Last lecture summary• Standard normal distribution, Z-distribution• Z-table• lognormal distribution, geometric mean

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

Nice applet:http://www.mathsisfun.com/data/standard-normal-distribution-table.html

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).

How to interpret QQ plot

How to interpret QQ plot

no outlier

no outlier

http://www.nate-miller.org/blog/how-normal-is-normal-a-q-q-plot-approach

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

Data 2013Population: 6,4,5,3,10,3,5,3,6,5,4,8,7,2,8,5,8,5,4,0

20 samples (n=3) and their averages

1. 10 3 5 … 6.0

2. 3 3 4 … 3.3

3. 4 4 8 … 5.3

4. 4 3 8 … 5.0

5. 5 5 6 … 5.3

6. 6 8 7 … 7.0

7. 3 8 8 … 6.3

8. 6 8 4 … 6.0

9. 8 8 4 … 6.7

10. 5 3 4… 4.0

11. 2 10 8… 6.7

12. 3 4 5 … 4.0

13. 5 6 5 … 5.3

14. 8 6 4 … 6.0

15. 4 8 4 … 5.3

16. 5 8 5 … 6.0

17. 4 4 3 … 3.7

18. 8 8 4… 6.7

19. 8 4 5… 5.7

20. 3 0 7… 3.3 http://blue-lover.blog.cz/1106/lentilky

Data 2014Population: 3,2,3,1,2,6,5,5,4,3,5,5,6,3,2,4,4,3,1,5

20 samples (n=3) and their averages

1. 5 1 4 … 3.3

2. 3 1 1 … 1.7

3. 6 6 5 … 5.7

4. 3 5 4 … 4.0

5. 4 1 4 … 3.0

6. 5 1 3 … 3.0

7. 2 5 4 … 3.7

8. 5 5 1 … 3.7

9. 3 3 5 … 3.7

10. 5 2 3 … 3.3

11. 5 3 4 … 4.0

12. 3 4 6 … 4.3

13. 2 5 5 … 4.0

14. 5 6 1 … 4.0

15. 2 2 5 … 3.0

16. 5 3 6 … 4.7

17. 1 5 3 … 3.0

18. 5 5 5 … 5.0

19. 3 5 4 … 4.0

20. 3 3 6 … 4.0 http://blue-lover.blog.cz/1106/lentilky

Sampling distribution, n = 3

Plot exact sampling distribution

sample_size <- 3data.set2014 <- c(3,2,3,1,2,6,5,5,4,3,5,5,6,3,2,4,4,3,1,5)samps <- combn(data.set2014, sample_size)xbars <- colMeans(samps)barplot(table(xbars))

Sampling distribution, n = 3• Calculate .• Calculate .

• Le’s create all possible samples of size 3.• Calculate .• Calculate .

𝑆𝐸=𝜎√𝑛

Sampling distribution, n = 3

Sampling distribution, n = 5

Central limit theorem• Distribution of sample means is normal.

• The distribution of means will increasingly approximate a normal distribution as the sample size increases.

• Its mean is equal to the population mean.

• Its standard deviation is equal to the population standard deviation divided by the square root of .• is called standard error.

𝑆𝐸=𝜎 𝑥=𝜎√𝑛

𝑀 ¿𝜇𝑥=𝜇

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

Another data1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,4,4,4,5,5,6,7,7,8,8,8,9,9,9,9,10,10,10,10,10,11,11,11,11,11,11

Sampling distribution

n = 2

Sampling distribution

n = 4

Sampling distribution

n = 6

Sampling distribution

n = 8

Sampling distribution applet

parent distribution

sample data

sampling distributions of selected statistics

http://onlinestatbook.com/stat_sim/sampling_dist/index.html