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Chapter 3

This

**preview**shows half of the first page. to view the full**3 pages of the document.**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

measurements.

•Standard Deviation: measures how closely the data are clustered about the

mean.

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

different.

•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

difference

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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