Bias. Bias and Propaganda Bias Bias- when a person has a strong feeling for or against something.
Last lecture summary Bias, Bessel's correction MAD Normal distribution Empirical rule.
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Transcript of Last lecture summary Bias, Bessel's correction MAD Normal distribution Empirical rule.
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Last lecture summary• Bias, Bessel's correction• MAD• Normal distribution• Empirical rule
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Standard deviation – empirical rule
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Standard deviation – empirical rule
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Standard deviation – empirical rule
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The nature of the normal distribution• In laboratory experiments, results vary due to several
factors: imprecise weighting of reagents, imprecise pipetting, nonhomogeneous suspensions of cells or membarnes ...
• Similarly, variation in a clinical value might be caused by many genetic and environmental factors.
• These random factors are independent and they tend to offset each other.
• Variation among values will approximate a Gaussian distribution• when there are many independent sources of variation• when individual sources add up to get the final result
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New stuff
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STANDARD NORMAL DISTRIBUTION
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Who is more popular?
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Who is more popular
s.d. = 36
s.d. = 60
Z = -3.53
Z = -2.57
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Standardizing
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Formula
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Quiz• What does a negative Z-score mean?
1. The original value is negative.
2. The original value is less than mean.
3. The original value is less than 0.
4. The original value minus the mean is negative.
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Quiz II• If we standardize a distribution by converting every value
to a Z-score, what will be the new mean of this standardized distribution?
• If we standardize a distribution by converting every value to a Z-score, what will be the new standard deviation of this standardized distribution?
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Standard normal distribution
N(,)
N(,)
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Z
Z – number of standard deviations away from the mean
If the Z-value is 1, how many percent are less than that value?
cca 84 %
0 +1 +2 +3-1-2-3
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Proportion of human heights
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+1-1-2 +20
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Quiz• Approximately what proportion of people is smaller than
168 cm?
173 178 183168163
16%
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Quiz• Approximately what proportion of people is higher than
183 cm?
173 178 183168163
2.5%
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Quiz• Approximately what proportion of people is between 163
cm and 178 cm high?
173 178 183168163
81.5%
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Quiz• Approximately what proportion of people is smaller than
180 cm?
173 178 183168163
ca 91.5%
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Quiz• What is the probability of randomly selecting a height in
the sample that is >5 standard deviations above the mean?1. 0.01
2. 0.3
3. 0.8
4. 0.99
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Quiz• What is the probability of randomly selecting a height in
the sample that is <5 standard deviations below the mean?1. 0.01
2. 0.3
3. 0.8
4. 0.99
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Quiz• What proportion of the data is either below 2 standard
deviations or above 2 standard deviations from the mean for a normal distribution?
95%
2.5% 2.5%
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Z-tableWhat is the proportion less than the point with the Z-score -2,75?
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Use Z-table
What proportion of people is smaller than 180 cm?
Z-value of 1.4 corresponds to 91.92%.
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Quiz – height data
• What proportion of people is smaller than you?• see Z-table
• What proportion of people is taller than you?• see Z-table• Table gives a value “less than”.• Note, that “greater than x” is the same as “less than -x”.
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Quiz – height data
• What proportion of people lie between you and you?
• How tall should you be to be in the top 5% of the highest people?
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An intriguing fact
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DISTRIBUTION, DISTRIBUTION, ARE YOU NORMAL?
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Life expectancy data – histogram
life expectancy
fre
qu
en
cy
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Making conclusions from a histogram• What can you tell about life expectancy data?
• how many modes?• where is the mode?• symmetric, left skewed or right skewed?• outliers – yes or no?
life expectancy
fre
qu
en
cy
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Making conclusions from a histogram• Where is the mode, the median, the mean?
life expectancy
fre
qu
en
cy
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Min. Q1 Median Q3 Max. 47.79 64.67 73.24 76.65 83.39
Five numbers summary
8.5>3.5
25.4>10.2
mean=69.9
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Lognormal distribution• Frazier et al. measured the ability of a drug isoprenaline to
relax the bladder muscle.• The results are expressed as the EC50, which is the
concentration required to relax the bladder halfway between its minimum and maximum possible relaxation.
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Lognormal distribution
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Geometric mean
Geometric mean – transform all values to their logarithms, calculate the mean of the logarithms, transform this mean back to the units of original data (antilog)
𝑥=1333𝑛𝑀 𝑥=2.71𝑥=102.71=513nM
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The nature of the lognormal distribution
• Lognormal distributions arise when multiple random factors are multiplied together to determine the value.• A typical example: cancer (cell division is multiplicative)
• Lognormal distributions are very common in many scientific fields.• Drug potency is lognormal
• To analyse lognormal data, do not use methods that assume the Gaussian distribution. You will get misleding results (e.g.,non-existing outliers).
• Better way is to convert data to logarithm and analyse the converted values.