Last lecture summary• Which measures of variability do you know?• What are they advantages and disadvantages?• Empirical rule

Statistical jargon

Population - parameterMean Standard deviation

Sample - statisticMean Standard deviation

Výběr - statistikaVýběrový průměr Výběrová směrodatná odchylka

population (census) vs. sampleparameter (population) vs. statistic (sample)

Statistical inference• A statistic is a value calculated from our observed data

(sample).

• A parameter is a value that describes the population.

• We want to be able to generalize what we observe in our data to our population. In order to this, the sample needs to be representative.

• How to select a representative sample? Use randomization.

New stuff

Random sampling• Simple Random Sampling (SRS) – each possible

sample from the population is equally likely to be selected.

• Stratified Sampling – simple random sample from subgroups of the population• subgroups: gender, age groups, …

• Cluster sampling – divide the population into non-overlapping groups (clusters), sample is a randomly chosen cluster• example: population are all students in an area, randomly select

schools and create a sample from students of the given school

Simple random sampling• sampling with replacement (WR)

• výběr s navrácením

• Generates independent samples• Two sample values are independent if that what we get on the first

one doesn't affect what we get on the second.

• sampling without replacement (WOR)• výběr bez navrácení

• Deliberately avoid choosing any member of the population more than once.

• This type of sampling is not independent, however it is more common.

• The error is small as long as 1. the sample is large

2. the sample size is no more than 10% of population size

Bias• If a sample is not representative, it can introduce bias into

our results.• bias – zkreslení, odchylka• A sample is biased if it differs from the population in a

systematic way.

• The Literary Digest poll, 1936, U. S. presidential election• surveyed 10 mil. people – subscribers• 2.3 mil. responded predicting (3:2) a Republican candidate to win• a Democrat candidate won• What went wrong?

• only wealthy people were surveyed (selection bias)• survey was voluntary response (nonresponse bias) – angry people or

people who want a change

Bessel’s correction

𝑠=√∑ (𝑥𝑖−𝑥 )2

𝑛−1www.udacity.com – Statistics

Sample vs. population SD• We use sample standard deviation to approximate

population paramater

• But don’t get confused with the actual standard deviation of a small dataset.

• For example, let’s have this dataset: 5 2 1 0 7. Do you divide by or by ?

Bessel's game

𝜇=2

Bessel's game• An important property of a sample statistic that estimates

a population parameter is that if you evaluate the sample statistic for every possible sample and average them all, the average of the sample statistic should equal to the population parameter.

We want: • This is called unbiased.

Bessel’s game1. List all possible samples of 2 cards.

2. Calculate sample averages.

SampleSample average

Population of all cards in a bag

Bessel’s game1. List all possible samples of 2 cards.

2. Calculate sample averages.

3. Now, half of you calculate samplevariance using /n, and half of you using /(n-1).

4. And then average all sample variances.

SampleSample average

Sample variance

0,2 1

0,4 2

2,0 1

2,4 3

4,0 2

4,2 3

0,0 0

2,2 2

4,4 4Population of all cards in a bag

𝑠2=∑ (𝑥𝑖−𝑥 )2

𝑛∨𝑛−1

Bessel’s game

SampleSample average

Sample variance (n-1) Sample variance (n)

0,2 1 2 1

0,4 2 8 4

2,0 1 2 1

2,4 3 2 1

4,0 2 8 4

4,2 3 2 1

0,0 0 0 0

2,2 2 0 0

4,4 4 0 0

average

𝜇=2 ,𝜎 2=83

Median absolute deviation (MAD)• standard deviation is not robust• IQR is robust• mean absolute deviation MAD – a robust equivalent of the

standard deviation

• Také your data, find median, calculate absolute deviation from the median, find the median of absolutes deviations

Median absolute deviation (MAD)Data Median deviation Absolute deviation

5

10

30

20

30

5

15

10

15

Median:

MAD:

NORMAL DISTRIBUTION

Playing chess• Pretend I am a chess player.• Which of the following tells you most about how good I

am:1. My rating is 1800.

2. 8110th place among world competitive chess players.

3. Ranked higher than 88% of competitive chess players.

Distribution

Distribution of scores in one particular year

We should use relative frequencies and convert all absolute frequencies to proportions.

Height data – absolute frequencies

http://wiki.stat.ucla.edu/socr/index.php/SOCR_Data_Dinov_020108_HeightsWeights

Height data – relative frequencies

Height data – relative frequencies

30%

What proportion of values is between 170 cm and 173.75 cm?

Height data – relative frequencies

What proportion of values is between 170 cm and 175 cm?

We can’t tell for certain.

• How should we modify data/histogram to allow us a more detail?1. Adding more value to the dataset

2. Increasing the bin size

3. A smaller bin size

Height data – relative frequencies

What proportion of values is between 170 cm and 175 cm?

36%

Height data – relative frequencies

Height data – relative frequencies

Normal distribution

1

√2𝜎𝜋𝑒𝑥𝑝 {− (𝑥−𝜇)2

2𝜎2 }

recall the empirical rule

68-95-99.7