# CHMB16 Chapter 3

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.

## Document Summary

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. The standard deviation measures the width of the gaussian curve. If the known answer does not lie within the 95% confidence interval, then the two methods give different results. Comparison of standard deviation with the f test. F test: tells us whether two standard deviations are significantly different from each other. o o o o. If f-calculated < f-table, use the first t-calculated equation. If f-calculated > f-table, use the second t-calculated equation. If f-calculated > f-table, then the difference is significant. If g-calculated is greater than g on p. 83, the questionable point should be discarded. Equation of a straight line: y = mx + b, where m is the slope and b is the y- intercept.