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)

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? Checking normality 1.Eyball histograms 2.Eyball QQ plots 3.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

QQ plot of left-skewed distribution

QQ plot of right-skewed distribution

SAMPLING DISTRIBUTIONS výběrová rozdělení

Histogram

Sampling distribution of sample mean výběrové rozdělení výběrového průměru

Sweet demonstration of the sampling distribution of the mean

průměr = 3.3 průměr = 1.7

Data 2015 Population: 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 averages 3,5,3,4,2,3,3,3,5,5,3,4,3,4,5,4,4,4,6,3,4,3,4,3,4

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 = samples

Sampling distribution, n = samples

Sampling distribution, n = samples

Central limit theorem (CLT)

Non-Gaussian distribution 1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,4,4,4,5,5,6,7,7,8,8,8,9,9,9,9,10,10,10,10,10,11,11,11,11,11,11

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

M and SE Let’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 SE pop_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