PROBABILITY AND STATISTICS WEEK 1 Onur Doğan

What is Statistics? Onur Doğan

Basic Terms Population: A collection, or set, of individuals or objects or events whose properties are to be analyzed. Two kinds of populations: finite or infinite. Sample: A subset of the population. Variable: A characteristic about each individual element of a population or sample. Data (singular): The value of the variable associated with one element of a population or sample. This value may be a number, a word, or a symbol. Data (plural): The set of values collected for the variable from each of the elements belonging to the sample. Experiment: A planned activity whose results yield a set of data. Parameter: A numerical value summarizing all the data of an entire population. Statistic: A numerical value summarizing the sample data. Onur Doğan

Example: A college dean is interested in learning about the average age of faculty. Identify the basic terms in this situation. The population is the age of all faculty members at the college. A sample is any subset of that population. For example, we might select 10 faculty members and determine their age. The variable is the “age” of each faculty member. One data would be the age of a specific faculty member. The data would be the set of values in the sample. The experiment would be the method used to select the ages forming the sample and determining the actual age of each faculty member in the sample. The parameter of interest is the “average” age of all faculty at the college. The statistic is the “average” age for all faculty in the sample.

Onur Doğan

Level of Measurement Nominal Scale Classifications, Set memberships, etc. Ordinal Scale Ordinal data Interval Scale Equal distances but no zero point Ratio Scale Absolute zero Onur Doğan

Data Presentation Basic Presentation Frequency Distributions Relative Frequency Class Intervals Onur Doğan Jan.Feb.Mar.Apr.MayJuneJulyAug.Sep.Oct

Measures of Central Tendency Mean Median Mode Onur Doğan

Mean Onur Doğan

Median Median: The value of the data that occupies the middle position when the data are ranked in order according to size To find the median: 1.Rank the data 2.Determine the depth of the median: 3.Determine the value of the median Onur Doğan

Mode Mode: The mode is the value of x that occurs most frequently Note:If two or more values in a sample are tied for the highest frequency (number of occurrences), there is no mode Onur Doğan

Measures of Variability Range Mean Absolute Error Standard Deviation and Variance Onur Doğan

Range Range: The difference in value between the highest-valued (X max ) and the lowest-valued (X min ) pieces of data: Range=X max - X min Onur Doğan

Mean Absolute Error Mean Absolute Error: The mean of the absolute values of the deviations from the mean: Onur Doğan    xx|| 1 error absoluteMean n

SD and Variance Onur Doğan

z-score z-Score: The position a particular value of x has relative to the mean, measured in standard deviations. The z-score is found by the formula: Notes: Typically, the calculated value of z is rounded to the nearest hundredth The z-score measures the number of standard deviations above/below, or away from, the mean z-scores may be used to make comparisons of raw scores

Example:A certain data set has mean 35.6 and standard deviation 7.1. Find the z-scores for 46 and 33: Example Solutions: 46 is 1.46 standard deviations above the mean z xx s      is 0.37 standard deviations below the mean. 0

Chebyshev’s Theorem: The proportion of any distribution that lies within k standard deviations of the mean is at least 1  (1/k 2 ), where k is any positive number larger than 1. This theorem applies to all distributions of data. Illustration: Chebyshev’s Theorem

Example Onur Doğan