1. summary

2. Distribution of scores in one particular year
My rating is 1800.
8110th place among world competitive chess players.
Ranked higher than 88% of competitive chess players.
We should use relative frequencies and convert all absolute frequencies to proportions.
Distribution of scores in one particular year

4. 𝑥 =173 cm
𝑠=5 cm -2 -1 +1 +2

5. Z-score
Standardize normal distribution by converting any point using this formula
Standard normal distribution … N(0,1)
𝑍= 𝑥−𝜇 𝜎

6. new stuff

7. Quiz Approximately what proportion of people is smaller than 168 cm?
173
178
183
168
163
16%

8. Quiz Approximately what proportion of people is higher than 183 cm?
173
178
183
168
163
2.5%

9. Quiz
Approximately what proportion of people is between 163 cm and 178 cm high?
𝑥 =173 cm
𝑠=5 cm
173
178
183
168
163
81.5%

10. Quiz Approximately what proportion of people is smaller than 180 cm?
173
178
183
168
163
ca 91.5%

11. Quiz
What is the probability of randomly selecting a height in the sample that is >5 standard deviations above the mean?
0.01
0.3
0.8
0.99

12. Quiz
What is the probability of randomly selecting a height in the sample that is <5 standard deviations below the mean?
0.01
0.3
0.8
0.99

13. Quiz
What proportion of the data is either below 2 standard deviations or above 2 standard deviations from the mean for a normal distribution?
95%
2.5%
2.5%

14. Z-table
What is the proportion less than the point with the Z-score -2,75?

15. Use Z-table What proportion of people is smaller than 180 cm?
Z−value = 180−173 5 = 7 5 =1.4
Z-value of 1.4 corresponds to 91.92%.
𝑥 =173 cm
𝑠=5 cm

16. Quiz – height data 𝑛=1000, 𝑥 =173, 𝑠=5.0
𝑛=1000, 𝑥 =173, 𝑠=5.0
What proportion of people is smaller than you?
𝑍 = ??−173 5 = , proportion = see Z-table
What proportion of people is taller than you?
𝑍= ??− = , proportion =1− see Z-table
Table gives a value “less than”.
Note, that “greater than x” is the same as “less than -x”.

17. Quiz – height data 𝑛=1000, 𝑥 =173, 𝑠=5.0
𝑛=1000, 𝑥 =173, 𝑠=5.0
What proportion of people lie between you and you?
𝑍 𝐴 = 𝑎−173 5 =, 𝑍 𝑏 = 𝑏−173 5 =, proportion=proportion 𝑍 𝑎 −proportion( 𝑍 𝑏 )=
How tall should you be to be in the top 5% of the highest people?
𝑍−𝑠𝑐𝑜𝑟𝑒=1.645, ×5.0≈181 cm

18. An intriguing fact
𝑥 =173 cm
𝑠=5 cm

19. sampling distributions

20. Tetrahedral die

21. Some statistics What is the population? What is the population mean?
1, 2, 3, 4
What is the population mean?
𝜇= = 10 4 =2.5
𝜇 is also called the expected value. We don’t really expect to get 2.5 – it’s not even the option. We expect to get somewhere around 2.5 if we take the sample from this population.
If we roll the dice twice, how many possible outcomes we can get? 16

22. 16 samples, 𝑛=2
1,1 𝒙 =𝟏
2,1 𝒙 =𝟏.𝟓
3,1 𝒙 =𝟐
4,1 𝒙 =𝟐.𝟓
1,2 𝒙 =𝟏.𝟓
2,2 𝒙 =𝟐
3,2 𝒙 =𝟐.𝟓
4,2 𝒙 =𝟑
1,3 𝒙 =𝟐
2,3 𝒙 =𝟐.𝟓
3,3 𝒙 =𝟑
4,3 𝒙 =𝟑.𝟓
1,4 𝒙 =𝟐.𝟓
2,4 𝒙 =𝟑
3,4 𝒙 =𝟑.𝟓
4,4 𝒙 =𝟒
What is the mean of the sample means? We design it 𝑀.
Some guess?
Copy sample means from my website (section data) to the WolframAlpha (link is also provided at my website).
𝑀=2.5=𝜇

23. Sampling distribution
Distribution of sample means – sampling distribution (výběrové rozdělení, výběrové rozdělení výběrového průměru)
What’s the shape?
Uniform
Bimodal
Normal
Skewed

24. Quiz
What’s the probability that the average of your two rolls will be 3 or more?
=0.375

25. Real life
We can easily calculate the probability for discrete samples in a discrete population.
What about in real life? We have huuuuge populations (e.g ). We can’t calculate means of all possible samples.

26. Standard deviation of the sample means
Calculate the population standard deviation 𝜎 for our population (1, 2, 3, 4).
Answer: 𝜎=1.118, do not forget to divide by 𝑛!
Now calculate the standard deviation of the sampling distribution 𝑆𝐸.
I did it for you. Before showing you the answer, did I divide by 𝑛 or by 𝑛−1?
By 𝑛. And 𝑆𝐸=0.790.
1,1 𝒙 =𝟏
2,1 𝒙 =𝟏.𝟓
3,1 𝒙 =𝟐
4,1 𝒙 =𝟐.𝟓
1,2 𝒙 =𝟏.𝟓
2,2 𝒙 =𝟐
3,2 𝒙 =𝟐.𝟓
4,2 𝒙 =𝟑
1,3 𝒙 =𝟐
2,3 𝒙 =𝟐.𝟓
3,3 𝒙 =𝟑
4,3 𝒙 =𝟑.𝟓
1,4 𝒙 =𝟐.𝟓
2,4 𝒙 =𝟑
3,4 𝒙 =𝟑.𝟓
4,4 𝒙 =𝟒

27. Relationship between 𝜎 and 𝑆𝐸
Is there any relationship between
𝜎=1.118 and
𝑆𝐸=0.790 ?
Be prepared for a big surprise!
What is the ratio 𝜎 𝑆𝐸 ? Someone calculate this number for me, please.
Population standard deviation
𝜎 𝑆𝐸 = 𝑛
Standard deviation of distribution of sample means (sampling distribution)

28. Central limit theorem Distribution of sample means is normal.
The distribution of means will increasingly approximate a normal distribution as the sample size 𝑛 increases.
Its mean is equal to the population mean.
Its standard deviation 𝑆𝐸 is equal to the population standard deviation divided by the square root of 𝑛.
𝑆𝐸 is called standard error.
Distribution we draw samples from can be of any shape. And still sampling distribution of the mean is normal.

29. Quiz As the sample size increases, the standard error
decreases
As the sample size increases, the shape of the sampling distribution gets
skinnier
wider
𝜎 𝑆𝐸 = 𝑛 ⟹𝑆𝐸= 𝜎 𝑛