in molarity, we need the uncertainty in moles delivered to the 500-mL flask.
• The concentrated reagent contains $0.899 (±0.003) g$ of solution per mL.
• Weight percent tells us that the reagent contains $0.280 (±0.005) g$ of $NH_3$ per gram of solution.
• Grams of $NH_3$ in concentrated reagent: $0.899(±0.003) \cdot 0.280 (±0.005) = 0.899(±0.334\%) \cdot 0.280 (1.79\%) = 0.2517(±1.82\%) g/mL$
• $\%e = \sqrt{(0.334\%)^2 + (1.79\%)^2} = 1.82\%$
• Next, we find the moles of ammonia contained in $8.46 (±0.04)$ mL of concentrated reagent. The relative uncertainty in volume is $0.04/8.46 = 0.473\%$.
• moles of $NH_3 = \frac{0.2517 (±1.82\%) g/mL \cdot 8.46 (±0.473\%) mL}{17.031 (±0\%) g/mol} = 0.12504(±1.88\%) mol$
• This much ammonia was diluted to $0.5000 (±0.0002) L$. The relative uncertainty in the final volume is $0.0002/0.5000 = 0.04\%$.
• The molarity is: $M = \frac{0.12504(±1.88\%) mol}{0.5000(±0.04\%) L} = 0.25008(±1.88\%) M$
• The absolute uncertainty is $1.88\%$ of $0.25008 M = 0.0047 M$.
• The uncertainty in molarity is in the third decimal place, so our final, rounded answer is $[NH_3] = 0.250(±0.005) M$

Example Problem 3: Volumetric vs. Gravimetric Dilutions

• Comparing the uncertainty resulting from a 10-fold volumetric dilution with a 10-fold gravimetric dilution.
• a. Volumetric dilution: standard reagent with a concentration of $0.04680 M$ (negligible uncertainty).
  • Dilute by a factor of 10, use a micropipet to deliver $1000 μL (= 1.000 mL)$ into a 10-mL volumetric flask and dilute to volume.
• b. Gravimetric dilution: standard reagent with a concentration of $0.046 80 mol reagent/kg solution$.
  • Dilute it by a factor close to 10, weigh out $983.2 mg (= 0.9832 g)$ of solution (≈1 mL) and add $9.0266 g$ of water (≈9 mL).
• For each procedure, find the resulting concentration and its relative uncertainty.

Example Problem 3a: Volumetric Dilution

• Tolerance for the volumetric flask is $10.00 ± 0.02 mL = 10.00 mL ± 0.2\%$,
• Tolerance for the micropipet is $1000 μL ± 0.3\%$.
• The dilution factor is: $\frac{10.00 (±0.2\%) mL}{1.000 (±0.3\%) mL} = 10.00 (±0.36\%)$
• $\%e = \sqrt{(0.2\%)^2 + (0.3\%)^2} = 0.36\%$
• The concentration of the dilute sample is $\frac{0.04680 M}{10.00(±0.36\%)} = 0.004680 (±0.36\%) M = 0.004680 ±0.000017 M$

Example Problem 3b: Gravimetric Dilution

• Dilute $0.9832 g$ of concentrated solution up to $(0.9832 g + 9.0266 g) = 10.0098 g$.
• The dilution factor is: $\frac{10.0098 g}{0.9832 g} = 10.1808$
• Suppose that the uncertainty in each mass is $± 0.3 mg$.
• Absolute uncertainty in the sum is $\sqrt{(0.0003 g)^2 + (0.0003 g)^2} = 0.00042 g$, which is $0.0042\%$.
• The uncertainty in the dilution factor is: $\frac{10.0098 (±0.0042\%) g}{0.9832 (±0.0305\%) g} = 10.1808 (±0.0308\%)$
• $\%e = \sqrt{(0.0042\%)^2 + (0.0305\%)^2} = 0.0308\%$
• The concentration of the dilute solution is $\frac{0.04680 mol/kg}{10.1808(±0.0308\%)} = 0.0045969(±0.0308\%) mol/kg = 0.0045969±0.0000014 mol/kg$
• Gravimetric dilution is 10 times more precise than volumetric dilution.
• Increased precision is the reason gravimetric titrations are recommended over volumetric titrations, though the latter are less tedious.

Propagation of Uncertainty: Exponents and Logarithms

• For the function $y = x^a$, the relative uncertainty in $y$ ($\%e<em>y$) is $a$ times the relative uncertainty in $x$ ($\%e</em>x$).
• If $y = \sqrt{x} = x^{1/2}$, a relative uncertainty of $±2\%$ in $x$ will result in $\%e_y = (\frac{1}{2})(2\%) = 1\%$.
• If $y = x^2$, a relative uncertainty of $±2\%$ in $x$ will result in $\%e_y = (2)(2\%) = 4\%$.

Example Problem 4

• If an object falls for $t$ seconds, the distance traveled is $d = \frac{1}{2}gt^2$, where $g$ is the acceleration due to gravity ($9.8 m/s^2$).
• If the object falls for $2.34 s$, then the distance traveled is $d = \frac{1}{2}(9.8 m/s^2)(2.34 s)^2 = 26.9 m$.
• If the relative uncertainty in time is $± 1.0\%$, the relative uncertainty in distance is calculated as follows:
• Since $d = \frac{1}{2}gt^2 \rightarrow \%e<em>d = a(\%e</em>t) = 2(1.0\%) = 2.0\%$.

Example Problem 5

• Consider the function $pH = -\log[H^+]$, where $[H^+]$ is the molarity of $H^+$.
• For $pH = 5.21 ± 0.03$, find $[H^+]$ and its uncertainty.
• $[H^+] = 10^{-pH}$
• This would tell us the function is $y = 10^x$ and that $e<em>y/y = (\ln10)e</em>x$ $e<em>{[H^+]}/[H^+] = (\ln10)e</em>{pH} = (\ln10)(0.03) = (2.3026)(0.03) = (0.0691)$
• The relative uncertainty in $[H^+]$ is $0.0691$.
• For $[H^+] = 10^{-pH} = 10^{-5.21} = 6.17 × 10^{-6} M$, we find $\frac{e<em>{[H^+]}}{[H^+]} = 0.0691 = \frac{e</em>{[H^+]}}{6.17 × 10^{-6} M} \rightarrow e_{[H^+]} = (6.17 × 10^{-6} M)(0.0691) = 4.3 × 10^{-7} M$
• The concentration of $H^+$ is $6.17 (±0.43) × 10^{-6} M = 6.2 (±0.4) × 10^{-6} M$.
• An uncertainty of 0.03 in pH gives an uncertainty of 7% in $[H^+]$.

Gaussian Distribution

• For an experiment repeated very many times with purely random errors:
  • The results tend to cluster symmetrically about the average value.
  • The more times the experiment is repeated, the more closely the results approach a Gaussian distribution.
  • Usually we repeat an experiment 3–5 times (not 400 times).
  • From small data sets we can estimate properties of a hypothetical large set

Mean and Standard Deviation

• Mean (average) ($\bar{x}$): the sum of a set of results divided by the number of values in the set.
  • $\bar{x} = \frac{\sum{x_i}}{n}$
• Standard deviation ($s$): measures how closely data are clustered about the mean.
  • $s = \sqrt{\frac{\sum{(x_i - \bar{x})^2}}{n-1}}$
  • as n increases, $\bar{x} \rightarrow \mu$
  • as n increases, $s \rightarrow \sigma$

Excel/Spreadsheet Applications

• Spreadsheets have built-in statistical functions:
  • Average: =AVERAGE(B1:B4)
  • Standard deviation: =STDEV.S(B1:B4)

Accuracy and Precision Revisited

• The smaller the standard deviation, s, the more closely the data are clustered about the mean.
• Precision: reproducibility
• Accuracy: nearness to the “truth”
• Experiments with a small standard deviation are more precise than experiments with a large standard deviation.
• Greater precision does not necessarily imply greater accuracy.
• Express the mean and standard deviation in the form $\bar{x} ± s$
• The average and the standard deviation should both end in the same decimal place.

Other Statistical Parameters

• Degrees of freedom (df): The number of independent values in a calculation that can vary without changing the overall result.
  • $df = n-1$
• Variance: square of the standard deviation
  • $Variance = s^2$
• Relative standard deviation (coefficient of variation): standard deviation expressed as a percentage of the mean
  • $RSD = \frac{s}{\bar{x}} \cdot 100$

Example Problem 6

• Find the average, standard deviation, and relative standard deviation for 821, 783, 834, and 855.
• Average: $\bar{x} = \frac{(821 + 783 + 834 + 855)}{4} = 823.2$
• Standard deviation: $s = \sqrt{\frac{(821-823.2)^2 + (783-823.2)^2 + (834-823.2)^2 + (855-823.2)^2}{(4-1)}} = 30.3$
• Relative standard deviation: $RSD = \frac{30.3}{823.3} \cdot 100 = 3.7\%$.

Standard Deviation and Probability

• Gaussian Curve:
  • The probability of observing a value within a certain range is proportional to the area of that range.
  • Express deviations from the mean value in multiples, $z$, of the standard deviation.
  • We transform $x$ into $z$: $z = \frac{x - \mu}{\sigma}$
  • $y ≈ \frac{e^{\frac{-(x-\mu)^2}{2\sigma^2}}}{\sigma \sqrt{2 \pi}}$
• Using $z$ table to determine probability.

Example Problem 7

• For many tosses of a set of 50 coins, probability theory predicts a mean of 25.00 heads and a standard deviation of 3.54.
• How many tosses are expected to have fewer than 15 heads if the 50 coins were tossed 400 times?
• We express the desired interval in multiples of the standard deviation and then find the area of the interval in the given table.
• Since $\bar{x} = 25.00, s = 3.54 \rightarrow z = \frac{15 - 25.00}{3.54} = -2.82 ≈ -2.8$
• From the table the area between the mean and z = −2.8 is 0.4974.
• The entire area from −∞ to the mean value is 0.5000, so the area from −∞ to –2.8 is 0.5000 − 0.4974 = 0.0026.
• The area to the left of 15 heads is only 0.26% of the entire area under the curve. If the class tosses the 50 coins 400 times, they would expect to see 15 or fewer heads only once (0.26% of 400 = 1.04).

Standard Deviation of the Mean

• The more times a quantity is measured, the more confident you can be that the mean is close to the population mean.
• $s_x = \frac{s}{\sqrt{n}}$
• as $n \rightarrow \infty$, $s_x \rightarrow$ constant value
• as $n \rightarrow \infty$, $\mu_x \rightarrow 0$

Standard Deviation and Probability

• The sum of the probabilities of all measurements must be unity.
• The area under the whole curve from z = −∞ to +∞ adds up to 1.
• The standard deviation measures the width of the Gaussian curve.
• The larger σ, the broader the curve.
• For any Gaussian curve:

Using a Spreadsheet to Find Area Under a Gaussian Curve

• For 400 tosses of 50 coins, how many tosses are expected to have between 20 and 27 heads?
• We need to find the fraction of the area of the Gaussian curve between $x = 20$ and $x = 27$heads and then multiply this fraction by 400 tosses.
• The function NORM.DIST in Excel gives the area under the curve from −∞ to a chosen value of x.
• Area from 20 to 27 = (area from −∞ to 27) − (area from −∞ to 20)
• NORM.DIST(x,mean,standard_dev,cumulative) are called arguments of the function.
  • cumulative = TRUE, NORM.DIST gives the area under the Gaussian curve.
  • cumulative = FALSE, NORM.DIST gives the ordinate (the y-value) of the Gaussian curve.