• Statistics I - a collection of numerical facts expressed as a summarizing statement .

• Population – complete set of individuals, objects, or measurements having same common observable characteristic

• Sample – subset or part of population

• Unit – single member of a population

• Random sample – sample in which all elements have an equal chance of being selected

• random sampling permits inferences about characteristics of the population from which the sample is selected

• Variable – any characteristic of a person, group, or environment that can vary or denote a difference

• (e.g. weight, political ideology, pollution count)

• Data – numbers collected as a result of observations, interviews, …

• Statistic – number describing a characteristic of a sample

• Parameter – any characteristic of a population

• Examples :

Population – CULS students

Sample – students of statistical course

Is it a random sample?

Unit – a concrete student

Variables – age, height, number of siblings, hair colour,

• Quantitative ( continuous , descrete )

• Qualitative (nominal , ordinal )

Measures of Central Tendency

• Measures that represent with a proper value the tendency of most data to gather around this value

• Number of different measures of central tendency

  • the arithmetic mean

  • the median

  • the mode

• The arithmetic mean (the sum of the values of a variable divided by the number of scores (by the sample size) )

Properties of the arithmetic mean

1. It is expressed in the same unit of measure as the observed variable

2. It is the point in a distribution of measurements about which the sum of deviations are equal to zero .

The value above and below which one-half of the frequencies fall

• n…odd number

median case number=(n+1)/2

• n…even number

the arithmetic mean of the two middle values

The value that occurs with greatest frequency

• for qualitative (nominal and ordinal) and quantitative discrete data

• from a statistical perspective it is also the most probable value

Use of mean, median and mode

• member of mathematical system in advanced statistical analysis

• preferred measure of central tendency if the distribution is not skewed

The median

• when the distribution is skewed

The mode

• whenever a quick, rough estimate of central tendency is desired

• The Range….R - it is the distance between the largest and smallest value

R=x_max-x_min

• It does not explain the variability inside the range !

• Very simple and straightforward measure of dispersion

• The Variance…s^{2 -} it is an average squared deviation of each value from the mean .

It is the sum of the squared deviations from the mean divided by n

• when computing the variation based on sample we correct the calculation

• the variance explains both

  • the variability of the values around the arithmetic mean

  • the variability among the values

• difficult interpretation

(it is expressed in the squares of the unit of measure)

• The Standard Deviation…s - it is the square root of variance

  • when computing the variation based on sample

• Properties of the standard deviation - it is expressed in the same unit of measure as the observed variable

• Coefficient of Variation…V - the ratio of the standard deviation to the mean .

• often reported as a percentage (%) by multiplying by 100

nominal variables – we can arrange the categories in any order:alphabetically, decreasing/increasing order of frequency

• ordinal variables – the categories should be placed in their naturally occuring order

• Pie chart – a circle divided into sectors

  • each sector represents a category of data

  • the area of each sector is proportional to the frequency of the category

• random experiment – repeated process leading to different outcomes based on random

• random event – outcome of a random experiment

• sample space S – collection of all possible outcomes

• Probability - describes how likely it is that some event will happen

Notation

• P…probability

• A, B, C…specific events

• P(A)…the probability of event A occuring

• 0 ≤ P(A) ≤ 1

• random variable – a variable that has a single numerical value, determined by chance

  • discrete – has a finite or countable number of values

  • continuous – has infinitely many values

Selected probability distributions

• Discrete

  • Alternative

  • Binomial

  • Poisson

  • Hypergeometric

• Continuous

  • Normal distribution

  • Student

  • Fisher-Snedecor

  • Χ²

• What is difference between Classical and Statistical approach of probability?

• What is distribution function? When we use it for?

• What is normal distribution? How it works?

• What is standard normal distribution?

• the goal of statistical inference is to use the information obtained from a sample and generalize the results to the population that is being studied

• Sampling - the goal in sampling is to obtain units for a study in such a way that accurate information about the population can be obtained

-the most basic sample survey design is simple random sampling

• Simple random sampling - a sample size n from a population N is obtained through simple random sampling if every possible sample of size n has an equally likely chance of occurring

• → simple random sample

• A parameter is a descriptive measure of a population

  • constant

• A statistic is a descriptive measure of a sample

  • random variable

Properties of point estimators ;

• unbiased

  • A point estimator is said to be an unbiased of a population parameter if the expected value is equal to that parameter

• consistency

  • as the sample size increases the estimator approaches the population parameter

• efficiency

  • the variance of the estimator among the samples is small

• sufficiency

complete information

Estimate ( point estimate , interval estimate )

• A point estimate is the value of a statistic that estimates the value of a parameter

Point estimate of the mean