Summary Five numbers summary, percentiles, mean Box plot, modified box plot Robust statistic – mean, median, trimmed mean outlier Measures of variability range, IQR, average absolute deviation, variation and standard deviation Average distance between each data value and the mean is zero.

Standard deviation – empirical rule

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

Bias, sampling

SRS sampling with replacement 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 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

Bessel’s game Sample Sample average Sample variance (n-1)Sample variance (n) 0,2121 0,4284 2,0121 2,4321 4,0284 4,2321 0,0000 2,2200 4,4400 average

Histogram revision Distribution – the pattern of values in the data Histogram – visualizing the distribution We can see whether the data tend to be close to the particular value whether the data varies a lot or a little about the most common values whether that variation tends to be more above or below the common values whether there are unusually large or small values in the data

Life expectancy data – histogram Use interactive histogram applet to generate histogram with bin size of 10, starting at 40. life expectancy frequency

Life expectancy data – histogram life expectancy frequency

Making conclusions from a histogram What all you can tell for life expectancy data? how many modes? where is the mode? symmetric, left skewed or right skewed? outliers – yes or no? life expectancy frequency

Making conclusions from a histogram Where is the mode, the median, the mean? life expectancy frequency

Min. Q1 Median Q3 Max Five numbers summary What is the position of the mean and the median?

symmetric, left or rigt skewed?

STANDARDIZING normování

Playing chess Pretend I am a chess player. Which of the following tells you most about how good I am: 1. My rating is th 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

Height data – relative frequencies

What proportion of values is between 170 cm and cm? 30%

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

Decreasing bin size Check out what happens with the smallest bin size for Physics Test Scores from

Height

Height data – relative frequencies

Normal distribution recall the empirical rule

Empirical rule

Z Z – number of standard deviations away from the mean If the Z-value is 1, how many percent are less than that value? cca 84 %

Who is more popular? Let’s demonstrate the importance of Z-scores with the following example.

Who is more popular s.d. = 36 s.d. = 60 Z = Z = -2.57

Standardizing

Formula

Quiz What does a negative Z-score mean? 1. The original value is negative. 2. The original value is less than mean. 3. The original value is less than The original value minus the mean is negative.

Quiz II If we standardize a distribution by converting every value to a Z-score, what will be the new mean of this standardized distribution? If we standardize a distribution by converting every value to a Z-score, what will be the new standard deviation of this standardized distribution?

Standard normal distribution

Meaning of relative frequencies

Histogram of these data

Probability density function Probability density function (PDF) Hustota pravděpodobnosti

Standard normal distribution