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Chapter 3

CHMB16 Chapter 3

Course Code
Kagan Kerman

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Chapter 4: Statistics
Gaussian Distribution
Gaussian Distribution: an ideal smooth curve that is a result of repeated
experimentation, as results tend to cluster about the average value.
Mean: the sum of the measured values divided by n, the number of
Standard Deviation: measures how closely the data are clustered about the
oSmaller the standard deviation, the more closely data are clustered
about the mean.
os = [(x – average)^2/n -1]
Degrees of Freedom: n – 1
Variance: s^2
Relative Standard Deviation/Coefficient of Variation: 100 x s/(average)
Gaussian Curve: y = 1/[σ(2π)] e^-[(x-µ)^2/2σ^2]
z = x (average or µ)/s or σ the probability of measuring z in a certain
range is equal to the area of that range.
The standard deviation measures the width of the Gaussian curve.
Standard Deviation of the Mean of Sets of n Values: σn = σ/n
oThe more times you measure a quantity, the more confident you can
be that the average is close to the population mean.
Confidence Intervals
Confidence Interval: average ± ts/n, where t is Student’s t, taken from a
table from the book at page 73.
o50% confidence interval means that if we repeated the experiment an
infinite number of times, 50% of the error bars would include the true
population mean.
o90% confidence interval means that if we repeated the experiment an
infinite number of times, we would expect 90% of the confidence
intervals to include the population mean.
Comparison of Means with Student’s t
t test: compares one mean value with another to decide whether there is a
statistically significant difference between the two.
Null Hypothesis: the mean values from two sets of measurements are not
We reject the null hypothesis if there is less than a 5% chance that the
observed difference arises from random variations.
If the “known” answer does not lie within the 95% confidence interval, then
the two methods give “different” results.
If t-calculated is greater than t-table at the 95% confidence level, the two
results are considered to be different.
ot-calculated = |average1 – average2|/s-pooled (n1n2/n1+n2)
ot-calculated = |average1 – average2|/(s1^2/n1 + s2^2/n2)
ot-calculated = |d|/s (n), where |d| is the absolute value of the mean
os-pooled = [s1^2 (n1 – 1) + s2^2 (n2 -1)/(n1 + n2 – 2)]
Two-Tailed t Test: 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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