Statistik, first introduced by Gottfried Achenwall (1749), originally designated the analysis of data about the state, signifying the "science of state“ .

S t a t i s t i c s

Topics

• Introduction to Statistics• History and Application• Two Kinds of Data• Statistical Presentations• Ranking, Percentile, and Percentile Rank• Basic Counting Principle,

Permutation and Combination• Probability• Inferential Statistics

Introduction to Statistics• Statistics can be found in:

– Business– Politics– Science and Technology– Education– Sports– Many other subjects

• Statistics is a branch of mathematics that deals with the collection, organization, and analysis of numerical data with such problems as experiment design and decision making.

• It has two branches, namely, descriptive statistics and inferential statistics

History and Application• Before 3000 BC, the Babylonians used small clay tablets to

record tablations of agricultural yields and of commodities bartered or sold.

• Egyptians analyzed the populations and material wealth of their country before beginning to build the pyramids in the 31st century BC.

• Biblical books of Numbers and 1st Chronicles are primarily statistical works.

• The ancient Greeks held censuses to be used as bases for taxation as early as 594 BC.

• At present, statistics is a reliable means of describing acurately the values of ec onomic, political, social, psychological, biological, and physical data and serves as a tool to correlate and analyze such data.

Two Kinds of Data

• Population Data– The term population refers to all measurements or

observations of interest.– This might be a population of people in a country, of

crystal grains in a rock, or of goods manufactured by a particular factory during a given period.

• Sample Data– A sample is simply a part of the population.– For practical reasons, rather than compiling data

about an entire population, one usually instead studies a chosen subset of the population.

Statistical Presentations

• Two methods of describing collection of Data:– Organizing Data is summarizing raw data in a

systematic manner to be come meaningful and useful.

– Frequency Distributions is a table which divides the data into various classes or categories. this can be done through construction of table or graphical

representations

Statistical Presentations• There are conventional rules in constructing table for

frequency distribution, one is:– Scan the raw data for the highest H and the lowest L values.– Calculate the range R of the values; R = H – L– Split the range into 5 to 15 classes, each covering the same

amount (the class interval).

Class size or width of class interval = R/number of classes– With all the data values, scan the raw data, item by item,

placing a tally mark next to a value each time it occurs (every 5th tally mark crosses through a group of four)

– Count the number of the tally marks for each value. This is its frequency, i.e., how many times it occurs.

Statistical Presentations

• Example for frequency distribution:– Construct a frequency distribution table for 30 grades

received on an examination of 30 students. The grades are:

30, 35, 43, 52, 61, 65, 65, 65, 68, 70, 72, 72, 73, 75, 75, 76, 77, 78, 78, 80, 83, 85, 88, 88, 90, 91, 96, 97, 100, 100.

• Cumulative Frequency Distribution is adding the frequency starting from the lowest class interval up to the frequency of the highest class interval.

Statistical Presentations

• Graphical representations of frequency distributions can be a histogram, frequency polygon, and frequency curve

In a cumulative-frequency graph, such as Fig. 1, the grades are marked on the horizontal axis and double marked on the vertical axis with the cumulative number of the grades on the left and the corresponding percentage of the total number on the right.

Statistical Presentations• Histogram is a series of rectangles with

bases equal to the interval ranges and areas proportional to the frequencies.– Analyze the grades received by 10 sections of

30 pupils each on four examinations, a total of 1200 grades.

Statistical Presentations• Frequency Polygon is a line graph of a

frequency distribution. It is easily obtained from a histogram by joining the midpoints of the bars with line segments to form a polygon.

The polygon in Fig. 3 is drawn by connecting with straight lines the interval midpoints of a cumulative frequency histogram.

Statistical Presentations• Description of the form of a frequency:

– skewness, which is its departure from symmetry and degree of its peakedness.negatively skewed, which is nonsymmetrical with longer tail of

the frequency curve on the left, symmetricalpositively skewed, which is nonsymmetrical with the loner tail of

the frequency curve on the right.

Statistical Presentations• Another description of the form of a

frequency: – Kurtosis is the extent to which a frequency

distribution is concentrated about the mean, a frequency curve can be.platykurtic, which is flat with number of observed values

distributed relatively evenly across the classes,mesokurtic, which is neither flat nor peaked with respect to the

general appearance of the frequency curve, leptokurtic, which is peaked with a large number of observed

values concentrated within a narrow range of the possible values of the variable being measured.

Ranking, Percentile and Percentile Rank

• Ranking– Ranks can have non-integer values for tied data

values. When there is an even number of the same data value, the statistical rank ends in ½

• Percentile is the value of a variable below a certain percent of observations fall.– Quartile is the 25th of percentile– Median is the 50th of percentile– Application: Physicians often use infant and children’s

height and weight percentile as a guage of relative health

Ranking, Percentile and Percentile Rank

• Percentile Rank– Percentiles are most often used for determining the

relative standing of an individual in population or the rank position of the individual.

– Percentile ranks are an easy way to convey an individual’s standing at graduation relative to other graduates.

– Example: If Jason graduated 25th out of a class of 150 students, then 125 students were ranked below Jason. Jason’s rank would be 125/150, which is equal to 83rd percentile.

Percentile rank = (number of scores below x) . 100

n

Ranking, Percentile and Percentile Rank

• Sample Problem– The math test scores were: 50 65 70 72 72 78

80 82 84 84 85 86 88 88 90 94 96 98 98 99. Find the percentile rank for a score of 84 on this test.

Basic Counting Principle, Permutations, Combinations

• Basic Counting Principle– Fundamental principle of counting is often

referred to as the multiplication rule– Theorem 1 (multiplication rule)

If an operation can be performed in “n1” ways, and if for each of these a second operation can be performed in “n2” ways, then the two operations can be performed together in “n1 n2” ways

Example:

How many sample points are in the sample space when a pair of dice is thrown once?

Basic Counting Principle, Permutations, Combinations

• Basic Counting Principle– Theorem II (generalized multiplication rule)

If an operation can be performed in “n1” ways, and if for each of these a second operation can be performed in “n2” ways, if for each of the first two a third operation can be performed in “n3” ways, and so on, then the sequence of k operations can be performed in n1 + n2 + ... + nk ways

Example:

How many lunches are possible consisting of soup, a sandawich, dessert, and a drink if one can select from 4 soups, 3 kinds of sandwiches, 5 desserts, and 4 drinks?

• Permutation– an arrangement of all part of a set of objects.– Any ordered arrangement of agiven set of

objects.– Example: Possible permutations of letters a b

and c are abc, acb, bac, bca, cab, cba– Represent the product by the symbol of n! read

as “n factorial” where n distinct objects can be arranged in n(n-1) (n-2) ... (3) (2) (1) ways.

Basic Counting Principle, Permutations, Combinations