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