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

2

3

3

1

2

6

5

5

4

3

5

5

4

2

6

3

4

3

1

5

2

3

3

1

2

6

5

5

4

3

5

5

4

2

6

3

4

3

1

5

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