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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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Last lecture summary• The nature of the normal distribution• Non-Gaussian distributions
New stuff
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=2.71 𝑥=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.
How normal is normal?
http://www.nate-miller.org/blog/how-normal-is-normal-a-q-q-plot-approach
Checking normality1. Eyball histograms2. Eyball QQ plots3. There are tests
QQ plot• Q stands for ‘quantile’. Quantiles are values taken at
regular intervals from the data. The 2-quantile is called the median, the 3-quantiles are called terciles, the 4-quantiles are called quartiles (deciles, percentiles).
Typical normal QQ plot
http://emp.byui.edu/BrownD/Stats-intro/dscrptv/graphs/qq-plot_egs.htm
QQ plot of left-skewed distribution
http://emp.byui.edu/BrownD/Stats-intro/dscrptv/graphs/qq-plot_egs.htm
QQ plot of right-skewed distribution
http://emp.byui.edu/BrownD/Stats-intro/dscrptv/graphs/qq-plot_egs.htm
SAMPLING DISTRIBUTIONSvýběrová rozdělení
Histogram
𝒙=𝟏𝟗 .𝟒𝟒
𝒙=𝟏𝟕 .𝟐𝟐
𝒙=𝟏𝟔 .𝟖𝟗
Sampling distribution of sample mean• výběrové rozdělení výběrového průměru
Sweet demonstration of the sampling distribution of the mean
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průměr = 3.3
průměr = 1.7
Data 2015Population:4,3,3,5,0,4,4,4,3,4,2,6,8,2,4,3,5,7,3,3
25 samples (n=3) and their averages3,5,3,4,2,3,3,3,5,5,3,4,3,4,5,4,4,4,6,3,4,3,4,3,4
http://blue-lover.blog.cz/1106/lentilky
Histogram of 2015 data
2015, n = 3, number of samples = 25
Going further• So far, we have generated 25 samples with n = 3.• To improve our histogram, we need more samples.• However, we don’t want to spend ages in the classroom.
• Thus, I have prepared a simulation for you. In this simulation, I use data from 2014 and I generate all possible samples, n = 3.
Sampling distribution, n = 3
1 540 samples
Sampling distribution, n = 5
42 504 samples
Sampling distribution, n = 10
20 030 010 samples
Central limit theorem (CLT)• The distribution of sample means is normal.
• The distribution of sample means is always normal irrespective of the underlying distribution.
• The distribution of sample means will increasingly approximate a normal distribution as a sample size increases.
Non-Gaussian distribution1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,4,4,4,5,5,6,7,7,8,8,8,9,9,9,9,10,10,10,10,10,11,11,11,11,11,11
Sampling distribution
n = 2
Sampling distribution
n = 4
Sampling distribution
n = 6
Sampling distribution
n = 8
Back to CLT• Once we know that the sampling distribution of the
sample mean is normal, we want to characterize this distribution.
• By which numbers you characterize a distribution?
mean
standard deviation
Back to CLT• Mean (sometime also denoted as ) of the sampling
distribution is equal to the population mean.
• Standard deviation (sometime also denoted as ) of the sampling distribution is equal to the population standard deviation divided by the square root of .• is called standard error (směrodatná chyba).
𝑆𝐸=𝜎 𝑥=𝜎√𝑛
𝑀 ¿𝜇𝑥=𝜇
M and SELet’s have a look at our demonstration data:
1. Calculate population mean, population standard deviation and standard error for n=3.
2. Take all our sample means and calculate their mean. It should be close to the population mean.
3. Take all our sample means and calculate their standard deviation. It should be close to the standard error.
M and SEpop_mean <- mean(data.set2015)pop_sd <- sd(data.set2015)*sqrt(19/20)se <- pop_sd/sqrt(3)
sampl_mean <- mean(prumery2015)sampl_sd <- sd(prumery2015)
Quiz• As the sample size increases, the standard error
• increases• decreases
• As the sample size increases, the shape of the sampling distribution gets• skinnier• wider
Sampling distribution applet
parent distribution
sample data
sampling distributions of selected statistics
http://onlinestatbook.com/stat_sim/sampling_dist/index.html
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