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Statistics gives us tools to accept conclusions that have a high probability of being correct and to reject conclusions that do not. Count, “normal” days 5.3× 10 6 cells/μL
5.1× 10
6 cells/μL 5.4× 10 6 cells/μL
4.8× 10
6 cells/μL 5.2× 10 6 cells/μL Today’s Count 5.6× 10 6 cells/μL Red Blood Cells Random error: Imagine four small random errors in a measurement that combine to give the overall error: Each error has the same magnitude, can cause the final result to be high or low by ± U, and has equal probability of occurring. 183. 74 Statistics CHEM 3000, Quantitative Analysis CHEM 3000, Quantitative Analysis Reading: Sections 4- through 4-
+U, +U, +U, +U +4U 1 combination -U, +U, +U, +U +2U 4 combinations +U, -U, +U, +U +U, +U, -U, +U +U, +U, +U, -U -U, -U, +U, +U 0U 6 combinations +U, +U, -U, -U +U, -U, +U, -U -U, +U, -U, +U -U, +U, +U, -U +U, -U, -U, +U +U, -U, -U, -U -2U 4 combinations -U, +U, -U, -U -U, -U, +U, -U -U, -U, -U, +U -U, -U, -U, -U -4U 1 combination What are the possible ways these errors can combine to cause deviations from the mean of a set?
Let’s now imagine 10 random errors, same magnitude, same probability of occurrence……. 0 50 100 150 200 250 Deviation from Mean Frequency of Occurance -8U -4U 0 +4U +8U 1:10:45:120:210:252:210:120:45:10: Only one way to combine to get +10U or – 10U 252 ways to get +5U & -5U (no deviation from mean; highest probability)
Gaussian curve or a normal error curve
_
If an experiment is repeated a great many times, and if the errors are purely random, then the results tend to cluster symmetrically about the average value. The more times the experiment is repeated, the more closely the results approach an ideal smooth curve called the Gaussian distribution (if fact a Gaussian curve describes an infinite set of data).
Gaussian Distribution for a “ Population ” Standard Deviation (σ): Mean (μ):
y =
e
$(x$ μ) 2 /2 " 2 68.3% (~2/3) of measurements fall within the range of μ ± σ 95.5% of measurements fall within the range of μ ± 2σ 99.7% of measurements fall within ± 3σ
Small, finite set of data: Mean (x): Standard Deviation (s): Data from flipping ten coins and observing the number of heads.
of heads
0 2 4 6 8 10 frequency 20 40 60 80 100 μ = 4. σ = 1.
An experiment that produces a data set with a smaller standard deviation, s, is said to be more precise than one that produces a larger standard deviation. The precision in each case says nothing about accuracy; each experiment could be equally accurate. The more times you measure a quantity, the more confident you are that your average value is close to the true mean (in the absence of systematic error). Variance = Precision is also measured using… Relative standard deviation (RSD) = Using Excel… “=AVERAGE(B1:B4)” “=STDEV(B1:B4) Calculator…………….“σ n
For example, if you want to decrease the uncertainty by a factor of 2 (cut the uncertainty in half), how many more measurements should you make? The uncertainty decreases in proportion to 1/√n, where n is the number of measurements. Instrumentation with rapid data acquisition allow us to average many experiments in a very short amount of time to increase the accuracy of a result.
Confidence Intervals Usually, we don’t know entire population due to a finite number of measurements. Therefore we cannot know the true mean or true standard deviation. The size of the confidence interval depends upon how closely our sample standard deviation, s, approximates the population standard deviation, σ. The confidence interval can be much narrower than if the uncertainty is based solely on s from a finite set of replicate measurements.
z = x " μ
Express deviations from the mean value in multiples of the standard deviation.
-4 -3 -2 -1 0 1 2 3 4 z = (x- μ )/! Relative Frequency -0.67σ +0.67σ 50%
-4 -3 -2 -1 0 1 2 3 4 z = (x- μ )/! Relative Frequency -1.64σ +1.64σ 90%
-4 -3 -2 -1 0 1 2 3 4 z = (x- μ )/! Relative Frequency +1.96σ 95% -1.96σ
-4 -3 -2 -1 0 1 2 3 4 z = (x- μ )/! Relative Frequency -2.58σ +2.58σ 99% We would assume that 90 times out of 100, the true mean, μ, will be within +/-1.64σ of any measurement that we make. s n is the standard error of the mean
A chemist obtained the following data for the alcohol content of a sample of blood: %C 2
H
5 OH: 0.084, 0.089, 0.079. Calculate the 95% confidence interval…. with no additional knowledge about the precision of the method. Average = 0.084% Standard Deviation = 0.005% t = 4. n = 3 Degrees of freedom = 2 on the basis of previous experience in which it is known that s → σ = 0.005% C 2
H
5
OH.
Average = 0.084% Standard Deviation = 0.005% t = 1. n = 3 Degrees of freedom = ∞
How many replicate measurements of the carbohydrate content are needed to decrease the 90% confidence interval to ±0.1 g?
Common estimates of uncertainty: 6.372 6.373 6.374 6.375 6.376 6. Mean ± Standard Deviation 90% Confidence 5 measurements 90% (^) 21 measurements 5 measured values of volume: 6.375, 6.372, 6.374, 6.377, 6.375 mL s = 0. 8 mL t = +/-0. 7 Report the volume as 6. 6
8 mL. Always report the number of replicate measurements.
