MATH 401Probability and Statistics

We distinguish between qualitative (blood types, colours of cars sold, letter grades in an exam etc.) and quantitative (i.e. numerical) data. Throughout this course we are mostly concerned with quantitative data. Basically, little can be done mathematically if the data are not numerical.

MATH 401 Final Exam Grades (2006).

A total of 297 exam grades (out of 50) are as follows.
2.5 7.5 22.5 23 28 16.5 27 31 … … 41 … … 32.5 41.5 … …
10.5 1
112 110
122 114 117 118
107 112 116
128 106 110 111 112 104 111 114 114
108 110
125 113
120 119 118
120 113
120 117
102 110
Inferential Statistics



Involves drawing conclusions from data. The point is to make inferences about a certain situation represented by a particular data set. The main tools for making such inferences are provided by the Probability Theory.
Types of statistical studies
Dependent on why the study is conducted, two types of statistical studies are distinguished: 1. Descriptive statistics. 2. Inferential statistics.
6 5 4 5 4 4 6 5 5 6 3 5 7 8 5 2 5 4 5 4 0 4 7 5 7 4 6 6 3 1
Large-range Numerical Data

ห้องสมุดไป่ตู้
The following data represent the life-times (in months, rounded to the nearest month) of 50 electronic color tubes for TV.

Math 401 exam results.
Device Life-time

The following data represent the life-time (in months) of 50 electronic color tubes for TV.
100 130 120 136 118 118 117 114 116 115 118 118 102 110 114 109 112 102 109


A major problem is to ensure that a selected sample is representative of the population.
Inferential Statistics (revisited)


So the major task of the inferential statistics is to make conclusions about the whole population on the basis of [analyzing] a sample taken from this population. We stress, again, that the Probability Theory is absolutely crucial for making such conclusions.
Populations


Statistical studies examine certain attributes/characteristics of a set of individuals: the population. Populations are normally so large that it is logistically impossible to examine all the individuals. So…
1.
2.
Categorical for qualitative and small quantitative data sets. Grouped for large quantitative data sets.
Populations and Samples
One takes a subgroup of a population and examines the desired characteristic for the subgroup. Such subgroups are called samples.
102 110 114
109 112 102 109
122 114 117 118
107 112 116
128 106 110 111 112 104 111 114 114
108 110
125 113
120 119 118
120 113
120 117
102 110
Types of Data



Data Organization.
Qualitative Data

The colors of 25 Toyota Corollas sold by a dealer in Maadi were recorded as follows (W = white, B = black, S = silver, R = red).
125 113
120 119 118
120 113
120 117
102 110
Frequency Distribution

A frequency distribution for a data set is a table listing groups into which the data are divided - classes, with a row for each class, the number of occurrences for each class – class frequency, sometimes the class relative frequency, i.e. the class frequency divided by the total number of values in the data set.
W Blue B S R
Quantitative Data
Data represent the number of requests per minute placed to a server, recorded during 30 consecutive minutes.
6 5 4 5 4 4 6 5 5 6 3 5 7 8 5 2 5 4 5 4 0 4 7 5 7 4 6 6 3 1
Relative Frequency
Relative frequency of a class is the frequency of the class divided by the total number of data. Relative frequencies are useful for comparing distributions of different sizes.
Frequency Distribution
Class Frequency (# of occurrences)
F1 F2 … Fn
Relative Frequency
Class 1 Class 2 ……… Class n
RF1 = F1/N RF2 = F2/N …………… RFn = Fn/N
Qualitative Data

The colors of 25 Toyota Corollas sold by a dealer in Maadi were recorded as follows (W = white, B = black, S = silver, R = red).
B S B W Blue B S W W W R R Blue W B Blue B W R B
39.5 34.5 45.5 26.5 37.5 38.5 40 … … … … … … … … … … … … … …
48.5 45.5 17
5
26
12
39.5 14
32
40
Descriptive Statistics


Involves only the collection, as well as organization, presentation and summarization of data. The point is to describe a certain situation as represented by a particular data set.