last lecture summary bias, bessel's correction mad normal distribution empirical rule

Last lecture summary• Bias, Bessel's correction• MAD• Normal distribution• Empirical rule

Standard deviation – empirical rule

Standard deviation – empirical rule

Standard deviation – empirical rule

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

New stuff

STANDARD NORMAL DISTRIBUTION

Who is more popular?

Who is more popular

s.d. = 36

s.d. = 60

Z = -3.53

Z = -2.57

Standardizing

Formula

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.

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?

Standard normal distribution

N(,)

N(,)

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

Proportion of human heights

+1-1-2 +20

Quiz• Approximately what proportion of people is smaller than

168 cm?

173 178 183168163

16%

Quiz• Approximately what proportion of people is higher than

183 cm?

173 178 183168163

2.5%

Quiz• Approximately what proportion of people is between 163

cm and 178 cm high?

173 178 183168163

81.5%

Quiz• Approximately what proportion of people is smaller than

180 cm?

173 178 183168163

ca 91.5%

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

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

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%

Z-tableWhat is the proportion less than the point with the Z-score -2,75?

Use Z-table

What proportion of people is smaller than 180 cm?

Z-value of 1.4 corresponds to 91.92%.

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”.

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?

An intriguing fact

DISTRIBUTION, DISTRIBUTION, ARE YOU NORMAL?

Life expectancy data – histogram

life expectancy

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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

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Making conclusions from a histogram• Where is the mode, the median, the mean?

life expectancy

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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

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.

Lognormal distribution

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

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.